Chapter IV Weighted Norm Inequalities

CHAPTER IV WEIGHTED NORM INEQUALITIES

The L p inequalities obtained in the previous chapters for seve ral kinds of operators remain true when Lebesgue measure dx is replaced by certain measures w(x)dx. This is not entirely new for us: In Chapter I we proved the Helson-Szeg5 theorem characterizing those w(x) for which the Hilbert transform is bounded in L2(w), while the boundedness of the Hardy-Littlewood maximal function in Lp(w) was established in Chapter I for every A1 weight w(x) and 1 < p < m .

We shall devote this chapter to a systematic study of this type of inequalities. F o r the maximal function and regular singular opera tors, it is possible to give a very precise and satisfactory answer to the question of finding those w for which either

or the corresponding weak type inequality hold. This answer is provided by the Ap theory, which has already reached a great level of perfection and has found applications in several branches of Analysis, from Complex Function Theory to Partial Differential Equations. The same problem for two weights is also considered (in sections 1 and 4), but here some unsolved questions remain which are probably far from being completely settled. Why should one be interested in inequalities like ( * ) ? We shall briefly sketch some answers 1) Conjugate functions, Hp spaces etc. can be defined in domains of the complex plane with a "reasonable" boundary a D . When estimating the Lp norms of operators appearing in this context, some of the problems that arise can be reduced, by a change of variables, to estimates for known operators on the line or on the torus, but with respect to a measure w(x)dx for a certain w. These ideas were actually used in CalderBn's original proof of his celebrated theorem on Cauchy integrals along Lipschitz Curves (see Chapter 1 1 , 7.12 for the precise statement). 385

386

IV. WEIGHTED NORM INEQUALITIES

2) Given a complete orthonormal system some interval I, does the series

{r$k}

in

L2(I),

for

converge to f(x) in the Lp norm, p # 2 ? For ordinary Fourier series, the partial sums are very closely related to the Hilbert transform, and M. Riesz theorem implies (Snf-flp +. 0, 1 < p < m . In some other interesting cases, the partial sums can be expressed for a certain in terms of the operator f(x) +. k(x)H(f k-l)(x) k(x), and then, convergence in Lp is equivalent to ( * ) with T = H, w(x) = Ik(x)lp. 3) Inequalities like ( * ) imply, when the structure of the weights satisfying them is sufficiently known, the following

for arbitrary u(x) 0, where N is (in the most desirable case) some kind of "maximal operator" which we can control. An inequality like ( * * ) was proved in Chapter 11, 2.12 for the Hardy-Littlewood operator, and a similar one for singular integrals will appear in corollary 3.8 of this chapter. Such inequalities are very easy to handle, and contain essentially all the relevant information about the boundedness properties of T (we shall come back to this point in Chapter VI).

4) There is an intimate connection between weights, BMO functions and C. Fefferman's duality theorem, which can be roughly described by saying that BMO consists of the logarithms of good weights. This connection is exploited in section 5, where we give a new proof o f the duality theorem and obtain, by real variable methods, a satisfac tory analogue in Rn of the Helson-Szeg8 theorem.

IV.1. THE CO!:DITION A

38 7

P

1 . THE CONDITION Ap

By a weight on a given measure space, we shall always mean a measurable function w with values in [O,m]. Our main problem is going to be the following PROBLEM 1 . G i v e n

p,

1 < p

<

i s of s t r o n g t y p e

M

f o r w h i c h t h e maximal o p e r a t o r

w(x)dx,

r e s p e c t to t h e m e a s u r e

determine those weights

a,

w

(p,p)

on Rn

with

t h a t i s , f o r w h i c h we h a v e an i n e -

quality:

We can also pose this more general PROBLEM 2 . G i v e n p , 1 < p < m , d e t e r m i n e t h o s e p a i r s o f w e i g h t s o n Rn (u,w), f o r w h i c h M i s of s t r o n g t y p e (p,p) with respect to t h e p a i r o f m e a s u r e s (u(x)dx, w(x)dx), t h a t i s , f o r w h i c h we h a v e an i n e q u a l i t y : ( 1 .2)

(IRn

(Mf(x))pu(x)dx)’/p

5

5 C(jRn (f(x)lPw(x)dx)’/P We can pose similar problems substituting weak type for strong type in the two problems above. For example: PROBLEM 3 . G i v e n

p,

1

2 p

<

m,

d e t e r m i n e t h o s e p a i r s of wi7:ghts

on Rn,

( u , w ) , f o r w h i c h ?I i s of weak t y p e (p,p) w i t h r e s p e c t to t h e p a i r o f m e a s u r e s (u(x)dx, w(x)dx), t h a t i s , f o r w h i c h we have t h e i n e q u a l i t y :

(1.3)

u(Cx € Rn : Mf(x) > tl) 5 t > O

IE

For a set E, u(E) stands for u(x)dx. This notation has been used in ( 1 . 3 ) and it will be used systematically.

IV. WEIGHTED NORM INEQUALITIES

388

We shall keep the usual conventions for multiplication in [O,m], namely m.t = t.m = m for 0 < t 5 m and m . 0 = 0.- = 0. Also - 1 = 0 and 0 - 1 = m when we consider w - 1 for a weight w. Let us start by analizing Problem 3. Suppose that the pair of weights (u,w) is such that (1.3) holds for a given p, 1 5 p < every function f and every t > 0 . Let f be a function LO. Let Q be a cube such that the average f = f(x)dx > 0. Q lQ\ Q Observe that f M(f.XQ)(x) for every x E Q. Then, for every

1

m,

Q -

t

with

0

t < f

Q C Et

Q'

by (1 .3): u(Q)

5 C

It follows that

t-p

=

1,

(fQ)Pu(Q)

(1.4)

{x E Rn : M(f.X

Q )(x)

> tl

so that,

f(x)pw(x)dx

5 C

JQ

f(x)pw(x)dx

We can actually write this inequality in a seemingly stronger form. If S i s a measurable subset of Q, we can replace f in (1.4) by f.XS, obtaining

'k 1s

(1 .6)

f (x)dx)Pu(Q)

2 C

f (x)pw(x)dx IS Of course (1.5) is just equivalent to (1.4), but ( 1 . 5 ) is more readily applicable sometimes. F o r f(x) E 1, (1.5) yields: (1.5)

(ISl/lQlIP u(Q) 5 C w(S)

From (1.6) we can draw some relevant information about the pair (u,w) : a) w(x) > o for a.e. x E R" (unless u(x) = o for a.e. x € Rn, trivial case which we shall exclude). b) u is locally integrable (unless w(x) = m for a.e. x E Rn, again a trivial case which we shall a l s o exclude). Let us prove a) and b). I'f w(x) = 0 on a set S with IS1 > 0, a set S which we could assume to be bounded, (1.6) would imply that u(Q) = 0 for every cube Q containing S, and consequently, u(x) = 0 for almost every x E En. If u(Q) = m for some cube Q and, consequently, f o r any cube containing 0, we would have w(S) = m f o r any set S with I S 1 > 0, and this implies w(x) = m f o r a.e. x E Rn. We are about to derive a necessary condition on the pair

(u,w)

for

IV.l. THE CONDITION A

( 1 . 3 ) to hold for every in the form:

f

If p

and t.

389

P

= 1,

(1.6)

can be written

(1.7) the inequality being valid for every cube Q and every set S C Q with I S 1 > 0. Fix Q and let a > ess.Qinf.(w), the essential infimum of w over Q, which is defined as the inf{t > 0 : ]{x 6 Q : w(x) < t}l > 0 1 . Then, Sa = {x E Q : w(x) < a} has lSal > 0, and ( 1 . 7 ) holds for S = Sa, from which we get: u(Q)/(QI S Ca. Since this is true for every a > ess inf. (w), we Q arrive finally at: (1.8)

1

1 IQI Q

u(x)dx

2 C ess. inf.(w) 2 C w(x),

Observe that the fact that ( 1 . 8 ) to :

Q

holds for every

Q

for a.e. xEQ is equivalent

Indeed, i t is clear that ( 1 . 9 ) implies ( 1 . 8 ) for every cube. Conversely if ( 1 . 8 ) holds for every Q, let us show ( 1 . 9 ) holds, that is, the set Cx E Rn : M(u) (x) > C w(x)} has measure 0. If M(u)(x) > C w(x), i t will be

for some cube Q containing x , and we can assume that Q has vertices with all coordinates rational. Then our set is contained in a denumerable union of sets o f measure 0 and, consequently has measure 0. Condition ( 1 . 9 ) is known as condition A, for the pair (u,w). When it holds, we also say that the pair (u,w) belongs to the class A,, viewing A 1 as a collection of pairs of weights (u,w). We often speak of the A 1 constant for the pair (u,w) which is the smallest C for which ( 1 . 8 ) , o r equivalently ( 1 . 9 ) , holds. We have just seen that (u,w) 6 A, is a necessary condition f o r to be of weak type ( 1 , l ) with resnect to the pair (u,w). It is very satisfactory to realize that this condition is actually sufficient. Indeed, let (u,w) € A 1 , s o that ( 1 . 9 ) holds. Then,

M

IV. WEIGHTED NORM INEQUALITIES

39 0

u s i n g i n e q u a l i t y ( 2 . 1 4 ) i n c h a p t e r 1 1 , we g e t

Now we s h a l l t r e a t t h e c a s e 1 < p m. We s t a r t by l o o k i n g f o r a n e c e s s a r y c o n d i t i o n . So f a r we know t h a t i f M i s o f weak t y p e (p,p) with r e s p e c t t o (u,w), then (1.5) holds f o r every f u n c t i o n Q and e v e r y m e a s u r a b l e s e t S C Q . Let us f L 0 , e v e r y cube choose f such t h a t f ( x ) = f ( x ) p w ( x ) . T h i s g i v e s f ( x ) = = w(x) - l ' ( p - l ) . A p r i o r i t h i s f u n c t i o n n e e d s n o t be l o c a l l y i n t e g r a b l e . F i x a cube Q and t a k e S = S . = { x E Q : w(x) > j - ' } I f o r j = l , Z ,... On e v e r y S . o u r f i s bounded, s o t h a t 1 f < m. With o u r c h o i c e f o r f , (1.5) gives:

Isi

or, since the integrals are f i n i t e ,

...

m

U

S = [ x 6 Q : w(x) > 0 1 , whose i=l j complement i n Q h a s m e a i u r e 0 , a s was p r e v i o u s l y o b s e r v e d . T h u s , l e t t i n g j + m , we g e t f i n a l l y

Now

SIC S 2 c

and

or We s h a l l s a y t h a t t h e p a i r (u,w) s a t i s f i e s t h e c o n d i t i o n A ? t h a t it belongs t o t h e c l a s s A i f and o n l y i f t h e r e i s a c o n s P' t a n t C s u c h t h a t ( 1 . 1 0 ) h o l d s f o r e v e r y cube Q . The s m a l l e s t s u c h c o n s t a n t w i l l b e c a l l e d t h e Ap c o n s t a n t f o r t h e p a i r ( u , w ) . i s n e c e s s a r y f o r M t o be o f weak We have p r o v e d t h a t (u,w) E A P type ( p , p ) with r e s p e c t t o t h e p a i r (u,w). Observe t h a t (u,w) E A i m p l i e s t h a t b o t h u and w - " ( p - ' ) are locally inteP g r a b l e . I n d e e d i f one o f t h e i n t e g r a l s i n (1 . l o ) were m y t h e same would happen f o r any cube c o n t a i n i n g Q , and t h a t would f o r c e t h e o t h e r f a c t o r t o be 0 . T h i s would imply e i t h e r u ( x ) = 0 f o r a . e . x E Rn o r w(x) = m f o r a . e . h a v e been e x c l u d e d b e f o r e h a n d . made i s t h a t t h e c o n d i t i o n A1

x E Rn. Both t r i v i a l s i t u a t i o n s Another o b s e r v a t i o n t h a t has t o be c a n be viewed a s a l i m i t c a s e o f

I V . l . THE C O N D I T I O N A

the conditions (1.11)

(- 1

p

+

1,

P

IQI

while

as

A

for

+

p

1.

39 1

P

I n d e e d , ( 1 . 8 ) c a n be w r i t t e n a s

u ( x ) d x ) . e s ~ . ~ s u p . (-1w)

<=

C

Q

i.e.

l/(p-1)

-+

m.

Thus ( 1 . 1 1 ) i s t h e r i g h t companion f o r ( 1 . 1 0 ) when p = 1 . A l s o n o t e t h a t (u,w) E A1 i m p l i e s t h a t u i s l o c a l l y i n t e g r a b l e and w - 1 i s l o c a l l y bounded. Our t a s k w i l l b e now t o show t h a t , e x a c t l y a s i n t h e c a s e p = 1 , when 1 < p < m, (u,w) € A is not only necessary, but a l s o P s u f f i c i e n t f o r M t o be o f weak t y p e ( p , p ) w i t h r e s p e c t t o t h e p a i r ( u , w ) . We h a v e o b t a i n e d c o n d i t i o n A from ( 1 . 4 ) . The f i r s t P s t e p w i l l b e t o show t h a t , c o n v e r s e l y , i f then (1.4) (u,w) € A p , h o l d s f o r e v e r y f 2 0 and e v e r y c u b e Q . T h i s i s a c t u a l l y t r u e

f o r 1 S p < m. I f p = 1 c u b e Q a n d e v e r y f => 0 :

1

(L IQI

Q for

and

f(x)dx)u(Q) =

(u,w) 6 A 1 ,

i,

f(x)dx

we h a v e , f o r e v e r y

u(Q)S IQ I

C

i,

f(x)w(x)dx,

which i s (1 . 4 ) p = 1 . I f 1 < p < m and (u,w) € A we h a v e , P' for e v e r y c u b e Q a n d e v e r y f => 0 , u s i n g H i i l d e r ' s i n e q u a l i t y w i t h p and i t s c o n j u g a t e exponent p ' = p / ( p - 1 ) ,

Thus

s o t h a t ( 1 . 4 ) h o l d s . We have e s t a b l i s h e d t h e e q u i v a l e n c e between ( 1 . 4 ) and A P'

I V . WEIGHTED NORM INEQUALITIES

392

Q

Now, s u p p o s e t h a t ( 1 . 4 ) h o l d s f o r e v e r y c u b e

and e v e r y

f 2 0.

We s h a l l o b t a i n ( 1 . 3 ) w i t h a p o s s i b l y b i g g e r C . O f c o u r s e , we h a v e ( 1 . 5 ) f o r e v e r y f L 0 , e v e r y c u b e Q and e v e r y s e t S C Q . Let f € Lp(w). We c a n o b v i o u s l y assume t h a t f 2 0 . O b s e r v e t h a t 1 L ~ o c ( w ) CLloc(Rn) a s f o l l o w s from ( 1 . 4 ) u s i n g Q s u c h t h a t u(Q) > 0 . Now we c a n a l s o assume t h a t f E L1 (R"). I n d e e d , we c a n a l w a y s w r i t e f = l i m f k where f k = f.X I f we have ( 1 . 3 ) Q(O,k) ' k+m f o r e v e r y f k i n p l a c e o f f , p a s s i n g t o t h e l i m i t we o b t a i n ( 1 . 3 ) f o r f . T h u s , a s s u m i n g f i n t e g r a b l e , we want t o e s t i m a t e u ( E t ) where E t = { x 6 Rn : Mf(x) > t l . We u s e t h e o r e m 1 . 6 from c h a p t e r 11, t o w r i t e E t C Q i , where t h e Q j ' s a r e n o n - o v e r -

0

'

l a p p i n g c u b e s f o r which t

t f(x)dx f Zn

Then, u(Et)

2

C u ( Q3. )

j

6

C 3"p

J

drip t - p c j

2

[

I

C C (- l3

j Qj

lQjl

f(x)dx)-P

f(x)Pw(x)dx

2

Qj

Qj

f(x)Pw(x)dx

I

2

C t-p

where t h e s e c o n d i n e q u a l i t y r e s u l t e d from a p p l y i n g ( 1 . 5 ) w i t h = QS and S = Q We have a c o m n l e t e p r o o f o f t h e f a c t t h a t t h e j' s o l u t i o n t o Problem 3 i s p r e c i s e l y t h e c l a s s A of p a i r s of P w e i g h t s . We c a n c o l l e c t o u r f i n d i n g s i n t h e f o l l o w i n g

Q

THEOREM 1 . 1 2 . Let

w

u

be w e i g h t s on

Rn

and l e t

1

2

p <

m.

Then, t h e f o l l o w i n g c o n d i t i o n s a r e e q u i v a l e n t : a) M that i s :

M

i s o f weak t y p e

Lp(w)

takes

there i s a constant and e v e r y t > 0

f

2

0

in

C

2

( p , p ) w i t h r e s p e c t t o (u,w), L!(u) boundedly o r , i n other.words,

such t h a t f o r every f u n c t i o n

u ( { x E Rn : Mf(x) > t l )

2

b ) T h e r e is a c o n s t a n t

C

Rn

and f o r e v e r y c u b e

f(x)dx)pu(Q) 5 C

C t-P

f

6 L;oc(Rn)

jRn I f ( x ) IPW(X)dX.

such t h a t f o r every f u n c t i o n Q

IV.l. THE CONDITION A

c)

(u,w) € A

t h a t f o r e v e r y cube

and, i n case

Besides,

Q

393

P

that i s , there i s a constant

P'

we h a v e , i n c a s e

1 < p <

C

such

m,

p = 1,

the constants

a p p e a r i n g i n a ) , b ) and c ) a r e of t h e

C

same o r d e r .

COROLLARY 1.13. L e t (u,w) € A . T h e n , f o r e v e r y q with P t h e maximal o p e r a t o r M i s bounded f r o m Lq(w) to p < q < m, Lq(u), t h a t i s , there e x i s t s a constant C such t h a t f o r ever2 f € Lloc(Rn) 1 : ~

IRn

IMf(x)lqulx)dx

S C

1,"

I f (x) 1 w'

(x) dx

Proof: We already known that F! is of weak type (p,p) with respect to (u,w), that is, M takes Lp(w) boundedly to L t ( u ) . We shall see presently that M is also bounded from Lm(w) to Lm(u). This is a consequence of the chain of inequalities: IMflm,U 5 I M f l ,

S

lfl,

2

Since ~Mf~,,u

=

sup{a : u({x

€

Rn : Mf(x) >

a } ) > 0}

the first inequality follows from the fact that u(E) > 0 implies [ E l > 0. Likewise, the last inequality follows from the fact that IEl > 0 implies w(E) > 0, because w(x) = 0 only for the points in a set of measure 0. Once we know that M is bounded from Lp(w) to L:(u) and from Lm(w) to Lm(u), we use Marcinkiewicz interpolation theorem to conclude that M is bounded from Lq(w) to Lq(u) provided p < q < m. 0 A particular instance of corollary 1.13 is the inequality

iRn

IMf(x) IPu(x)dx

=< C

jRn [ f(x) IPMu ( W x

valid for 1 < p < m, which appeared in Chapter 11, (2.13). I t is contained in our corollary because (u,Mu) € A 1 .

394

IV. WEIGHTED NORM INEQUALITIES

The following theorem contains some simple basic facts about the classes A P'

THEOREM 1.14. a ) Let b)

Let

1 < p < q <

1 5 p <

m,

Then A I C A C A P 9

m,

0 < E <

1

( u , ~ )6

A P'

Then -

(uE,wE) 6 A E P + l - E .

1 < p < =. T h e n (u,w) E A P i f and o n l y i f u7'p-l)) E A ~ ' , pl being, as u s u a l , t h e exponent c o n j u g a t e t o p, t h a t i s p ' = p/(p-l). c) L e t

(w-1'(p-'),

Proof: a) We just need to observe that

6 ess .Qsup. (w- 1 ) The first inequality follows from Jensen's o r H6lder's inequality, since (q-l)(p-l)-' > 1. The second one is obvious. b) For

r

=

EP+~-E

is

r-1

=

~ ( p - 1 ) . Then

if C is the A constant for the pair (u,w). Again Jensen's P inequality has been used to deal with the integral containing u . The case p = 1 is included by interpreting the second factor properly. c ) Suppose

since (p-i)(pf-l) quality as:

=

(u,w) E A

P'

Thus:

pp'-p-pi+l = 1, we can write the previous ine-

IV. 2 . REVERSE H O L D E R ' S

and conclude t h a t that

(u,w) 6 A

39 5

A c t u a l l y we s e e

(w - l ' ( p - l ) , is equivalent t o

P

INEQUALITY

.o

(w

EXAMPLE 1 . 1 5 . We s h a l l g i v e h e r e a n e x a m p l e w h i c h shows t h a t c o r o l l a r y 1 . 1 3 c a n n o t be improved s o as t o i n c l u d e a l s o t h e case q = p . We s h a l l g i v e w e i g h t s u,w s u c h t h a t (u,w) € A a n d , however, M P i s n o t b o u n d e d f r o m Lp(w) t o L P ( u ) . I f p = 1 , we j u s t n e e d t o t a k e u = w E 1 . We a l r e a d y know t h a t i f g 2 0 i s i n t e g r a b l e a n d is not 0 a t a.e. x , t h e n Mg i s n e v e r i n t e g r a b l e ( s e e t h e r e m a r k T h i s same F a c t l e a d s t o a n f o l l o w i n g theorem 2.6 i n Chapter 11). e x a m p l e f o r p > 1 . Let g 2 0 , i n t e g r a b l e a n d non t r i v i a l , i n s u c h a way t h a t

Mg

e

Take

L1.

g

bounded s o t h a t you c a n guarantee

and hence Mg(x) i s a l w a y s f i n i t e . Then ( g , Mg) € A I C A p l , ( P ' ' ) , g " ( p ' l ) ) = ((Mg)'-p, g l - p ) € Ap. I f we t a k e ((Mg) u = ( b l g ) l - P a n d w = g l - p , we h a v e a p a i r ( u , w ) € A f o r which P the inequality that

jlMf(x) IPu(x)dx cannot hold, since f o r I j l f l p w = J g < m.

f

5

C

=

g

I

l f ( x ) J p w(x)dx we have:

Mg =

11.f

I n t h i s way, we h a v e s e e n t h a t t h e c o n d i t i o n s o l v e Problem 2 . I t i s n e c e s s a r y f o r ( 1 . 2 ) t o p l i e s ( 1 . 3 ) . However, i t i s n o t s u f f i c i e n t .

m

and

E Ap d o e s n o t s i n c e (1.2) i m

2 . THE REVERSE H ~ L D E R ' S INEQUALITY A N D THE C O N D I T I O N A ~ .

The t h e o r y d e v e l o p e d f o r t h e case situation:

u

THEOREM 2 . 1 .

Let

=

i n s e c t i o n 1 becomes p a r t i c u l a r l y i n t e r e s t i n g w. F i r s t of a l l , t h e o r e m 1 . 2 r e a d s a s follows i n t h i s

w

be a weight on

Rn,

and l e t

1

2 p

<

m.

Then,

t h e foZlowing c o n d i t i o n s are e q u i v a l e n t :

a) M i s of weak t y p e

( p , ~ ) with respect t o

w({x € R n : b l f ( x ) > t } ) 5 C t - p b ) 2 e r e i s a constant

C

iRn I

f

( X I I pw

w,

&

(x) dx

such t h a t , for every f u n c t i o n

I V . WEIGHTED NORM INEQUALITIES

396

f > 0

and f o r e v e r y c u b e

cl

(w,w) 6 A

Q,

f o r e v e r y cube constants

C

P'

Q

1 < p <

that i s , i n case

and, i n c a s e

appearing i n a ) , bl

m

p = 1, Mw(x) 5 C w ( x ) p.e. c) a r e of t h e same o r d e r .

&

The

When w s a t i s f i e s c ) , we s a y t h a t w s a t i s f i e s t h e c o n d i t i o n *P' and w r i t e w E A We a l s o s p e a k o f t h e Ap c o n s t a n t f o r w , w i t h P' t h e n a t u r a l meaning. N o t i c e t h a t t h e c l a s s A, i s t h e same w h i c h appeared i n Chapter I1 ( a f t e r theorem 2 . 1 2 and a l s o i n theorem 3 . 4 ) . We saw ( i n 1 . 1 5 ) t h a t a p a i r o f w e i g h t s

( u , w ) may b e i n A and P In contrast t o y e t M may n o t b e b o u n d e d f r o m Lp(w) t o L p ( u ) . t h i s s i t u a t i o n , f o r p > 1 , i t s u f f i c e s t h a t w i s i n Ap f o r M t o b e b o u n d e d i n Lp(w) . T h i s f a c t depends on a b a s i c p r o p e r t y e n j o y e d by t h e A weights: P t h e r e v e r s e H z l d e r ' s i n e q u a l i t y ( R . H . I . ) a p p e a r i n g i n lemma 2 . 5 b e l o w . F i r s t we p r e s e n t a c o u p l e o f s i m p l e p r o p e r t i e s o f t h e weights.

We s t a r t w i t h a n

a cube

Let w

and e v e r y

X

b e ay.

A

1

w(QX) where

of

Qx

Q.

LEMMA 2 . 2 . Q

w-measure o f t h e d i l a t e d

estimate for the

Ap

C

weight i n

P

2

C Xnp

is of t h e same o r d e r

as

Then, f o r every cube

w(Q) the

P r o o f : I n b) o f t h e o r e m 2 . 1 , t a k e

Rn.

f = Xs

A

constant f o r

P

with

S C Q,

w

and

Q

a

c u b e . Then (2.3)

Using (2.3) w i t h Q x w(Q ) 2 C X n p w ( Q ) .

( / s ~ / ~ Q ~ ) ~ w 5( Qc) W(S) in place of

0

S

and

Qx

i n p l a c e of

Q

we g e t

IV.2. REVERSE H ~ ~ L D E RINEQUALITY ~S

39 7

In particular the lemma implies that for an A weight w, the P measure 1~ given by du(x) = w(x)dx is a doubling measure. Actually, what we have shown is that property b) in theorem 2.1 implies that p is a doubling measure. Observe also that the same pro perty b) implies that 5 C1’p IMu((f(p)(x)?”p

Mf(x)

a.e.

where Mv is the operator introduced in Chapter I1 (after theorem 1.12). We showed there that, for 1~. doubling, Mu is of weak type (1,l) with respect t o u . We can rely upon this fact to prove that b) implies a) in theorem 2.1. Indeed w(Cx E R” : Mf(x) > tj)

where the last in b).

C

<=

w({x

E Rn : CMp(lflP)(x)

LEMMA 2.4.

Let

B

1

w € A

P’

Then, f o r e v e r y p o s i t i v e

depending on

r a b l e s e t contained i n a cube follows that

tP3)1

contains the doubling constant and the constant

The next lemma is a comparison between the measure besgue measure

exists

>

a

such t h a t , whenever

Q

and s a t i s f g i n g .

w(x)dx

and

< 1,

there

c1

A /A1

Lg

i s a measu-

2

alQl,

w(A) 5 B w(Q).

Proof: We start from (2.3) where, of course, it is always C b 1 (set S = Q). If we u s e in (2.3) S = Q\A where \ A \ 5 a \ Q l , we get:

We shall use the previous lemma to establish o u r basic inequality LEMMA 2.5. on -

p

w E A

and o n t h e

A

P

P’

Then, t h e r e e x i s t s

constant for

E

> 0 , depending o n l y

w , such t h a t , f o r every cube Q

IV. WEIGHTED NORM INEQUALITIES

398

C

w i t h a constant

Q.

n o t d e p e n d i n g on

The o p p o s i t e i n e q u a l i t y h o l d s , w i t h C = 1 , f o r e v e r y f u n c t i o n w and i s a p a r t i c u l a r c a s e o f H b l d e r ' s i n e q u a l i t y . T h i s i s why t h e lemma i s c a l l e d t h e r e v e r s e H o l d e r ' s i n e q u a l i t y ( R . H . I . ) . P r o o f : We s h a l l f i x t h e cube with

and

E

...

ho < h 1 <

Q

and we s h a l l g e t t h e i n e q u a l i t y

&

i n d e p e n d e n t o f Q . We t a k e an i n c r e a s i n g s e q u e n c e < w i t h A, = wQ = w(x)dx a n d , f o r

C

...

< Ak

JQ

we make t h e C a l d e r h - Z y g m u n d d e c o m p o s i t i o n o f Q f o r t h e each h k , f u n c t i o n w and t h e v a l u e A k ; t h a t i s , we c o n s i d e r t h o s e maximal (the d y a d i c s u b c u b e s o f Q o v e r which t h e a v e r a g e o f w i s > A k dyadic subcubes of s i d e o f Q i n 2N

.

r Q k , j j =1 , 2 , . . . 'k < wQk . <= 2"Ak, =

Dk

is

1 1

a r e t h e c u b e s r e s u l t i n g from d i v i d i n g e a c h e q u a l p a r t s , N = 0 , l , 2 , . . . ) . L e t them be I t follows t h a t , f o r each j is Q

while f o r a . e .

w(x) 2 X k .

Since

Ak+l

x

not belonging t o

> Xk,

i n Qk,i f o r some i , i n s u c h a way t h a t what p o r t i o n o f Q k , i c a n b e c o v e r e d by

each

Q k + l ,i

._

D k + l C Dk.

Dk+l.

y

Qk,j =

i s contained

Let u s s e e

We know t h a t :

w(x)dx = Qk,in

Dk+l

I Q k + l ,j

w(x)dx

' 'k+l

lQk,i"

Dk+ll

IQk,il

Let us t a k e t h i s r a t i o equal t o

c1

< 1,

t h a t i s A k + l = 2 n a - 1 A k , A k = ( 2 na - 1 ) k A., I f we c o n s i d e r t h e @ a s s o c i a t e d t o a a c c o r d i n g t o t h e p r e v i o u s lemma, we s h a l l have w ( Q k , i n D k + l ) 5 6 W(Qk,i) a n d , summing o v e r i , w ( D ~ + ~I) Bw(Dk), which l e a d s t o

lDk+l1 S a\Dkl m

I n

k=o

Dkl

w(Dk) S and

= l i m lDkl

k+m

kw ( D ) .

\Dkl B =

0.

Of c o u r s e , we a l s o have

R CY ID,\,

Then:

which i m p l i e s t h a t

399

IV. 2. REVERSE HOLDER'S INEQUALITY

I f we take E > 0 s m a l l enough to have (2" a will have a finite sum and we shall get: I

w(x)'+'ddx

2 C X:(w(Q\Do)

+

-') B < 1 ,

w(Do))

the series

C wE w Q) Q

=

Thus +c

'

0

Lemma 2.5 has far reaching consequences which we shall presently see w 6 A with 1 < p < THEOREM 2.6. I-f P s u c h t h a t w € Aq; that i s , for every Ap= A 4' 9
u

m,

t h e n t h e r e is some

p,

1 < p

m,

q

ue h a v e

Proof: Theorem 1.14(c) for the special case u = w tells us that On the other hand, from lemma w 6 A implies w -1/(P-1) 6 A P' . P 2.5 for the weight w -l'(p-l) we know that there exist c > 0 , C > 0 such that, for every cube Q:

But

1+E p-l >

1 p-1

- implies

l+E p-l-q-l for some

Actually, since w itself satisfies following stronger result. T I E O R E M 2.7. i fw 6 A 1+E p E > 0 such t h a t w € Ap.

1 5

11

3

K.Il.I.,

<

m,

1

c

q < p.

Then

we obtain the

then,

Ih~7i-e c x i s L s

c

p

400

I V . WEIGHTED NORM INEQUALITIES

Proof: I f enough :

p = 1

1,

it s u f f i c e s t o observe t h a t f o r

w(x)l+'dx

p > 1,

I

5 (A w ( x ) d x ) 1 + E 2 c w(x) 1 + E IQI

Q

it suffices t o take

s m al l

for a.e. x € Q

w 1+E 6 A 1 .

i n s u c h a way t h a t If

> 0

E

small enough t o h a v e , a t t h e

> 0

E

same t i m e :

and

O f c o u r s e , theorem 2 . 7 combined w i t h p a r t b) o f theorem 1 . 1 4 , g i v e s

theorem 2 . 6 . Now, w i t h t h e h e l p o f t h e o r e m 2 . 6 , w e c a n i m p r o v e t h e o r e m 2 . 1 a s anticipated, obtaining

Let w

THEOREM 2 . 8 .

be a w e i g h t on

R"

and l e t

1 < p <

m.

Then,

t h e foZZowing c o n d i t i o n s a r e e q u i v a l e n t :

a ) M i s o f weak t y p e ( p , p ) w i t h r e s p e c t t o w, t h a t is, t h e r e i s a c o n s t a n t C s u c h t h a t f o r e v e r y f u n c t i o n f t Lloc(Rn) 1 and e v e r y

t > 0

w({x € Rn : Mf(x) > t j ) 6 C t - P b ) There i s a c o n s t a n t

f

>=

0

in

Rn

and e v e r y c u b e

c) w E A every cube

Q

P'

s u c h ttut f o r e v e r y

If(x)IPw(x)dx

such t h a t f o r every f u n c t i o n

Q

t h a t is: t h e r e i s a c o n s t a n t

d ) ?I is b o u n d e d i n

C

C

iRn

Lp(w),

f € Lp(w):

C

such t h a t f o r

t h a t i s , t n e r e is a c o n s t a n t

I V . 2 . REVERSE H O L D E R ' S I N E Q U A L I T Y

401

P r o o f : A l l t h a t remains t o be proved i s t h a t c ) i m p l i e s d ) . Here i s t h e p r o o f . We h a v e that

w 6 A

9

Since 1 < p < m, theorem 2.6 t e l l s us P' q < p . T h en M i s o f weak t y p e ( q , q ) w i t h

w E A

f o r some

i s a l s o bounded i n e q u a l i t y f o l l o w s from t h e f a c t t h a t 0 < w(x) < m respect t o

w

and,

since

M

Lm(w)

=

(this

La

for a.e.

x), the

Marcinkiewicz i n t e r p o l a t i o n theorem allows us t o conclude t h a t

i s bounded i n

Lp(w)

.

M

0

A l s o , t h e r e v e r s e H E l d e r ' s i n e q u a l i t y a l l o w s u s t o g i v e a more p r e c i s e v e r s i o n o f lemma 2 . 4 .

THEOREM 2 . 9 . 6 > 0,

Lf

C > 0

w E A f o r some p 6 [ l , m ) , then there e x i s t P s u c h t h a t , e v e r y t i m e we h a v e a m e a s u r a b l e s e t A

contained i n a cube

Q,

the following inequaZity holds:

(2.10)

P r o o f : The k e y f a c t i s t h a t

w

s a t i s f i e s a n i n e q u a l i t y l i k e t h e one

a p p e a r i n g i n lemma 2 . 5 f o r some

( R . H . I . ) . We s t a r t b y u s i n g 1 + ~a n d i t s c o n j u g a t e ( l + E ) / f ,

> 0

E

Hijlder's i n e q u a l i t y w i t h exponents and t h e n w e a p p l y t h e R.H.I.

which i s (2.10) w i t h

We g e t

6 = €/(I+€).

C o n d i t i o n ( 2 . 1 0 ) i s known a s

Am

s o o n . We a l s o s p e a k o f t h e c l a s s

f o r r e a s o n s which w i l l a p p e a r v e r y Am

which i s , n a t u r a l l y , t h e c l a s s

f o r m e d by t h o s e l o c a l l y i n t e g r a b l e w e i g h t s

w

satisfying the

Am

condition. For t h e n e x t r e s u l t ,

pl

and

u2

a r e going t o be doubling

measures, t h a t i s , both s a t i s f y a doubling c o n d i t i o n l i k e (1.13) i n Chapter 11. For t h e s e measures, w e g i v e t h e f o l l o w i n g d e f i n i t i o n :

I V . WEIGHTED NORM I N E Q U A L I T I E S

402

p,

p2

i s comparabze t o

when t h e r e e x i s t

a,B

A

t h a t , e v e r y t i m e we h a v e a m e a s u r a b l e s u b s e t

Q with

uZ(A)/p2(Q) 5 a,

i t foZZows t h a t

< 1

of a c u b e

p1(A)/p,(Q)5B.

W ith t h i s d e f i n i t i o n we c a n w r i t e

THEOREM 2 . 1 1 , rable set

The f o l Z o w i n g c o n d i t i o n s a r e e q u i v a l e n t

P 2 (A) p21Q)

' '(v) ~1 (A)

bl p2

i s comparable t o

p1

el p l

is comparable t o

p2

dl dp2(x) = w(x)dpl(x)

f o r some

E

that and

with:

is c l e a r . Indeed, i f

I C a6.

p2(A)/p2(Q)

6

> 0

P r o o f : a ) =*b)

a

s u c h t h a t f o r e v e r y measu-

C > 0 a ) There e x i s t 6 > 0 , A contained i n a cube Q

pl(A)/pl(Q)

Ca6 < 1 a n d we o b t a i n 6 B = Ca .

p2

comparable t o

it w i l l be a > 0 such

6 a,

I t s u f f i c e s t o s t a r t w i t h some p1

with constants

b ) 3 c ) . T o s a y t h a t p l ( . 4 ) / v l ( Q ) <= a i m p l i e s p 2 ( A ) / p 2 ( Q ) is equivalent t o saying t h a t p2(A)/p2(Q) > implies

U ~ ( A ) / I J ~ ( Q )> a .

Then i f

u2(A)/u2(Q)

5 (1-8112 < 1 - B ,

2

f3

i t w i l l be

p2(Q\A)/p2(Q) > 6, i n s u c h a way t h a t p 1 ( Q U ) / p l ( Q ) > CI a n d , c o n s e q u e n t l y p l ( A ) / p l ( Q ) < 1 - a . T h u s , we h a v e s e e n t h a t y l i s comparable t o p2 with constants a' = ( 1 - B ) / 2 and B ' = 1 - a . I t becomes c l e a r t h a t , a c t u a l l y , b ) an d c ) a r e e q u i v a l e n t . Le t u s s e e now t h a t b )

We s t a r t f r o m t h e f a c t t h a t

p2 i s compa6 . We s e e , f i r s t o f a l l , t h a t p2 is absolutely continuous with respect t o p , , that is: U l ( E ) = 0 => p 2 ( E ) = 0 . Once t h i s i s p r o v e d , t h e Radon-Nikodym theorem g u a r a n t e e s t h a t d u 2 ( x ) = w ( x ) d p l ( x ) w i t h w l o c a l l y i n tegrable with respect t o p l . Let p l ( E ) = 0 a nd s u p p o s e t h a t v 2 ( E ) > 0 . S i n c e t h e measures a r e r e g u l a r , t h e r e w i l l b e an open s e t R s u c h t h a t R 3 E and p 2 ( R ) < B - l u 2 ( E ) . Let R = Q.

rable t o

pl

3

d).

with constants

a

and

uj

. I

HBLDER~SINEQUALITY

IV. 2. REVERSE

403

where the

Q J s are non-overlapping cubes. Since for each j is o = E ) 5 C X U ~ ( Q ~ ) ,we shall have w2(Qj fl E ) <= B U ~ ( Q ~ ) and, adding in j : u2(E) $ @p2(G), which contradicts the election of R. We have used the fact that the faces or edges of the cubes have measure p 2 (or p, for that matter) equal to 0. This follows easily from the doubling condition. Indeed, if 1~ is doubling, there is a constant k < 1 such that if Q is a cube and R is a half of Q (that is R is an interval contained in Q having one side equal to a half the corresponding side of Q and the other sides coinciding with those of Q) then u ( R ) $ ku(Q). The proof o f this is essentially the same that the argument we used in chapter I1 (after theorem 1.12) to guarantee that the Calderdn-Zygmund decomposition makes sense for a doubling measure. Viewing a face of Q as intersection of the R!s resulting from repeatedly dividing by 2 a side 3 of Q , we see that a face has 1-1 measure 0 for u doubling.

ul(~jn

Let dp2(x) = w(x)dul(x). I t remains to s e e that the inequality in d) holds. All we have to do is to repeat the proof of lemma 2.5. with p l in place of Lebesgue measure. Observe that in the proof of lemma 2.5 we just used these two facts: w(x)dx is comparable to Lebesgue measure and Lebesgue measure is doubling. These hypotheses still hold for du2(x) = w(x)dp,(x) and dul(x). Thus, we obtain the inequality in d). Finally we have to see that d) implies a ) . But this is done exactly as in the proof of theorem 2.9. 0 COROLLARY 2.12. T h e c o m p a r a b i Z i t y of m e a s u r e s is a n e q u i v a l e n c e r e Zation.

Proof: The equivalence between b) and c) in 2.11 tells us that comparability is a symmetric relation. 'Transitivity is proved very simply by using the characterization given by a ) in 2.11. COROLLARY 2.13. __ L e i w(x) 2 0 b e localZy i n t e g r a b l e i n following conditions arc? e q u i v a l e n t : il w € A

P

w(E) I Bw(Q)

whenever

a,R

E

is

T&

p € [1,m)

for some

iii Them, exist

Rn.

1

< (i

such t h a t

IE( 5 n\Q(

implies

m e a s u r a b l e subset of t h k c u b e

Q.

404

I V . WEIGHTED NORM INEQUALITIES

i i i l There e x i s t

and

> 0

E

C > 0

such t h a t f o r e v e r y

cube Q -

Proof: A l l t h e implications i )

3

i i ) =*

iii)

-

iv)

have a l -

r e a d y b e e n p r o v e d . Observe t h a t t h e p r o o f o f lemma 2 . 5 a c t u a l l y yields the fact that i i ) i i i ) . I t o n l y remains t o s e e t h a t i v ) d i ) . Let u s s e e i t . We know from theorem 2 . 1 1 t h a t w E A,

is

e q u i v a l e n t t o s a y i n g t h a t t h e m e a s u r e s dx and w(x)dx a r e compar a b l e a n d , t a k i n g i n t o a c c o u n t t h a t dx = w ( x ) - ' w ( x ) d x , the following R.H.I. must h o l d : T h e r e e x i s t q > 0 and C > 0 s u c h t h a t f o r e v e r y cube Q

5

&-i,

w(x)-lw(x)dX =

c

i4.7

Hence

Setting

rl =

-,1

p

that is:

Thus we have shown

A,

given t o condition (2.10).

=

=

u

l+rl - 1 > 1 ,

1 _-< p < m

Ap,

we o b t a i n

w

E A

P'

which e x p l a i n s t h e name

0 A,

A c t u a l l y , t h e name A, i s j u s t p e r f e c t , s i n c e , a s we s h a l l p r e s e n t l y show, Am c o i n c i d e s w i t h t h e f o r m a l l i m i t o f c o n d i t i o n AP as p tends t o

The l a s t i d e n t i t y above i s a s i m p l e e x e r c i s e i n m e a s u r e t h e o r y ( s e e , f o r example Rudin [ l ] to

,

c h a p t e r 111, p r o b l e m 5 ) .

Thus, t h e c o n d i t i o n o b t a i n e d by p a s s i n g t o t h e l i m i t a s in condition A is: P l o g ( w ( x ) - ' ) d x ) 5- C (1w ( x ) d x ) IQI Q

m

1

p

tends

405

1v.2. REVERSE H~LDER'S INEQUALITY

I

or, equivalently:

1

(2.14)

IQI

Q

w(x)dx

5

C exp(- 1

IQI

Q

log w(x)dx)

The exponential in the right hand side of (2.14) is the geometric mean of w on Q, which is, of course, dominated by the arithmetic mean wQ (Jensen's inequality). Thus (2.14) implies that the arithmetic and the geometric means of w on every cube, are equivalent. The equivalence between this condition and Am is contained in the f01 lowing

L e t w(x) 2

THEOREM 2 . 1 5 .

0

Rn.

be l o c a l l y i n t e g r a b Z e i n

Then,

t h e foZZowing c o n d i t i o n s a r e e q u i v a l e n t :

a,B e (0,l)

i l There e x i s t I{X E

Q

: W(X)

5

such t h a t , f o r every cube

"WQ}~

Q:

5 BlQl

i i l w e Am i i i l There e x i s t s

C,

such t h a t , f o r e v e r y cube

Q:

Proof. Suppose i) holds. Let us prove ii). After the proof of theorem 2 . 1 1 , it will be enough to see that, for appropriately chosen y,6 e ( O , l ) , the following property holds: If E is a subset of a cube Q such that w(E)/w(Q) <= y , then lEl/lQl <= 6. To prove this property, assume w(E)/w(Q) 5 y , to be chosen later. Then we split E = E l u E2, where E l = Cx e E : w(x) > a w 1 and Q E2 = Ix e E : w(x) 5 a w 1 . For E2, i) gives the estimate Q lE21 5 BlQl. For E, we use Chebichev's inequality to get:

If we choose Adding up the two estimates, we have IEl 5 ( B + ): I Q I . y s o small that 6 + < 1, we get what we wanted with 6 = B + (y/a). To see that ii) implies iii) is quite easy. Indeed, if w e A m , it follows from corollary 2.13 and theorem 1.14 a), that there is a constant C such that, for all p large enough and for all cubes Q:

Letting

1

p

tend to

m

we obtain iii).

IV. WEIGHTED NORM INEQUALITIES

406

Finally, assuming iii), we are going to see that i) holds. Take a cube Q. Dividing w by an appropriate constant, we see that,

JQ

without loss o f generality, we may assume that wQ 5 C.

and, consequently, Then, with I{x e

Q

A >

: W(X)

I

<

still undetermined, we have:

0

5 All

Q

=-

log(1

IIx

= +

E

since, by assumption,

w(x)-l)dx =

=

1 lOg(l+X-’)

log w(x)dx

By using the simple inequality < C, we get:

2 log

: l o g ( 1 + w(x)-’)

Q

1 +w (x) log 7 dx w x

-

w

log w(x

JQ

log(1 +

0.

=

log(l+w) 5 w

and the hypothesis

Q =

C

log(l+A-l) In particular: I{X

if

ct

e

Q

: W(X)

(Q(

5

6 1 ( Q ( if X

C1 Wg)(

2

[IX E

is small enough.

Q

: W(X)

2 C 0.11

is small enough. We have obtained i) with

6

=

(l/Z)lQ( 1/2.

0

In chapter I1 (theorem 3.4) we gave examples of A, weights, namely those functions w of the form w ( x ) = (Mp(x))Y where 1-1 is a positive Bore1 measure such that Mp(x) < m for a.e. x e R” and 0 < y < 1. We used this result to show that 1x1‘ is an A1 weight in R” if and only if - n < ct 5 0 . for W(X)

Starting with A1 weights one can easily generate Ap weights 1 < p < m . Let wo,wl E A, in Rn, and let 1 < p < a. Then = w0(x).wl (x)’-~ is and A weight. Indeed P

IV.2 , REVERSE H ~ ~ L D E RINEQUALITY ~S

407

We shall show in section 5 that every Ap weight w is actually of the form w(x) = wo(x)w,(x)l~p for some wo,wl e A1 (factorization theorem). F o r the time being, we shall content ourselves with giving examples of A weights. If - n < c1 5 0 and - n < B 2 0, (x(alx/B(l-P) is an A weight in Rn. Hence lxl’ is an P AP weight in Rn if and only if - n < c1 < n(p-1) since 1x1’ and have to be locally integrable.

(IX~~)-~’(~-~)

By using the R.H.I. we get a converse of theorem 3 . 4 in Chapter 11, giving the following characterization of A1 weights: THEOREM 2 . 1 6 . Let w(x) be LO and f i n i t e a . e . Then, w E A1 i-f a n d o n l y if w(x) = k(x)(Mf(x))’, w h e r e k(x) 0 i s such that k,k-’ E Lm, f i s l o c a l l y i n t e g r a b l e and 0 < y < 1 . Proof. Theorem 3 . 4 of chapter I1 implies that every function of the given form is an A1 weight. Conversely, let w e A 1 . We know that w satisfies a R.H.I.:

I

(L w(x)~+€ dx) o

IQI

C <- -

l/(l+E)

IQI

-

1 +E

1

0

w(x)dx

5 C w(x)

a.e.

~ h u s w(x) z ( ~ ( w )(x)~”(’+~) 5 c w(x). We can write W(X) = k(x)(M(~”~)(x))”(~+~) with C-l 5 k(x) 2 1 and we obtain the representation required with f(x) = w(x)~+€ and y = l / ( i + ~ ) . n U There is a relation between weights and B . M . O . functions. We have already seen in Chapter I 1 (theorem 3 . 3 ) that the logarithm of an A, weight is a B.M.O. function. We shall see presently that the same is true for any Am weight. Of course this follows trivially after the factorization theorem, but a simple proof can be given without appealing to that result which we have not proved yet. First of all, we give a characterization of A weights in terms of their P logarithms. THEOREM 2.17. a) Let Rn and l e t 1 < p <

@ 03.

be a r e a l l o c a l l y i n t e g r a b l e f u n c t i o n on

TheM

e@

two c o n d i t i o n s a r e s a t i s f i e d :

ii

b)

1 -

I91

1,

@

e

(@(XI

-aQ) dx

as i n a),

5- C ,

e@

E Am

E

A

P

i f and o n l y i f t h e f o l l o w i x

i n d e p e n d e n t of t h e c u b e

Q

U n d o n l y if il h o l d s . N o t e

IV. WEIGHTED NORM INEQUALITIES

408

that for

p

c o n d i t i o n i i l becomes e m p t y , so t h a t b l i s j u s t an

= m,

e x t e n s i o n of a) t o

p

= m

c) I t foZZows from a) and b ) t h a t pnd w-”(p-l) a r e i n A,.

both w

w

is in

Ap

i f and onZy i f

Proof. a) It is clear that the two conditions i) and ii) imply together that e $ e A since P’

Conversely, suppose that

e$ e A

Then

P’

Also

I

b) Theorem 2.15 implies that

e@ e A,

if and only if

1

e@(x)dx 5 C e@Q, which is equivalent to condition i). Q c) I t follows from b) that, f o r w = eo, condition i) is equivalent to saying that w e Am and condition ii) is equivalent to saying that w -’/(P-’) e Am, 0 IQI

In case

-+I

p e

=

$(XI

2,

-4

conditions i) and ii) become:

Qdx 5 C

and

J-1 IQI

e

-

(4

-4Q)

IQI Q Q These two inequalities together are equivalent to

We can write:

dx 5 C.

IV. 2 . REVEP.SE H ~ L D E R I S INEQUALITY

COROLLARY 2 . 1 8 .

Rn.

Then

Let

409

b e a r e a l l o c a l l y i n t e g r a b l e f u n c t i o n on

eQ e A 2 i f and o n l y i f t h e r e i s a c o n s t a n t Q c Rn

C

such t h a t

f o r e v e r y cube

-

The r e l a t i o n b e t w e e n w e i g h t s and B . M . O . COROLLARY 2 . 1 9 .

w

E

A,

f u n c t i o n s i s now c l e a r .

l o g w e B.M.O.

P r o o f . L e t w f: A, and w r i t e w = e Q , t h a t i s : w e A 2 , w e know from c o r o l l a r y 2 . 1 8 t h a t

so t h a t

Q

=

Q

=

log w.

If

l o g w e B.M.O.

w e A, w e A f o r some p E [ l , , ) . If p P we h a v e w e A C A 2 a n d , a s w e have j u s t s e e n , l o g w E B.M.O. P I t follows t h a t p > 2 we l o o k a t w - l ’ ( p - l ) E A ,C A 2 . P 1 1 0 g ( w - ” ( ~ - ~ ) )= -logw eB.M.0. Thus, i n any case, P- 1 l o g w e B.M.O. In general

Observe t h a t , i f the

A

P If

constant for

w w.

A

E

l l o g wI*

P’

d e p e n d s o n l y on

p

2

2, If

a n d on

Q e B . M . O . , w e know t h a t e X 4 E A 2 f o r X > 0 small enough ( 0 < X < C 2 / l Q l * w i t h t h e n o t a t i o n u s e d i n C h a p t e r 11, 3 . 1 0 ) . I f we s e t e X Q = w , we g e t Q = X - ’ l o g w . Thus B.M.O.

= {a

log w : a

A c t u a l l y , t h e same i s t r u e f o r a n y

p

2

0,

with

2

w

E

A2)

1 < p

5

w e A 1 P We a l r e a d y know t h a t t h i s i s t r u e i f p 2 2 . For 1 < p < 2 , if e B . M . O . , we c a n w r i t e Q = a l o g w w i t h a 2 0 and w E A 2 . But u = wP-’ E A s i n c e 2 ( p - l ) + l - ( p - l ) = p ( s e e theorem 1 . 1 4 P p a r t b ) ) . Therefore Q = a log w = a log(u l / ( P - l ) ) = (a/(p-1))logu. B.M.O.

= {a

log w : a

0,

I n c o n t r a s t t o t h i s s i t u a t i o n , we h a v e { a log w :

a 2 0,

w

f

A1) = B . L . O .

I n f a c t , we a l r e a d y know t h a t

ci

$

B.M.O.

log w e B.L.O.

when

a

2 0,

( s e e t h e proof of theorem 3 . 3 . i n Chapter 1 1 ) . Conversely, l e t

w

E

A1

410

IV. WEIGHTED NORM INEQUALITIES

$ e B.L.O. Then, a c c o r d i n g t o C o r o l l a r y 3 . 1 0 ( i i ) i n C h a p t e r 11, we h a v e f o r E > 0 s m a l l enough, e v e r y cube Q and g i v e n C:

which i m p l i e s

IQ

1

e E @ ( X ) d x5 C e x p (E$ ) < C expIE(C + e s s . Q i n f . < c -

Q -

ess. inf.(eE$)

e E @f A 1 .

I t follows t h a t

$11

Q

Thus,

$ =

log w

E-'

with

w = e E @E A1.

Combining t h i s w i t h theorem 2 . 1 6 , which t e l l s u s t h a t e v e r y

w E A1 c a n b e w r i t t e n a s w(x) = k ( x ) ( M f ( x ) ) y , w i t h k ( x ) 2 0 s u c h t h a t l o g k e Lm and 0 < y < 1 , we a r e l e d t o : B.L.O.

=

{ h + 8 log(Mf) : h

f

Lm,

f e Lloc, 1

6 2 0)

We f i n i s h t h i s s e c t i o n by o b s e r v i n g t h a t t h e Lp i n e q u a l i t y e s t a b l i s h e d i n c h a p t e r I1 ( t h e o r e m 3 . 6 ) b e t w e e n t h e H a r d y - L i t t l e w o o d maximal f u n c t i o n Mf and t h e s h a r p maximal f u n c t i o n f ' , also h o l d s when Lebesgue measure dx i s r e p l a c e d by t h e measure w ( x ) d x , where w i s any Am w e i g h t . The c o n c r e t e s t a t e m e n t i s a s f o l l o w s THEOREM 2 . 2 0 .

Mf E L

PO

(w)

such t h a t

L e t

w e A,

f o r some

po 5 p <

po

in

Rn and l e t

with 0

< po <

f m.

be such t h a t

w ,f o r

every p

m

P r o o f . I t i s a r e p e t i t i o n o f t h e argument we u s e d t o p r o v e t h e o r e m 3 . 6 i n c h a p t e r 11. F i r s t of a l l , t h e Calder6n-Zygmund d e c o m p o s i t i o n can be c a r r i e d o u t f o r o u r f . I n d e e d i f QIC Q2C ... i s a n i n c r e a s i n g f a m i l y o f d y a d i c c u b e s f o r e a c h o f which 1 -

I

IQk Qk

since {Qk}

f(y)dy > t ,

w(x)dx

then

w(Qk)

i s bounded i n d e p e n d e n t l y o f k a n d ,

i s a doubling measure, t h i s implies t h a t t h e sequence

is finite. We c o n s i d e r as i n t h e o r e m 3 . 6 . o f c h a p t e r I 1

cubes

{Qt,j}.

Given

t > 0

we f i x

Qo = Q - n - l 2

t h e family of t,jo

and

take

41 1

I V . 3 . WEIGHTS FOR SINGULAR INTEGRALS

A > 0.

P r o c e e d a s i n c h a p t e r 11. Then a f t e r we have o b t a i n e d t h e

estimate

I:

j : Q t , j C Qo we u s e t h e f a c t t a t

c

l Q t , j \ 5 2A-’IQol

t o c o n c l u d e t h a t , f o r some

w E Am

6 > 0

and

> 0,

Adding i n a l l t h e p o s s i b l e

Qots we o b t a i n t h e e s t i m a t e

A f t e r t h i s , t h e proof c o n t i n u e s p r a c t i c a l l y unchanged.

0

3 . WEIGHTED NORM INEOUALITIES FOR SINGULAR INTEGRALS Only f o r r e g u l a r s i n g u l a r i n t e g r a l s ( s e e 5 . 1 7 and 5 . 1 i n c h a p t e r I 1 f o r t h e d e f i n i t i o n ) , a s a t i s f a c t o r y t h e o r y can be given i n terms of t h e A c l a s s e s . We s t a r t by u s i n g i n e q u a l i t y 5.20 ( i ) from P chapter I 1 t o prove t h e following THEOREM 3 . 1 . L e t T be a r e g u l a r s i n g u l a r i n t e g r a l o p e r a t o r , 1 < p < m and w E A T h e n T i s bounded i n Lp(w). More p r e c i P‘ s e l y , t h e r e is a c o n s t a n t C d e p e n d i n g o n l y on T, w a n d p _ .

m

Lc ( t h a t is: f w i t h compact s u p p o r t ) , t h i s i n e q u a l i t y h o l d s : t h a t , f o r every function

1,.

f

E

I T f ( x ) l P w(x)dx 5 C

P r o o f . We s h a l l show below t h a t i f

i s a bounded f u n c t i o n

jRn I f ( x ) I P w ( x ) d x f e L,:

then

M(Tf) e L p ( w ) .

Once t h i s i s g r a n t e d , we c a n u s e t h e o r e m 2 . 2 0 t o w r i t e :

jRn I T f ( x ) I p

w(x)dx I

(M(Tf) ( x ) I p w(x)dx 5

q > 1 small p > 1 ( t h i s c a n h e done a c c o r d i n g t o enough f o r h a v i n g w E A P/q, a theorem 2 . 6 ) . Then t h e l a s t i n e q u a l i t y f o l l o w s r e a d i l y .

We have u s e d i n e q u a l i t y 5.20 ( i ) from c h a p t e r I 1 w i t h a

412

I V . WEIGHTED NORM INEQUALITIES

To f i n i s h t h e p r o o f we j u s t need t o c h e c k t h a t M(Tf) e L p ( w ) f o r f e L z . I t s u f f i c e s t o show t h a t Tf e Lp(w), T h i s c a n b e done i n t h e f o l l o w i n g way. L e t R > 0 b e s u c h t h a t f ( y ) = 0 f o r a l l y w i t h ( y ( 2 R . Then f o r 1x1 > 2 R , t h e f o l l o w i n g e s t i m a t e holds :

since

'r

Now we w r i t e Rn

ITf(x) (pw(x)dx

=

Tf(x) (pw(x)dx

i,X,<2R

t

T f ( x ) pw (x) dx and s e e t h a t b o t h i n t e g r a l s i n t h e r i g h t hand s i d e a r e f i n i t e . F o r every M > 0 p (1 +1/ E)

E/

(1 + E)

j[xl 0 i s c h o s e n i n s u c h a way t h a t w s a t i s f i e s a r e v e r s e H U l d e r ' s i n e q u a l i t y w i t h exponent 1 + E ( s e e 2 . 5 ) , s i n c e we know t h a t T i s bounded i n L4(dx) f o r e v e r y q w i t h 1 < q < m , in particular for q = P(l+l/E). Now, i n view o f t h e e s t i m a t e o b t a i n e d f o r I T f ( x ) J 1x1 > 2 R , we j u s t need t o o b s e r v e t h a t i f w E A and P have : (3.2)

dx <

JO
when p > 1,

m

Indeed I 2 k- M<

5 <_ ( t a k i n g

1 < q < p

m

C

1

k= 1

so t h a t

(2kM)-nP w(ZkB(0,M)) 5 w

E A

9

and u s i n g lemma 2 . 2 )

we

IV.3. WEIGHTS FOR SINGULAR INTEGRALS

Since LT is dense i n t o t h e whole s p a c e Lp(w).

Lp(w),

T

413

c a n b e e x t e n d e d by c o n t i n u i t y

Theorem 3 . 1 s a y s n o t h i n g a b o u t t h e c a s e p = 1 . Of c o u r s e f o r p = 1 , t h e e s t i m a t e 5.20 ( i ) used i n t h e p r o o f , i s t o t a l l y u s e l e s s , s i n c e i t i n v o l v e s a q > 1 . However, we a r e g o i n g t o show t h a t i f w E A1, t h e n T i s o f weak t y p e ( 1 , l ) w i t h r e s p e c t t o w . The p r o o f w i l l b e b u t a r e p e t i t i o n o f t h e Calder6n-Zygmund p r o o f g i v e n f o r Lebesgue measure i n c h a p t e r 11. We s t a r t by e x t e n d i n g lemma 5 . 2 i n chapter I 1 t o our s i t u a t i o n . --____ LEMMA 3 . 3 . __ Let

a

w e i g h t and

K E

be a r e g u l a r s i n g u l a r i n t e g r a l k e r n e l ,

L 1 (w),

T h e n i f we s e t , a s u s u a l ,

with

C

Q

supported i n a cube

0=

Q2fi

K

&

d e p e n d i n g onZy o n

with

(Q

w

~n

A1 a(y)dy = O.

the following inequality holds:

w.

P r o o f . We c a n a s s u m e , f o r s i m p l i c i t y , t h a t

Q

is centered a t the

o r i g i n . Then, e x a c t l y a s we d i d i n c h a p t e r I 1 ( r i g h t b e f o r e lemma 5 . 2 ) , we c a n w r i t e

(note t h a t

a(x)

i s a c t u a l l y i n t e g r a b l e and t h e c o n v o l u t i o n makes

sense). Then we j u s t n e e d t o o b s e r v e t h a t I K ( x - y ) - K ( x ) Iw(x)dx 2 CMW(Y) 1x1 > 2 I Y I T h i s i n e q u a l i t y i s p r o v e d e x a c t l y a s ( 5 . 2 1 ) i n c h a p t e r 11, c h o o s i n g (3.4)

y instead of 0 t o evaluate

Mw.

S i n c e w E A 1 , we have Mw(y) 5 Cw(y) t y o f t h e lemma i s c o m p l e t e l y p r o v e d . 0

a.e.

and t h e i n e q u a l i -

We a r e now r e a d y t o p r o v e a s u b s t i t u t e r e s u l t o f t h e o r e m 3 . 1

for

p = 1.

THEOREM 3 . 5 .

w

f

A1.

Let T

Then

precisely,

T

b e a r e g u l a r s i n g u l a r i n t e g r a l o p e r a t o r , and

i s of

weak t y p e

t h e r e is a c o n s t a n t

such t h a t f o r every f u n c t i o n

C

(1,l)

with respect t o

d e p e n d d i n g o n l y on

f E L1(w)

and e v e r y

T

t > 0:

w.

&

More -

w,

I V . 3 . WEIGHTED NORM INEQUALITIES

41 4

Proof. We are assuming w E A 1 , This means that there is a constant C such that, for every cube Q, w(Q) 5 c w(x) for a.e. x e Q . Let f e L 1 (w) and t > 0. We ca ry out the corresponding Calder6n-Zygmund decomposition. If Q is a cube for which

Therefore, every dyadic cube over which the average of If1 is > t will be contained in a maximal one. These maximal cubes are the Cal der6n-Zygmund cubes { Q j ) . F o r each of them,

t
I f(x)

Idx

<=

2"t,

and outside of their union we know

that f(x) 5 t a.e. Next, w e obtain the Calder6n-Zygmund decomposi tion o f the function f. We write f = g+b where g, the "good part", is given by: g(x) = f(x) if x d R = Qj, and J

f(y)dy J

can be split as

b(x)

fQ

= =

1 j

x e Qj, while

if

j b.(x)

with

J

bj(x)

=

b,

(f(x)-f

In chapter 11, the corresponding result for w 5.9) was proved by using the following properties: a) T

is bounded in

the "bad part"

E

Qj

1

)X

Qj

(x).

(inequality

L'

d) lemma 5.2 in chapter 11.

J

I t turns out that these four properties extend to the case of a general w e A, i n the following way:

a!) Since w E A I C A for every p > 1 , theorem 3 . 1 tells us P that T is bounded in Lp(w) for every p > 1 .

IV.3. WEIGHTS FOR SINGULAR INTEGRALS

415

Now we c a n w r i t e a p r o o f e n t i r e l y a n a l o g o u s t o t h e o n e g i v e n i n t h e unweighted c a s e : w({x E R"

: [Tf(x)l > t ) )

+ w ( 6 ) + w({x

d

5

5 w({x

: ITb(x)l

e Rn

: lTg(x)] > t/2))

+

> t/Z})

We e s t i m a t e e a c h o f t h e s e t h r e e t e r m s . F o r t h e f i r s t one we u s e p r o p e r t i e s a ' ) ( w i t h some p > 1 ) t h a t [ g ( x ) l 2 2"t a . e . We g e t : w({x e Rn

'3 t 1,

: (Tg(x)l > t/Z}) 2

Ig(x)lpw(x)dx =

C

C

and

tp C -

Inn

b ' ) together with the f a c t

lRn

lg(x)[

ITg(x) Ipw(x)dx 5

(w)'-'

w(x)dx 2

C

lf(x)Jw(x)dx i,n The s e c o n d term o f t h e sum h a s b e e n a l r e a d y e s t i m a t e d , w i t h t h e same bound a s t h e f i r s t , i n c ' ) . As f o r t h e t h i r d , we h a v e :

5

w({x k

lg(x)lw(x)dx

fi

5 t

: lTb(x)l > t/Z))

2 C

lRn,,

[ T b ( x ) Iw(x)dx

2

The n e x t t o l a s t i n e q u a l i t y i s a c o n s e q u e n c e o f lemma 3 . 3 and t h e l a s t o n e f o l l o w s from b ' ) s i n c e

b(x) = f(x)

-

g(x).

n

I f T is a regular singular integral operator with kernel K f E Lp(w) with w e A and 1 p < a, then t h e truncated P o p e r a t o r s T E , E > 0 , c a n b e d e f i n e d a c t i n g d i r e c t l y on f by means o f t h e f o r m u l a and

416

IV. WEIGHTED NORM INEQUALITIES

TEf(x)

K(X-Y) f(Y)dY

=

Indeed, this integral converges absolutely, since

I K(x-y) I

w (y) - p ' I p dy)

and it is clear that the right hand side is finite, since the second factor is bounded by a constant times W(X-Y) -"(p-l) Iy[-np' dy)l/pl, which is finite because (IlYl>E

W(X-Y) -"(p-l)

is, as a function of y , an A p l weight, and ( 3 . 2 ) holds for this function in place of w and p ' in place of p . Of course after theorem 3.1, Tf may be defined by density. But it is natural to ask whether a more explicit definition of Tf as limit of TEf f o r E + 0 is possihle. We already know that the answer to this question rests upon the study o f the boundedness properties of the maximal operator T*, defined as T*f(x) = s u p ITEf(x)[, with respecto to the weight w. E>O

We can use inequality 5 . 2 0 (ii), from chapter 11, obtaining the following result THEOREM 3 . 6 . Let 1 c p < m & w

T be a r e g u l a r s i n g u l a r i n t e g r a l o p e r a t o r , A

Then t h e c o r r e s p o n d i n g maximal o p e r a t o r a r i s i n g f r o m t h e t r u n c a t e d o p e r a t o r s TE i s bounded i n Lp(w). p r e c i s e l y , t h e r e is a c o n s t a n t C d e p e n d i n g on T & w, t h a t f o r e v e r y f e Lp(w):

iRn

f

P'

(T*f(x))p

w(x)dx

f

5 C J R n If(x)

Proof. We observe that inequality for every f f: Lp(w) provided q just need to take q such that f c L:~~(R") as soon as w e A ~ / be granted by taking q > 1 close

Ip

5 C(

More

w(x)dx.

ii) in chapter I1 is valid is close enough to 1. Indeed, we e LToc(Rn), but I fIq E Lp'q(w) and we already know that this can ~ , enough to 1 . 5.20

Now we can write: (T*f(x))Pw(x)dx

T*

(Mqf(x))pw(x)dx

+

IV.3. WEIGHTS FOR SINGULAR INTEGRALS

by 2.8 and 3.1, if

q

has been chosen so that

The weak type inequality, p = 1, of theorem 3.6 is also V, 4.11. It is now clear that valid for f e Lp(w) with w

w

417

E A

P/q' 0 corresponding to the limit case, true. This will be proved in Chapter corollary 5.22 of Chapter I1 remains e AD, 1 5 p < m.

We have just seen that regular singular integrals satisfy the same weighted inequalities as the Hardy-Littlewood maximal function. The question arises naturally whether o r not the conditions imposed on w in theorems 3.1 and 3 . 5 are really necessary. We shall see next that the answer is positive. THEOREM 3.7. Let

w

pose t h a t each o f t h e

(p,p),

weak t y p e

b e a w e i g h t i n En a n d l e t 1 5 p < co. Supn R i e s z t r a n s f o r m a t i o n s R,, ,Rn i s of

...

with respect t o

w.

Then, necessariZy

w e A

P'

Proof: Let Q be a cube, and let f be a non-negative function living in Q, with f > 0. Denote by Q' the cube touching Q Q having the same side length as Q and such that ;j 2 yj for any x E Q', y E Q and j=1,2 , . . . ,n. Call Rf(x) = 1 R.f(x). Then, j=1 J for every x E Q' n ~ Q x.-y.) ( I 3 Ix-yl -n-l f(Y)dY 2 Rf(x) = cn . J=1

It follows that, for every

0 < t <

C f

9'

Q ' C {x E Rn : [Rf(x) I > t}. Now, since R is of weak type (p,p) with respect to w , we shall have: w(Q') 5 C t - p fp w. We get

5 C

(fQ)P w(Q')

lQ

(f(x))p

w(x)dx.

IQ

In particular, the choice

f = X Q leads to w(Q') 2 C w(Q). By the same token we have w(Q) 5 C w(Q'), from which we conclude that (fQ)p w(Q) 5

' IQ c2

(f(x))p

w(x)dx.

In other words:

w

E A

P'

n

The information contained in our previous results on weighted inequalities for singular integral operators is very often applied in the following useful formulation:

COROLLARY 3 . 8 . Let f o r every

E

> 0

T

b e a r e g u Z a r s i n g u l a r i n t e g r a l o p e r a t o r . Then,

& 1

j/Tf(x)lP u(x)dx

< p <

w,

then inequality

5 CE,p Ilf(x)

Ip

M 1 + E u(x)dx

418

IV. WEIGHTED NORM INEQUALITIES

holds f o r arbitrary functions

f and u

0

f o r which t h e r i g h t

hand s i d e i s f i n i t e . T h e same i n e q u a l i t y i s v a l i d f o r t h e a s s o c i a t e d maximal o p e r a t o r

T".

'+'

is locally integrable, since otherProof: We assume that u(x) wise M,+E u(x) = +a at every point x. Then, by 2.16, M 1 + € u f: A , C A with A constant depending only on E , and P P theorem 3.1 gives jlTf(x)lP

5 I/Tf(x)Ip M 1 + E u(x)dx

u(x)dx

< C E,P IIf(X)

The inequality for

T"

Ip

5

MI+€ u(x)dx

follows in the same way from theorem 3.6. 0

The novelty in the inequality just proved (which should be compared with 2.12 in Chapter 11) stems from the fact that we try to estimate 11 Tf 11 for an arbitrary u(x) 2 0. In spite of that, LP(U) we are able to remove the T , the only price to pay for this being the change of u by another function, M 1 + € u , which is essentially of the same size (observe that M1+' is bounded in Lp for p > 1+~). A lot of information is contained in corollary 3.8; for example, by using it only when p = 2, all the Lp boundedness properties of T follow from Hslder's inequality plus duality. In this connection, see section V.6. We shall end this section by giving a weighted version of the Hsrmander-Mihlin multiplier theorem. THEOREM 3.9. Let a b e an i n t e g e r s u c h t h a t n/2 < a 2 n. S u p p o s e m f: Lm(Rn) i s o f c l a s s Ca o u t s i d e t h e o r i g i n and s a t i s f i e s

that

the condition:

(R-l

jR
f o r every multi-index

lDam(S)I2 ci

dC)'l2

such that

5 CR-Ial, (a1

(0 < R <

m)

5 a. Then:

i l f-I n/a < p < w and w e ApaIn, m i s a m u l t i p l i e r f o r Lp(w), i n o t h e r w o r d s : t h e o p e r a t o r Tm g i v e n by (Tmf)"(t) = = m(c)?(S), i s bounded i n Lp(w).

ii) I f

1 c p < (n/a)l b o u n d e d i n Lp(w) . i i i ) ~fwnIa e A1,

W.

Tm

and w - 1 / (p-1 1 i s o f weak t y p e

f:

Ap,a/n,

(1 ,1)

Tm

i s again

w i t h r e s p e c t to

419

IV.3. WEIGHTS FOR SINGULAR INTEGRALS

Proof: i) We look for an a priori inequality for Tmf with f € S (:Bn). The notation will be as in section 6 of Chapter 11. serve that TNf(x) -+ Tmf(x) uniformly as N a. Indeed:

Oh

-+

IITmf - T

N

fl ,

=

Il((m-m

mN ( x )

N

If) [ I m 5 " v

N m , since -+ m(x) IIrnNIlm 5 C independently of N.

as

-+

II(m-m

as

N

-+

N A

)fU1

m

+

0

for every

x # 0

and

In general, we shall denote by Ilg(l the norm of the function PS W g in Lp(w). Since TNf(x) + Tmf(x) uniformly as N m, it will be sufficient to obtain an estimate N (3.10) IT flpyw 2 c Ifllp,w -f

with

C

independent of

N.

In the proof of theorem 6.10 in Chapter I1 we obtained a pointwise estimate N # (3.11) (T f ) (XI 2 Cq Mqf(x) valid for every q > n/a, with Cq independent of N. Even though this estimate was not written down explicitly, it was the key to prove the corresponding estimate for (Tmf)# appearing in the theorem. From (3.11) and theorem 2.20 we shall obtain (3.10). First we shall make sure that TNf e Lp(w), s o that theorem 2.20 can actually be applied. This is very simple because T N f is a continuous function and we only have to check that it has the right behaviour at m . For 1x1 large we have ,-

since (TNf)^ = m N f has derivatives in it is enough to observe that

L1

up to order

a.

Now

as it follows from ( 3 . 2 ) , since

w e ApaIn. we can write, using theorem Once we know that TNf E Lp(w), 2.20 and (3.11) with some n/a < q < p

(M( I f 19) (xj)p/q w(x) dx) l/p Of course p/q < pa/n. However q that w e A With this choice P/q'

can be chosen s o close to n/a

420

IV. WEIGHTED NORM INEQUALITIES

and

(3.10)

follows.

i i ) i s o b t a i n e d by d u a l i t y . W r i t i n g that IlTmfll p,w =

IqTmf(x)

SUP

u

=

m dx

l l ~ l l p l

sup llgllpl ,cr5

=

'C

IIfllp,w

since p < t h e weight

n

($I

u

E

implies t h a t Apla/n.

p' >

n

a,

and we c a n a p p l y p a r t i ) t o

The p r o o f o f i i i ) i s v e r y s i m i l a r t o t h a t o f theorem 3 . 5 . L e t f e L 1 ( w ) . We a r e g o i n g t o s e e t h a t w({x e Rn : I T N f ( x ) I > t ) ) <= C t - ' Ilflll,w w i t h C i n d e p e n d e n t o f f , t > 0 and N . T h i s i s c l e a r l y enough. S i n c e w e A 1 , g i v e n t > 0 , we c a n p r o c e e d e x a c t l y a s i n t h e p r o o f o f theorem 3 . 5 , obt a i n i n g t h e CalderBn-Zygmund c u b e s { Q j l f o r f a t h e i g h t t . The c u b e s { Q . ) a r e d i s j o i n t and s a t i s f y : J

and (f(x)1 5 t Next we w r i t e

-

I

f = g+b,

f o r almost every where

g(x) = f ( x )

x d if

U Qj.

j x d n =

u

Qi, j J 1 b.(x) j J

f(y)dy = fQ i f x e Q j , and b ( x ) = j IQjI Q ; J w i t h b . ( x ) = ( f ( x ) - f ) XQ (x) J Qj j Now we j u s t h a v e t o c h e c k t h a t t h e f o u r p r o p e r t i e s a t ) t o d ' ) i n t h e p r o o f o f t h e o r e m 3 . 5 c o n t i n u e t o h o l d f o r o u r w and t h e o p e r a t o r T~ g(x)

=

a ' ) T N i s bounded i n Lp(w) p r o v i d e d p > n / a . This follows from i ) s i n c e w E A I C ApaIn b l ) and c l ) a r e p r o v e d a s b e f o r e u s i n g t h e f a c t t h a t w e A 1 .

F o r d l ) we n e e d t o s e e t h a t lemma 3 . 3 h o l d s f o r o u r w and t h e k e r n e l K N o f T N . T h i s i s s e e n by showing t h a t KN s a t i f i e s (3.4).

IV.3.

WEIGHTS FOR SINGULAR INTEGRALS

421

The proof of [3.4) for KN and w such that wnla E A 1 will be be based upon inequalities (6.11) of Chapter I 1 with some q > n/a such that wq e At. We know that such a q exists after theorem 2 . 7 . With this choice of q we have:

II

XI > 2

I YI

IKN(x-y) - KN(x)

m

5 C

-

since

1

j=1

(2-')j

M(w~)(Y)

M w(y)

=

C M w(y)

5 ess inf w(x)q

lXl5lYl Once these four properties have been checked, the proof is idell tical to that o f theorem 3.5. 0 In the problem o f convergence of Fourier series and integrals in one variable, one is interested in the multipliers xI, I an interval of 8 , which give rise to the partial sum operators SI, (SIf)" = 3 xI. We know, see Chapter I1 (6.15), that S I is a multiplier in Lp(R) f o r all l < p c m (this fact is actually equi valent to M. Riesz theorem). Strictly speaking, S I is neither a singular integral operator nor a multiplier operator satisfying the conditions of theorem 3.9, due to the lack of regularity of XI at the end points of the interval I. Nevertheless, the weighted inequalities for SI are really an easy consequence o f known estimates : COROLLARY 3.12. Let

W(X)

0

be a weight i n R .

Then

a) If 1 < p < m, t h e p a r t i a l sum o p e r a t o r s SI a r e u n i f o r r n l x bounded ( f o r a l l i n t e r v a l s I ) Lp(w) i f and o n l y i f w e A

P'

b) The o p e r a t o r s SI a r e u n i f o r m l y of weak t y p e r e s p e c t t o w i f and o n l y i f w f A 1 . c) F o r a l l

f e Lp(w), Ils[_R,R]

with w

'

E

A

1 < p <

and

P - fllp,w * O ,

*

(1,l)

m,

with

IV. WEIGHTED NORM INEQUALITIES

422

P r o o f : Let

be t h e a n a l y t i c p r o j e c t i o n , d e f i n e d by ( A f ) ^ ( E ) = o r A f ( x ) = f ( x ) + i H f ( x ) , where H i s t h e H i l b e r t t r a n s f o r m . D e n o t i n g b y Ma t h e i s o m e t r y c o n s i s t i n g i n m u l t i p l y i n g e a c h f u n c t i o n f ( x ) by e 'Tiax, we have i S [a,-) = 'Z Ma AM-a =

A

t(5) X[,,,)(5),

= s[a,-) - s [b,m)

'[a,b)

=

i

7 (Ma

- Mb AM-b)

S i n c e H i s a r e g u l a r s i n g u l a r i n t e g r a l o p e r a t o r , t h e o r e m s 3 . 1 and 3 . 5 imply t h e s u f f i c i e n c y o f t h e c o n d i t i o n w E A i n p a r t s a ) and P b ) . F o r t h e n e c e s s i t y , we o b s e r v e t h a t Hf = i ( S ( - m ,o] f-S[o , m ) f ) and a p p l y theorem 3 . 7 ( s i n c e t h e r e i s o n l y one R i e s z t r a n s f o r m i n R', and i t i s p r e c i s e l y t h e H i l b e r t t r a n s f o r m ) . F i n a l l y , c ) f o l lows from a ) b e c a u s e t h e r e s u l t i s t r i v i a l l y t r u e f o r t h e f u n c t i o n s

f E L1(R) whose F o u r i e r t r a n s f o r m h a s compact s u p p o r t , and t h e s e a r e d e n s e i n Lp(w). 0

4 . TWO-WEIGHT.NORM INEQUALITIES FOR MAXIMAL OPERATORS

The main g o a l o f t h i s s e c t i o n i s t o g i v e a s o l u t i o n o f p r o b l e m 2 ( p o s e d i n s e c t i o n 1 ) . T h i s s o l u t i o n w i l l b e b a s e d on i d e a s o f

E . Sawyer [ l ] and B . J a w e r t h [ l ]

.

I n o r d e r t o i n t r o d u c e t h e main i d e a , and a l s o f o r i t s own s a k e , we s h a l l p r e s e n t f i r s t a g e n e r a l r e s u l t o f B . J a w e r t h , which c o n t a i n s B . Muckenhoupt's theorem 2 . 8 w i t h a s i m p l e and e l e g a n t p r o o f . By a b a s i s i n Rn we s h a l l s i m p l y mean a c o l l e c t i o n 8 o f open s e t s i n R n . For a b a s i s 8 we s h a l l c o n s i d e r w e i g h t s w s u c h t h a t w(B) < 03 f o r e v e r y B E 8. Given B and w , w e s h a l l d e n o t e by M B S w t h e maximal o p e r a t o r s e n d i n g t h e m e a s u r a b l e f u n c t i o n f into the function defined as follows: I f x E B, Bet3 M f ( x ) = sup

u

,w

where t h e sup i s t a k e n o v e r t h o s e B E B s u c h t h a t x E B and w(B) > 0 . O t h e r w i s e MB,wf(x) = 0 . When w 2 1 , w e s i m p l y w r i t e Ma. S i n c e t h e s e t s o f B a r e o p e n , w e s e e t h a t M a f i s a l o w e r semicontinuous f u n c t i o n . We s h a l l s a y t h a t t h e w e i g h t

1 5 p

my

w

is in the class

i f and o n l y i f , t h e r e i s a c o n s t a n t

C

A

P,B' such t h a t :

423

IV.4. TWO-WEIGHT NORM INEQUALITIES

f o r every

B e B

(with t h e u s u a l i n t e r p r e t a t i o n of t h e second f a c -

t o r as

e s s . s u p (w-') when p = 1 ) . B Jawerth's r e s u l t is the following:

THEOREM 4 . 1 , Let 1 < p c m . CalZ

w a weight f o r B & T h e n , t h e foZZowing c o n d i -

B be a b a s i s i n Rn, U ( X ) = W ( X )- " ( p - ' ) .

t i o n s are equivaZent:

a ) MB

i s bounded b o t h i n

b ) w e A p,B;

in

MB,u

and

LP(w)

Lp'(u)

MB,w

Lp(u)

i s bounded i n

i s bounded

~"(w).

P r o o f : We s h a l l s t a r t by showing t h a t b ) i m p l i e s a ) . A c t u a l l y w e j u s t n e e d t o s e e t h a t b ) i m p l i e s t h a t MB i s bounded i n Lp(w). i f and o n l y i f u E A The r e a s o n i s t h a t w E A P ' ,B' s o , P,B a s s u m i n g b ) , we want t o s e e t h a t MB i s bounded i n Lp(w) . Let

f E Lp(w) . We may assume t h a t MBf(x) < w e x c e p t f o r a s e t o f n u l l w-measure ( t a k e , f o r i n s t a n c e , f b o u n d e d ) .

For every i n t e g e r Sk = {x

k,

we s h a l l c o n s i d e r t h e s e t k k+l Rn : 2 < MBf(x) 5 2

E

From t h e d e f i n i t i o n o f

MB,

SkC

tisfies

Define

Ek,l

=

Bk,ln

and, f o r

Sk

Ek,j

=

(Bk,j \

u

Bk

j

,j'

where

F.

B

sa-

j > 1:

u

s
Bk,s)

n

'k'

The s e t s Sk form a d i s j o i n t c o l l e c t i o n and e a c h for varying j . j o i n t union o f t h e s e t s E k , j

where

Bk,j

Sk

i s the d i s -

IV. WEIGHTED NORM INEQUALITIES

424

gk,j

=

(v I jBk,j

1

f(Y) I U(Y) -'~o~)dY)P

1 pk,j gk,j as an integral on a measure, space k,j ( X , p ) built over the set X = {(k,j)) by assigning to each (k,j) the measure pk,j ' For X > 0 , call

We view the sum

r(X)

=

G(X)

=

I

{(k,j)

We use this to estimate

: gk,j >

u

C

XI

Bk,j

(k,j)er(h) p(r(A))dX.

1 pk,j gk,j' k,j 0 valent to saying that there is Then

x

2 '

Observe that

w

E

A

such that, for every

P,B B e

is equi

B:

'k,j

Next we shall use the boundedness of

to estimate

p(r(X))

IV.4. TWO-WEIGHT NORM INEQUALITIES

425

= C I R n ( f ( x ) ( p w(x)dx

s i n c e u l - p -- w . plies a).

T h i s f i n i s h e s t h e proof o f t h e f a c t t h a t b) i m -

L e t u s p r o v e now t h a t a ) i m p l i e s b ) . F i s t t h e p r o o f t h a t t h e boundedness o f MB i n Lp(w) i m p l i e s w f A p , B , i s e x a c t l y a s t h e one g i v e n i n s e c t i o n 1 f o r t h e b a s i s o f c u b e s . Then, o b s e r v e i s bounded i n t h a t , by symmetry, i t i s enough t o p r o v e t h a t M B , u L P ( u ) . The p r o o f o f t h i s f a c t i s v e r y s i m i l a r t o t h e one g i v e n f o r t h e o t h e r h a l f o f t h e t h e o r e m . We t a k e f n i c e i n L P ( u ) and d e and E w i t h MB,a i n p l a c e o f MB. Then fine Sk, B k , j k,j

=

w(G(X)) 5 w({x

Finally

E

Rn : ( M B ( f a ) ( X ) ) p > A } )

4 26

IV. WEIGHTED NORM INEQUALITIES

5 C

I

m

w({x

E

Rn : (M (fo)(x))p

B

0

=

C

iRn

5 C

IMB(fa)(x)lPw(x)dx

> h1)dX

I

R"

=

If(x)lPu(x)Pw(x)dx

=

1 < p < m. Suppose t h a t t h e b a s i s 8 i s s u c h COROLLARY 4 . 2 . implies that M i s bounded i n Lp(u) that w e A 89 0 P 38 MB,w i s bounded i n Lp'(w). Then MB i s bounded i n Lp(w) Ji and o n l y i f w E A P,B'

A particular case of corollary 4 . 2 in Muckenhoupt's theorem, which was proved in section 2 (theorem 2 . 8 ) and is derived here in a completely different way. COROLLARY 4.3.

Then M

Let

1 < p <

i s bounded i n

w

Lp(w)

and l e t

w

i f and o n l y i f

be a weight i n w e A P'

Rn.

Proof: The reason i s that the basis Q formed by all the open cubes in Rn satisfies the hypothesis of corollary 4 . 2 . Indeed, if w E Ap,!! = Ap, then u e A and s o , lemma 2 . 2 guarantees that P' ' both w(x)dx and u(x)dx are doubling measures. Consequently, theorem 2 . 6 in Chapter I 1 implies that = Mu is bounded in LP(u) and M = Mw is bounded in Lp (w). 0 2,w Corollary 4 . 2 also applies to the basis D formed by the open dyadic cubes. (See Chapter I1 section 1 for the definition). Since the dyadic maximal functions will play a basic role in the sequel, it will be convenient to introduce short notations for them. We shall write N = MD and, in general Nw = M . Also, for R > 0 , we shall denote by N(R1, M(R), N F ) , M$Rypw the maximal operators obtained by taking in the corresponding definition just those cubes o f the basis whose side length is 2R. Here is a simple but useful observation about the dyadic maximal functions. LEMMA 4 . 4 . Let 1 < p 5 m and l e t w be a l o c a l l y i n t e g r a b l e weight i n Rn. Then Nw i s bounded i n Lp(w). Proof: The fact that w is locally integrable implies (as in the proof o f corollary 1 . 1 3 ) that Nw is bounded in Lm(w). Then, we

IV.4. TWO-WEIGHT NORM INEQUALITIES

427

just need to prove that Nw is of weak type ( 1 , l ) with respect to w and appeal to the Marcinkiewicz interpolation theorem to complete the proof of the lemma. Let us prove the weak type ( 1 , l ) inequality. Let f E L 1 (w) . Since Nwf(x) = lin NrSR) f(x) and NrSR) f(x) R+m increases with R, it will be enough to prove the inequality for N(R) with constant independent of R. But, after fixing R > 0 W and X > 0, observe that {x E Rn :NiR) f(x) > A } = Qj, where

u

Qj

are the maximal dyadic cubes of side length

$R

j

for which

J

These maximal dyadic cubes do exist because of the restriction on their size. Since they are disjoint, we get W({X

e Rn : N\SR)f(x)

J

>

XI) 6

6

w(Qj)

1

0

as we wanted to show.

COROLLARY 4.5. Let 1 < p < m , and l e t w b e a w e i g h t in Rn. T h e n N i s bounded i n Lp(w) i f and o n l y i f w e A that i s , P,U' w s a t i s f i e s t h e Ap c o n d i t i o n o v e r d y a d i c c u b e s . proof: Lemma 4.4 shows that corollary 4.2 applies to the basis

U . 0

Jawerth's approach to Muckenhoupt's theorem provides a proof which is independent of the reverse HElder's inequality. An even shorther proof has been given by M. Christ and R. Fefferman [ l ] . It is very similar to Jawerth's, but it uses Calder6n-Zygmund cubes. Here is an account o f it. ALTERNATIVE PROOF OF COROLLARY 4.3. p m and let w be an A weight in Rn. Take P hoose a constant a > 2". Then f o r each integer k , be the Calder6n-Zygmund cubes of f at height ak . Then, calling as usual a = w - 1 / (p-1) Qk,j \ Q,,,.

rt;lk

w(x)dx

5 ap

1

k,j

(---

1

lQk,j I

1

Qk,

f(y)dyIP

w ( E ~ , ~ )5

428

IV. WEIGHTED NORM INEQUALITIES

since

w

E

A

P’

Thus

The following general result of Jawerth [l] obtain a solution to problem 2 of section 1.

will allow us to

THEOREM 4.6. Let 8 b e a b a s i s , ( u , w ) a coupZe of w e i g h t s i n Rn and 1 < p < m . S u p p o s e t h a t a(x) = W(X)-”(~-’) i s such t h a t Mg,‘ i s bounded i n LP(a). Then t h e f o l l o w i n g two c o n d i t i o n s a r e equivalent:

a) Mg

LP(w)

i s bounded f r o m

2

Lp(u)

b) T h e r e i s a c o n s t a n t a union o f s e t s in

I,

8,

C such t h a t f o r every s e t t h i s inequaZity holds:

IMg(a xG)(x)

Ip

u(x)dx

G

which i s

<= c ‘ ( G I .

Proof: Implicit in the assumptions is the fact that o(B) < m for every B e B. That a) implies b) is almost immediate. If a) holds, we have an inequality

IRn

IMgf(x)IP u(x)dx 5 C If(x) IP In.(x)dx IRn valid for every function f. If we apply this inequality to the function f = o XG, we obtain:

jRn IMg(u XG)(x)lp

u(x)dx

5 C

IG

o(x)p

w(x)dx

=

C o(G)

and

b) follows. Next, assuming b), we shall prove a). Let f be a nice function in L p ( w ) . We proceed exactly as in the proof of theorem 4.1. Just like there we define the sets Sk, Bk,j and Ek,j and write

IV.4. TWO-WEIGHT NORM INEQUALITIES

429

where

1

gkyj = The d e f i n i t i o n s o f

r(A)

I f(y) 1 0

jBkYj and

G(A)

(y)

for

-'

A > 0

0 (y)dyIp

a r e e x a c t l y as i n

t h e v r o o f o f t h e o r e m 4.1. Then

5 C u({x

Rn : ( M , , , ( f / ~ ) ( x ) ) ~ > A ) )

E

"

where t h e n e x t t o l a s t i n e q u a l i t y i s t h e r e s u l t o f a p p l y i n g c o n d i t i o n b) t o the set

5

G(A)

I

=

( k , j )eT(A)

Bk,j'

Now

m

C

u ( { x E Rn

: ( M B y , ( f / a ) ( x ) ) P > A1)dA =

0

=

c

=

C

j R n I M B , ( ~f / o j ( x )

IRn

Ip

I f ( x ) l p w(x)dx.

a(x)dx 5

c

jRn [ f ( x ) I P a ( x ) l - p d x

We h a v e o b t a i n e d a ) .

=

n

Theorem 4.6 i s n o t i m m e d i a t e l y a p p l i c a b l e t o t h e b a s i s o f c u b e s b e c a u s e we c a n n o t c o n c l u d e from b ) t h a t Ma i s bounded i n L p ( o ) (a d o e s n o t h a v e t o b e d o u b l i n g ) . However f o r t h e b a s i s o f d y a d i c c u b e s D , t h e b o u n d e d n e s s o f Nu i s a u t o m a t i c , a c c o r d i n g t o lemma 4.4. I t w i l l b e t h r o u g h t h e d y a d i c c a s e t h a t we s h a l l b e a b l e condit o s o l v e problem 2 o f s e c t i o n 1 . A c t u a l l y , f o r t h e b a s i s D , t i o n b ) c a n b e w r i t t e n i n a s i m p l e r form. Here i s t h e s o l u t i o n o f t h e two w e i g h t s p r o b l e m f o r t h e d y a d i c maximal o p e r a t o r N .

Q

I V . WEIGHTED NORM INEQUALITIES

430

Let 1 < p c m and l e t ( u , w ) b e a c o u p l e o f w e i g h t s Then, t h e f o l l o w i n g two c o n d i t i o n s a r e e q u i v a l e n t :

THEOREM 4 . 7 .

in

Bn.

a) N i s bounded f r o m

Lp(w)

b) T h e r e i s a c o n s t a n t IN(o XQ) (XI I p

u

where

=

LP(u) s u c h t h a t , f o r e v e r y d y a d i c cube

C

Q:

u(x)dx 5 C a ( Q ) <

w‘Ql,(p-ll.

1,

P r o o f : F i r s t , we have t o s e e t h a t a ) i m p l i e s o ( Q ) < a f o r e v e r y Q . Indeed i f a(Q) = a we have (w(x)-lJ]rw(x)dx = m . I t follows that there exists =

1,

f(x)dx =

m.

f

E

Lp(w)

But t h e n

such t h a t

Mf(x)

=

1,

w(x)-lf(x)w(x)dx

f o r every

x

E

Q,

=

and i t

f o l l o w s from a ) t h a t u(Q) = 0 . A c t u a l l y u ( R ) = m f o r e v e r y cube R 3 Q , s o t h a t u E 0 , a t r i v i a l c a s e which we b e t t e r e x c l u d e . Now, t o prove t h e theorem, w e j u s t need t o s e e t h a t c o n d i t i o n b ) e x t e n d s from d y a d i c cubes t o a r b i t r a r y open s e t s ( i . e , , u n i o n s o f d y a d i c c u b e s ) . Suppose b) h o l d s and l e t G be an open s e t . We s h a l l prove an i n e q u a l i t y P U

with

{Qk,jI

length

C

x ) d x 5 C u(G)

i n d e p e n d e n t o f G and R . T h i s i s c l e a r l y s u f f i c i e n t . L e t be t h e maximal d y a d i c c u b e s c o n t a i n e d i n G and h a v i n g s i d e 2 R and s u c h t h a t 1

S i n c e a l l t h e cubes i n t h e doubly indexed f a m i l y { Q k , j I have s i d e length zR, e v e r y cube w i l l b e c o n t a i n e d i n a maximal o n e . Let {Qi}

be t h e s u b f a m i l y formed by t h e s e maximal c u b e s . Then

431

I V . 4 . TWO-WEIGHT NORM INEQUALITIES

What m a k e s i t p o s s i b l e t o e x t e n d t h e o r e m 4 . 7 t o t h e o r d i n a r y

i s t h e f o l l o w i n g e s t i m a t e by a n a v e r a g e o f t r a n s l a t i o n s o f t h e d y a d i c m a xim a l o p e r a t o r N . The

H a r d y - L i t t l e w o o d max i mal o p e r a t o r

M

i d e a goes back t o C. Fefferman and S t e i n [ l ] . LEMMA 4 . 8 . f

in

Rn

For e v e r y i n t e g e r

k,

e v e r y ZocaZZy i n t e g r a b Z e f u n c t i o n

x e Rn:

and e v e r y

-rtg(x) = g ( x - t ) .

where

Proof: ( T - t 0 N o - r t ) f ( X) =

(X+t) =

sup x e Q - t , QeD

1

IQI

Q-t

f(z)dz

T - ~ O N * T i~s s i m p l y t h e o p e r a t o r

Thus

D-t

formed by t h e c u b e s

t r a n s l a t e s by - t

k, f

Given e x i s t a cube

Let

1

N('ftf)

1 sup f(y-t)dy = x+teQeQ I Q I Q

-

sis

=

j

course

j 5 k.

2j+'

with

Q

of t h e d y a d i c c u b e s . and

x,

by d e f i n i t i o n o f

of side length

5 2k

be an i n t e g e r such t h a t

t E Q(O,Zk+')

equal t o

R

Q-t

associated t o the ba-

k M(2 ) f ( x ) ,

such t h a t

x

E

R

2j-l < s i d e length of

Consider t h e s e t

R

I!c Q.

there w i l l

and

3f

R 2 2j.

consisting of those

f o r w h i c h t h e r e i s some and s u c h t h a t

dyadic, t h a t i s , the

Q e D-t For e v e r y

with side length t E R,

w e have:

Now i t i s a s i m p l e g e o m e t r i c a l o b s e r v a t i o n t h a t t h e m e a s u r e o f

R

432

IV. WEIGHTED NORM INEQUALITIES

'

L

3n+ 1

I

(Q(0,2k+2) 1

(T -

jQ(0,zkt2)

O N O T

t) f(x) dt .

0

We are finally ready to give our promised solution to the two weights problem for M. THEOREM 4.9. Let 1 < p < a , and l e t (u,w) b e a coupZe o f w e i g h t s i n Rn. T h e n , t h e f o Z l o w i n g t w o c o n d i t i o n s a r e e q u i v a Z e n t : a) M

i s bounded f r o m

LP(W)

b ) There i s a c o n s t a n t

where

1,

IM(0 XQ)(x)IP

C

LP(u)

to

such t h a t , f o r e v e r y cube

u(x)dx

2

C

Q:

o(Q) <

u = w- 1 / (p-1)

Proof: All we have to do is to show that b) implies a). But, after lemma 4.8, the boundedness of M is equivalent to the uniform boundedness of the operators T - ~ O N O T for ~ t E Rn. Indeed, once this uniform boundedness has been established, we can use Minkowski's integral inequality and the monotone convergence theorem to get the boundedness of M . Now, since

and JRn we see that the uniform boundedness of the operators T - ~ O N O T ~ between Lp W) and Lp(u), is equivalent to the fact that the couples ( T ~ u , T ~ w )satisfy condition b) in theorem 4.7 with a constant independent of t . But this fact follows quite easily from our condition b). Indeed, for any t e Rn and any dyadic cube Q, we have :

4 33

IV.5. FACTORIZATION AND EXTRAPOLATION

=

C(T~U)(Q).

0

This finishes the proof.

Condition b) in theorem 4.9 will be called condition on Sp the couple (u,w). It is obvious that implies Ap, since

s?

But, of course, S is stronger than . However, for u=w, P AP is equivalent to A Actually, it is very easy we know that S P P' to see directly that, in the case of equal weights, A implies P This is an observation of Hunt, Kurtz and Neugebauer [ l ] which Sp. we shall presently explain. It is clear that, for a function f living in a cube Q, the Hardy-Littlewood maximal function at a point x e Q coincides with the supremum of the averages of If1 over cubes contained in Q. Now, if R is a cube contained in Q and w E A we have

P'

so that IM(a XQ)(x)lp $ Clb!w(w-l XQ)(x) sequently, since w (x)dx is "doubling"

and (w,w) e S P' theorem.

I

for

x

E

Q,

and, con-

This gives yet another proof of Muckenhoupt's

5. FACTORIZATION AND EXTRAPOLATION

We have already seen (immediately before theorem 2 . 1 6 2 that if we have two A 1 weights wo and w 1 and if 1 < p < m , then w(x) = wo(x) W , ( X ) ~ - ~ is an Ap weight. Now we are going to show that, conversely, every weight w e AD can be written in this form for certain wo,wl e A 1 . This factorization theorem will have important consequences. The proof will be based on a simple lemma, which, as we shall see, provides a strikingly powerful method to deal with several problems about weights.

I V . WEIGHTED NORM INEQUALITIES

434

LEMMA 5 . 1 . Let S b e a s u b l i n e a r o p e r a t o r bounded i n L p ( u ) , w h e r e p 2 1 p i s a n a r b i t r a r y p o s i t i v e m e a s u r e o n some m e a s u r a b l e s p a c e . S u p p o s e t h a t S f 2 0 f o r e v e r y f f L p ( p ) . Then, f o r e v e r y

u 2 0

k

Lp(p)

i ) u(x)

there i s

6 v(x)

i i ) llvllp 2 2

v

2 0

@

Lp(p)

such t h a t :

x

for a . e .

IlUU,

i i i ) Sv(x) 2 C v ( x )

x

f o r a.e.

Proof: I t s u f f i c e s t o take

v

v e r g e s c l e a r l y i n norm, a n d

m

=

Ivll,

which i s i i ) .

1

j=o

5

( C = 2llSII

i s enough).

( 2 ~ ~ S ~ ~ ) - j STj hu i.s s e r i e s c o n -

,I ( 2 l l S l l ) - j l l s l l j l ~ u ~ l =P

J=O

211ullp,

On t h e o t h e r h a n d , u ( x ) 5 v ( x ) b e c a u s e a l l t h e p a r t i a l sums 2u as a c o n s e q u e n c e o f t h e f a c t t h a t S f 2 0 f o r e v e r y f .

are

Finally, since

i s s u b l i n e a r , we h a v e :

S

A c t u a l l y , w i t h t h e h e l p o f t h i s lemma, w e c a n g i v e a g e n e r a l f a c t o r i z a t i o n t h e o r e m w h i c h i n c l u d e s t h e o n e we were s e e k i n g f o r Ap

weights.

THEOREM 5 . 2 .

T

be a p o s i t i v e s u b l i n e a r o p e r a t o r a c t i n g on

(X,dx) ( t h i s means and a l s o t h a t If1 6 g i m p l i e s

m e a s u r a b l e f u n c t i o n s o n some m e a s u r e s p a c e

that

I T ( f + g ) I 5 l T f [ + lTgl

l T f l 2 Tg ) . For 1 < p < m, l e t us call W = {w : 0 5 w(x) < m a . e . and T i s b o u n d e d i n P = Lp(w(x)dx)). Also, we c a l l W,

=

{w: 0 5 w(x) <

independent

a . e . and

XI

0 : '

Tw(x)

5

C w(x)

a.e.

for some

C

n

m, we h a v e Wp W;ipC W1 . W i - p , that i s : 1 w and a l s o w / ( P - 1 ) then there e x i s t P" such t h a t w = B e s i d e s , t h e c o n s t a n t s C for

Then, f o r every Q

m

Lp(w) =

w E W

1 < p <

P w o , w l e W1 w 0 -and w i n t h e c l a s s W1 depend o n l y upon t h e c o n s t a n t s f o r -l'(p-l) i n Wp g& W r e s p e c t i v e l y , t h a t i s , on t h e w and w P' r e s p e c t i v e norms of T 3 L P ( ~ ) LP'(~-''(?-') 1. P r o o f : We j u s t n e e d t o c o n s i d e r t h e c a s e lows f r o m

W

P

n

WAyPc

W1.W;-p

1

p

2

2,

by r i s i n g t o t h e power

since it f o l 1-pl,

that

IV.5. FACTORIZATION AND EXTRAPOLATION

W

P'

Wi-p'~

n

n

W1

W1-p'.

1 i.e.,

5 2,

1 < p

So, l e t

435

and s u p p o s e t h a t

w e Wp WA;p, w e W and a l s o w - 1 / ( P - 1 ) w We want P P I . t o see that w = w i t h wo,wl e W1. After writing v - l = w ; - ~ , w e s e e t h a t t h i s i s e q u i v a l e n t t o f i n d i n g v such t h a t :

<=

t h a t i s : T(vw)

a ) vw e W , ,

C vw,

and a l s o

1/(p-1)) 5

b) v valently (T(v Suppose t h a t f o r e v e r y so that:

u

i n some

c

v "(P-')

or

s p a c e we c a n f i n d

Lq

equi Su

IT(u w)I IS(U)W

If the operator

S

s a t i s f i e s t h e h i p o t h e s e s o f lemma 5 . 1 . , we s h a l l

b e a b l e t o f i n d v >= 0 s u c h t h a t b e c a u s e t h e n we s h o u l d h a v e

S(v) 5 C v .

T h i s would s u f f i c e ,

T ( v w) 5 S ( V ) W 5 C vw T (V l / ( p - l ) p - l

A l l w e h a v e t o do i s t o l o o k f o r

5 S(v)

S

5 c v

and make s u r e t h a t i s s a t i s f i e s

t h e h y p o t h e s e s o f t h e lemma. The n a t u r a l c a n d i d a t e f o r operator sending t h e function

u

S ( u ) = IT(u w ) l w - l

F i r s t o f a l l we o b s e r v e t h a t

S

into +

Su

S

is the

g i v e n by

T(lu1 1I ( P - 1 ) ) P - l

is s u b l i n e a r ( f o r t h e f i r s t term i n

t h e sum, s u b l i n e a r i t y i s c l e a r , and f o r t h e s e c o n d i t f o l l o w s from t h e f a c t t h a t , b e i n g 1 < p 5 2 , we have l / ( p - 1 ) 2 1 , and i t i s a s i m p l e e x e r c i s e t o c h e c k t h a t u I-+ T( IuI a) ' l a i s s u b l i n e a r provided a 2 1). B e s i d e s , S i s bounded i n L p l ( w ) . I n d e e d :

w e W From t h e d e f i n i t i o n o f S , i t i s e v i d e n t t h a t P' f o r every u e L p ' ( w ) . Thus, S s a t s f i e s a l l t h e c o n d i t i o n s r e q u i r e d i n lemma 5 . 1 . Note t h a t C i n lemma S . l ( i i i ) d e p e n d s

because Su

2

0

o n l y on t h e norm o f S i n L p ' ( w ) , and t h i s d e p e n d s o n l y on t h e norms f o r T i n Lp(w) and i n L P ' ( w - 1 / ( p - 1 1 . T h i s f i n i s h e s t h e proof. 0

I V . WEIGHTED NORM INEQUALITIES

436

COROLLARY 5 . 3 . (P. J o n e s ' f a c t o r i z a t i o n t h e o r e m ) . For 1 < p < i f and o n l y i f t h e r e e x i s t Ap = A1.Ai-P, that i s : w e A P wo,wl f A1 s u c h t h a t w = w 0 w ; - ~ . P r o o f : I f we t a k e

m,

T = M = H a r d y - L i t t l e w o o d maximal o p e r a t o r i n

and W1 = A 1 . B e s i d e s W 1 i P = theorem 5 . 2 , we know t h a t W = A P P -l/(p-l)e A P AP' i P = AP b e c a u s e w e AP i f and o n l y i f w P' + T h e r e f o r e , theorem 5 . 2 a p p l i e s giving us A C A,.A;-p. The i n c l u P h a s b e e n a l r e a d y e s t a b l i s h e d i n s e c t i o n 2. sion A1.A;-PC A 0 P By combining t h e f a c t o r i z a t i o n t h e o r e m w i t h t h e c h a r a c t e r i z a t i o n =

o f A1 for A

w e i g h t s g i v e n by theorem 2 . 1 6 , we o b t a i n a g e n e r a l expression w e i g h t s i n t e r m s o f maximal f u n c t i o n s . T h i s i s t h e n a t u r a l

P e x t e n s i o n t o p > 1 o f t h e o r e m 2 . 1 6 . Then, by u s i n g t h e J o h n - N i r e n berg theorem, t h i s y i e l d s an e x p r e s s i o n f o r B.M.O. f u n c t i o n s i n

terms o f maximal f u n c t i o n s : COROLLARY 5 . 4 .

a)

Then, w

e A

a.e.

Let w

be a w e i g h t i n Rn s u c h t h a t i f a n d o n l y i f i t can b e w r i t t e n a s

P

w(x) = k ( x )

w(x) <

( M f ( x ) I a (Mg(x)) B(1 - P I

1 f , g e Lloc(Rn), k bounded away f r o m 0 m and 0 < a , 6 < 1 . I n t h i s r e p r e s e n t a t i o n , k c a n be t a k e n b e t w e e n two p o s i -

with

t i v e bounds which depend o n l y o n t h e

and

b ) T h e r e a r e c o n s t a n t s C1 mension n, s u c h t h a t e v e r x I$

with

f,g

f

C2

B.M.O.

E

constant f o r

~

w.

d e p e n d i n g o n l y on t h e d i Rn c a n be w r i t t e n a s :

+ ( x ) = b ( x ) + y l o g Mf(x) - n l o g Mg(x) 1 L. , y J n 2 0 & lbl, + Y + n 5 C1ll+(l*.

*

ConverseZy, e v e r y B.M.O.

Ap

II+I*2

I$

w h i c h can be. w r i t t e n a s a b o v e , b e l o n g s t o

C2(llbIlm

+

Y

+

n).

c ) we can w r i t e a s t a t e m e n t l i k e B.M.O. and n = 0 . -

with

b)

B.L.O.

i n p l a c e of

function d) As a consequence of b) & c ) , every B.M.O. can be w r i t t e n a s a d i f f e r e n c e o f t w o B . L . O . f u n c t i o n s . I n s h o r t : B.M.0.C B . L . O . - B . L . O . P r o o f : a ) The " i f 1 ' p a r t i s c l e a r , s i n c e it f o l l o w s from theorem 2 . 1 6 t h a t b o t h (Mf(x))a quently w e A P'

and

( M ~ ( X ) ) a~r e

A1

weights and, conse-

437

I V . 5 . FACTORIZATION AND EXTRAPOLATION

corollary 5.3 implies t h a t Conversely i f w e A P’ w = w wlPp with wo,wl e A1. Then we j u s t need t o a p p l y t h e o r e m 0’ 1 2.16 t o o b t a i n t h e d e s i r e d r e p r e s e n t a t i o n . Observe t h a t , i n t h e p r o o f o f t h e o r e m 2 . 1 6 t h e l o w e r bound o b t a i n e d f o r t h e f u n c t i o n k d e p e n d s o n l y upon t h e c o n s t a n t C i n t h e r e v e r s e H o l d e r ’ s i n e q u a l i t y f o r t h e A1 w e i g h t , and t h i s , i n t u r n , d e p e n d s o n l y upon i t s A1 c o n s t a n t . The u p p e r bound o b t a i n e d f o r k i n t h e p r o o f o f t h e o r e m 2.16 i s j u s t 1 . I n o u t p r e s e n t s i t u a t i o n , t h e f a c t o r i z a t i o n theorem t e l l s u s t h a t t h e A1 c o n s t a n t s f o r wo and w1 depend o n l y upon the A constant f o r w . Therefore i n our r e p r e s e n t a t i o n f o r t h e P A weight w , t h e f u n c t i o n k i s bounded away from 0 and m P w i t h bounds d e p e n d i n g o n l y upon t h e Ap c o n s t a n t f o r W .

B . M . O . w i t h norm i n d e p e n d e n t o f f . C o n s e q u e n t l y , i f $ r e p r e s e n t a t i o n e x h i b i t e d i n b ) , we have @ E B . M . O . with

11$11* 5

n)

C2(nbllm + y +

Conversely, i f

f o r some a b s o l u t e c o n s t a n t

C2.

i t f o l l o w s from c o r o l l a r y 3 . 1 0 i n

e B.M.O.,

@

is i n has t h e

l o g Mf(x)

b ) A c c o r d i n g t o C o r o l l a r y 3 . 5 o f C h a p t e r 11,

C h a p t e r I 1 and c o r o l l a r y 2.18 t h a t , t a k i n g

A = C/U$ll*

with

C

an

i n t h e n o t a t i o n of c o r o l l a r y 3.10 absolute constant (say C = C z / 2 C h a p t e r 1 1 ) t h e f u n c t i o n w(x) = e A @ ( x ) i s i n A 2 w i t h a n A 2 c o n s t a n t independent of

( l o o k a t t h e end o f t h e p r o o f o f t h e

Q

f o r e m e n t i o n e d c o r o l l a r y 3 . 1 0 and r e a l i z e t h a t , w i t h t h e c h o i c e t h e A 2 c o n s t a n t f o r w t u r n s o u t t o depend o n l y on C1 i n t h e n o t a t i o n u s e d t h e r e ) . A p p l y i n g p a r t a ) t o o u r W , we o b t a i n

A = C2/2,

log W(X) = log k(x) Since

Q = 1 - l log w ,

b = A-l

Observe t h a t t h e s i n c e A-1

we g e t t h e d e s i r e d d e c o m p o s i t i o n w i t h

log k ,

Lm

y = A

norm o f

C-’ll@l]* and

=

a l o g Mf(x) - B l o g Mg(X)

+

0

Ilbllm

log k

5 a +

Y

< 1, +

rl

-1

11 = A - l

a,

d o e s n o t depend on 0

5

with

is actually in B.L.O.,

Ilbllm <

m

and

y

Then,

5 C o n s t a n t IIQII*

s o t h a t any

a r e a l number

@.

we have

B < 1,

c ) A s we o b s e r v e d i n t h e p r o o f o f t h e o r e m 3 . 3 l o g Mf(x)

B

20,

Q

=

i n C h a p t e r 11, b

+ y

l o g Mf

w i l l a l s o belong t o

B.L.O.

For t h e c o n v e r s e , t h e p r o o f i s v e r y much l i k e t h e one i n p a r t b ) . The d i f f e r e n c e i s t h a t , a s we n o t e d i n t h e s e c o n d remark f o l l o w i n g c o r o l l a r y 2 . 1 9 , i f Q e B . L . O . , t h e w e i g h t w ( x ) = e A@(x) i s

IV. WEIGHTED NORM INEQUALITIES

438

a c t u a l l y i n A,, b e f o r e , b u t now

n o t j u s t i n A2. Then we c a n u s e p a r t a ) a s p = 1 , s o t h a t we o b t a i n t h e r e p r e s e n t a t i o n w i t h

q = 0.

F i n a l l y d ) f o l l o w s o b v i o u s l y from b ) and c ) .

Next we a r e g o i n g t o a p p l y theorem 5 . 2 t o s t u d y t h e f o l l o w i n g q u e s t i o n : Let RC Rn b e a m e a s u r a b l e s e t w i t h IRI > 0 . L e t f be a m e a s u r a b l e f u n c t i o n on 5 2 . We want t o know how f h a s t o b e i n o r d e r t h a t t h e r e e x i s t s F d e f i n e d on t h e whole s p a c e b e l o n g i n g t o B . M . O . , and s u c h t h a t F(x) = f ( x ) f o r a . e .

Rn,

x

E

R.

I n o t h e r w o r d s , we want t o c h a r a c t e r i z e t h e s p a c e c o n s i s t i n g o f t h e

R o f t h e f u n c t i o n s i n B.M.O. We s h a l l o b t a i n t h e s o l u t i o n t o t h i s problem from t h e s o l u t i o n t o t h e c o r r e s p o n d i n g r e ? t r i c t i o n problem f o r w e i g h t s .

restrictions t o

We s h a l l u s e t h e o p e r a t o r g r a b l e o v e r bounded s u b s e t s o f

defincdon functions

VR,

0 , by

where t h e s u p i s t a k e n o v e r a l l t h e c u b e s o f We s h a l l s a y t h a t

w(x) $ 0 ,

1 5 p .: m , c l a s s Ap,R, s u c h t h a t f o r e v e r y cube

0

(only t h e cubes

inte-

g

d e f i n e d on

Rn

containing

R,

x.

belongs t o t h e

i f and o n l y i f t h e r e i s a c o n s t a n t Q o f Rn

intersecting

i n a s e t o f p o s i t i v e measure

R

w i l l be r e a l l y r e l e v a n t h e r e ; f o r

C

p = 1,

t h e second f a c t o r has t h e

u s u a l i n t e r p r e t a t i o n a s t h e e s s e n t i a l supremum o v e r w(x) - ' ) .

QnQ

of

The f o l l o w i n g theorem i s p r o v e d e x a c t l y a s theorem 1 . 1 2 .

THEOREM 5 . 5 . L e t

1

2

p <

(u,w)

m

g a t i v e measurabZe f u n c t i o n s on

R.

b e a c o u p l e of n o n - n e -

Then, t h e f o l l o w i n g t h r e e condi-

tions are equivalent:

a ) Ma

i s bounded f r o m

i s a constant

C

LP(R;w)

s u c h t h a t , for e v e r y

L;(Q,u), f

on 0 -

that is, there and e v e r 2

> 0:

I V . 5 . FACTORIZATION AND EXTRAPOLATION

u ( { x E 0 : M,f(x) b) There e x i s t s f o r e v e r y cube

5 C A-p

> A])

C

J,

439

I f ( x ) I P w(x)dx

on

2 0

f

such t h a t , f o r every

and

fi

B":

Q

C

c ) There e x i s t s

s u c h t h a t , f o r e v e r y cube

Q

0 f -

Bn:

w i t h t h e u s u a l i n t e r p r e t a t i o n f o r t h e second f a c t o r i n case

In p a r t i c u l a r ,

w,

i s o f weak t y p e

Mn

i f and o n l y i f

w

E A

>=

0

p = 1

with respect

(p,p)

to

PlQ' Here i s t h e s o l u t i o n t o t h e r e s t r i c t i o n p r o b l e m f o r w e i g h t s .

THEOREM 5 . 6 . Let

w

on

&

R

1 5 p <

Then, t h e f o l l o w i n g

m.

conditions are equivalent:

i ) There e x i s t s W(x) = w ( x )

W

for a.e.

i i ) There i s

E

> 0

E

A

P

(W

d e f i n e d on

x e R. s u c h that

w(x)

1 +E

E

A

R"),

such t h a t

P,R'

Proof: I t is c l e a r t h a t i) implies i i ) . Indeed, i f w is t h e r e s f o r some triction to 0 of W f A then t h e r e w i l l be W1+€ e A P P' E > 0. On r e s t r i c t i n g t o R , we s h a l l h a v e w l + € E A P,R' Next w e w a n t t o show t h a t i i ) i m p l i e s i ) . Let p > 1 . S u p p o s e t h a t 1+E f o r some E > 0 . T h e n , a s i n 1 . 1 4 b ) , we c a n f i n d E A P,R It 6 , q with 0 < 6 < E , 1 < q < p , s u c h t h a t w ~ E+ A ~ q,R' f o l l o w s t h a t MR i s o f weak t y p e ( q , q ) in 0 with respect t o

w1+6.

Since it i s also of type

( s e t s o f measure z e r o a r e

(m,=)

e x a c t l y t h e same a s t h o s e f o r L e b e s g u e m e a s u r e ) , i t f o l l o w by i n t e r polation that that

'+'

i s bounded i n

MQ

LP(w1+')

o r , i n o t h e r words,

w b e l o n g s t o t h e c l a s s Wp corresponding t o the operator we s h a l l On t h e o t h e r h a n d , s i n c e ( w ' + ~ ) - ' ' ( ~ - ' ) E .4 MR. 9,' l+6 -l/(p-l) f o r some 1 < r < p . Then MQ i s o f h a v e (w ) *r,Q weak t y p e ( r , r ) w i t h r e s p e c t t o ( w l + [ ' ) - l ' ( p - l ' a n d , by i n t e r p o 1+6 - l / ( p - l ) w Th eo r em 5 . 2 c a n b e a ppl i e d , obtaining lation, (w ) P' . 1+6 1 -p w - wo w , with wo,wl e W 1 , t h a t i s : MQ W . 5 C w i , j = 0,l. J We e x t e n d e a c h w . m a k i n g i t 0 o u t s i d e oi R a n d c o n s i d e r t h e J f u n c t i o n V . = I\l(w.) d e f i n e d o n R". Of c o u r s e , f o r a . e . x t: a , w e have

y'

1

1

w - ( x ) 5 M-(x) = M w . ( x ) .l J 1

= hl

K.(x) 5 C

R I

!a:.(?().

1

Therefore,

IV. WEIGHTED NORM INEQUALITIES

440

f o r a . e . x e R , Mj(x) ;w j ( x ) , s o t h a t w ( x ) ~ + = ~ = Mo(x)Ml(x) l - p k ( x ) , where k i s a p o s i t i v e f u n c t i o n d e f i n e d on R

and bounded away from 0 and

We c a n w r i t e

m.

Observe t h a t , a c c o r d i n g t o theorem 3 . 4 i n C h a p t e r 11, t h e f u n c t i o n s d e f i n e d on R n , M]/(1+6) = M(wj) 1 / ( 1 + 6 ) a r e A1 w e i g h t s . I f we f o r x e R and K(x) = 1 f o r x L 0 , d e f i n e K(x) = k ( x )' 1 / ( 1 + 6 ) and w r i t e , f o r x e Rn W(x) = Mo(x) 1/(1+6) ( M l ( x ) 1 / ( 1 + 6 ) ) 1 - P ~ ( x ~ , we s e e t h a t W e A s i n c e K i s o b v i o u s l y bounded away from 0 and P a. C l e a r l y W(x) = w(x) f o r a . e . x e a. In the case

p = 1,

t h e p r o o f i s d i r e c t , w i t h o u t need o f f a c -

0

torization. COROLLARY 5 . 7 .

Per f

R,

d e f i n e d on

the following conditions are

equivalent:

i ) There e x i s t s F(x) = f ( x ) f o r a . e . i i ) There e x i s t a constant

CQ,

F

d e f i n e d on

Rn,

such t h a t

F

E

B.M.O.

&

Q

Rn

x e R. A > 0,

C > 0

c&,

f o r e v e r y cube

f0

such t h a t :

J-1 IQI

Qfln

e

A

P r o o f : T h a t i ) i m p l i e s i i ) f - l l o w s i m m e d i a t e l y from C h a p t e r 1 1 , c o r o l l a r y 3.10 i i ) . L e t u s s e e t h a t i i ) i m p l i e s i ) . I t f o l l o w s from i i ) , e x a c t l y a s i n c o r o l l a r y 2.18, t h a t e Af e A2,R. T a k i n g 0 < X o c A , we 'Ofll+E A have ( e f o r some E > 0 . A c c o r d i n g t o theorem 5 . 6 , 2 ,Q t h e r e w i l l e x i s t W E A 2 s u c h t h a t e x O f ( x ) = W(x) f o r a . e . x e s2.

I f we w r i t e

G e B.M.O.

Thus

F = AilG.

0

tions

W = eG

f(x) =

we h a v e , by c o r o l l a r y 2 . 1 9 , t h a t

h i iG(x)

for a.e.

x

E

R.

This is i) with

I t seems n a t u r a l t o d e n o t e by B . M . O . (a) t h e s p a c e o f f u n c f satisfying condition i i ) i n corollary 5.7.

We c a n g i v e a v e r s i o n o f t h e o r e m 5.2 v a l i d f o r some l i n e a r , n o t n e c e s s a r i l y p o s i t i v e o p e r a t o r s , provided

p

=

2.

The c a s e o f t h e

441

I V . 5 . FACTORIZATION AND EXTRAPOLATION

Hilbert transform class

H,

is particularly interesting.

the We know from t h e o r e m s 3 . 1 and 3 . 7 t h a t , f o r 1 < p < m, Wp(H) a s s o c i a t e d t o H a s i n t h e o r e m 5 . 2 , c o i n c i d e s w i t h

the class

A

Now we g i v e t h e f o l l o w i n g :

P'

DEFINITION 5 . 8 . Me shaZZ d e n o t e b y

O
those functions tions:

a)

-m

w(x)

>=

R

0

W1(H) t h e c Z a s s f o r m e d b y s a t i s f y i n g t h e foZZowing c o n d i -

q d x < m l+x

b) T h e r e e x i s t s $ s u c h t h a t I $ ( x ) I 5 C ~ ( x ) for a . e . w i t h C i n d e p e n d e n t of x, and t h e f u n c t i o n F ( x + i t ) = = Pt*(w+i$)(x) i s h o Z o m o r p h i c i n R.+. 2 I t f o l l o w s from b ) t h a t

Pt*$(x)

x

i s a c o n j u g a t e harmonic f u n c

I t can be s e e n ( a s i n theorem 1.20 o f Chapter N T I ) t h a t P t * $ ( y ) -P $ ( x ) ( a s y+itx) f o r a . e . x , s o t h a t I$ would d e s e r v e t o be c a l l e d t h e H i l b e r t t r a n s f o r m o f w . Actually tion for

Pt*w(x).

f o r w s a t i s f y i n g a ) , Hw i s a l w a y s w e l l d e f i n e d a s a n e q u i v a l e n c e c l a s s modulo c o n s t a n t s , s i n c e Pt*w a l w a y s h a s a h a r m o n i c c o n j u g a t e and t h i s harmonic c o n j u g a t e h a s n o n - t a n g e n t i a l b o u n d a r y v a l u e s because t h e corresponding holomorphic f u n c t i o n

F

does.

( T h i s f o l l o w s from t h e f a c t t h a t Re F 2 0 . We j u s t n e e d t o c o n s i d e r t h e h o l o m o r p h i c f u n c t i o n e - F = G which s a t i s f i e s I G I 5 1 a n d u s e F a t o u ' s t h e o r e m 4 . 1 9 i n C h a p t e r 1 1 ) . We c a n s e e b ) above a s a c o n c r e t e form o f t h e s t a t e m e n t l H w l 2 C w. Here i s t h e f a c t o r i z a t i o n t h e o r e m f o r t h e H i l b e r t t r a n s f o r m . THEOREM 5 . 9 . E v e r y

w i t h wo

,Mi,

P r o o f : For

E W1 (H)

.

w e A2

u e L'(w),

@

R

can be w r i t t e n a s

w

=

w w- 1 0 1

define

Su(x) = IH(u.w)(x)

1

w(x)-'

+

IHu(x)I

T h i s makes p e r f e c t l y good s e n s e , s i n c e we know ( t h e o r e m 3 . 1 ) t h a t H i s bounded b o t h i n L2(w) and i n L 2 ( w - l ) , and we h a v e u e L 2 (w) 2 A l s o , c l e a r l y , S i s bounded i n L 2 ( w ) . By and u.w E L ( w - ' ) . 2 a p p e a l i n g t o lemma 5 . 1 , we c o n c l u d e t h a t t h e r e e x i s t s v E L (w)

x . T h i s i m p l i e s IHv(x) 1 <= such t h a t Sv(x) 5 C v(x) f o r a . e . 5 Cv(x) and a l s o IH(v.w)(x) I 5 C v ( x ) w ( x ) f o r a . e . x . Note a l s o t h a t we h a v e v e L 2 (w) and vw e L 2 ( w - l ) a n d , s i n c e w and w - 1

442

are

IV. WEIGHTED NORM INEQUALITIES

A2

against against

weights, it follows t h a t both

v

and

vw

are integrable

( 1 + x 2 ) - l . A c t u a l l y , more i s t r u e : t h e y a r e i n t e g r a b l e ( 1 + l x l a ) - ’ f o r some 0 < c1 < 1 . I n d e e d , f o r example:

i f we t a k e

c1

c l o s e enough

t o 1 (see 3 . 2 ) .

-1 I f we w r i t e w o = vw and w 1 = v , we h a v e w = w 0w 1 and we c a n s e e t h a t b o t h wo and w1 b e l o n g t o W1(H). The r e a s o n i s t h a t w e have IHwj(x)l. 2 C w j ( x ) a . e . f o r j = O , l and P t * ( w . + i H w . ) ( x ) i s a n a l y t i c a s c a n be s e e n from t h e i d e n t i t y 1 3 I n f a c t , t h i s i d e n t i t y i s obvious Pt*(w.+iHw.)(x) = (Pt+iQt)*wj(x). 1 1 f o r n i c e ( s a y , Schwartz) f u n c t i o n s , and, i n o r d e r t o extend i t t o a r b i t r a r y f E L 2 (w) ( i n p a r t i c u l a r t o f = w . ) i t s u f f i c e s t o 3 o b s e r v e t h a t , f o r f i x e d x e R and t > 0 , b o t h f k P * f j f ( x ) and f Q t * f ( x ) a r e bounded l i n e a r f u n c t i o n a l s on L (w) , because

\ \ P t ( x - . ) \ \2 , w <

m

and

IIQt(x-.)

\12,w

2

<

m.

CI

Next we s h a l l s e e t h a t t h e f u n c t i o n s b e l o n g i n g t o W1(H) s a t i s f y t h e c o u n t e r p a r t i n t h e l i n e o f t h e Helson-SzegG c o n d i t i o n appearing i n theorem 8.14 of Chapter I . THEOREM 5 . 1 0 . E v e r w(x) = e u(x)+H+ [VI,

w E W,(H) c a n b e w r i t t e n i n t h e form o r some u v s a t i s f y i n g IIullm <

m

and

< a/Z.

w E W1(H). A c c o r d i n g t o d e f i n i t i o n 5 . 8 t h i s i m p l i e s t h a t t h e r e i s a f u n c t i o n F ( x + i t ) , holomorphic i n t h e upper h a l f p l a n e , s u c h t h a t F ( x + i t ) = P t * ( w + i $ ) ( x ) w i t h I $ ( x ) ) <= C w(x) f o r a . e . x . Now I P t * $ ( x ) l <= C P t * w ( x ) . Observe a l s o t h a t a l w a y s Pt*w(x) > 0 , and l A r g F ( x + i t ) 1 = ( a r c t g ( P t * $ ( x ) ) ( P t * w ( x ) ) - l 1 , $ 2 a r c t g C < a / 2 . I t f o l l o w s t h a t we c a n w r i t e F ( x + i t ) = e G ( x + i t ) w i t h G h o l o m o r p h i c s u c h t h a t IIm G I 6 a r c t g C < a / 2 . T h i s c o n d i t i o n , a l l i e d t o F a t o u ‘ s t h e o r e m , g u a r a n t e e s a s u s u a l t h e existence o f boundary v a l u e s G(x) f o r a . e . x . We s h a l l h a v e , modulo c o n s On t h e t a n t s , G(x) = Hv(x) - i v ( x ) w i t h IIvII, I a r c t g C < n / 2 . o t h e r h a n d , s i n c e I F ( x ) I = Iw(x) + i $ ( x ) I , we h a v e : w(x) s I F ( x ) I 6 w(x) + I $ ( x ) I 5 ( 1 + C ) w ( x ) , s o t h a t we c a n w r i t e w(x) = ko(x) IF(x) I = k ( x ) e H v ( x ) where k and ko a r e > O and bounded away from 0 and a . A f t e r w r i t i n g u = l o g k , we g e t what we w a n t e d . Proof: Let

n

By combining theorem 5 . 9 w i t h theorem 5 . 1 0 . we g e t a weak form

443

I V . 5 . FACTORIZATION AND EXTRAPOLATION

o f t h e h a r d p a r t o f t h e Helson-Szeg8 theorem, namely: THEOREM 5 . 1 1 . E v e r x w e A 2 = eu ( x ) + H v ( x ) f o r some u

iivn,

and

R v

w(x)

can be w r i t t e n as satisfying

jlullm <

=

< n.

-1 P r o o f : A f t e r t h e o r e m 5 . 9 , we c a n w r i t e w = w 0 w 1 with w e w r i t e T h e n , u s i n g t h e o r e m 5 . 1 0 , w o , w l e W1(H). uo (XI +Hvo ( X I wo(x) = e

and

[ujll,

with

w

=

w 0 w-1 1

=

<

e

m

and

Ivjllm < n / 2

u -ul+H(v0-vl) 0

for

j=O,l.

I t follows t h a t

= e

A s we s a i d , t h i s i s a weak f o r m o f t h e H e l s o n - S z e g B t h e o r e m b e c a u s e we a r e n o t a b l e t o g e t t h e bound )Ivllrn < n / 2 c o n t e n t o u r s e l v e s w i t h t h e w o r s e e s t i m a t e IIvllrn < T I . o b t a i n a s a consequence COROLLARY 5 . 1 2 . B.M.O.(R) f e B.M.O.(R)

a c o n s t a n t and

=

Lm(R) + HL"(R).

can be w r i t t e n a s

I$II,

+

IIqII,

2

and w e have t o H o w e v e r , we

More c o n c r e t e l y , e v e r y

f ( x ) = c + $ ( x ) + H$(x)

C l!fll*

where

c

with

c

i s an a b s o l u t e cons-

tant. P r o o f : Let

f e B.M.O.(R).

Then, as w e e x p l a i n e d i n t h e Droof o f

c o r o l l a r y 5 . 4 . ( b ) , t a k i n g A = C/llfll* c o n s t a n t , t h e f u n c t i o n w(x) = e X f ( x ) t a n t independent of

f.

with

C a certain absolute i s i n A 2 w i t h a n A2 c o n s -

Then, keeping t r a c k o f t h e c o n s t a n t s i n t h e

p r o o f s o f theorems 5.9 and 5 . 1 0 , we r e a l i z e t h a t w e c a n w r i t e u ( x ) + Hv(x) w(x) = K e with K z 0, gives

IIullrn

<=

C,

f(x)

=

c

an a b s o l u t e c o n s t a n t , and +

$ ( x ) + H$[x),

llvll,

<

TI.

This

where

c = ( l o g K ) llfll*/C, $ ( X I = u ( x ) llfll*/C a n d @ ( X I = v ( x ) llfil*/C. C l e a r l y II@(l, + ~ $ l l r n 2 C Ilfll* f o r some a b s o l u t e c o n s t a n t C . We a l r e a d y know ( P r o p o s i t i o n 5 . 1 5 i n C h a p t e r 11) t h a t , c o n v e r -

L"

sely,

+ HL"CB.M.0.

w i t h t h e r i g h t e s t i m a t e f o r t h e B.M.O.norm

We h a v e e s t a b l i s h e d t h e i d e n t i t y b e t w e e n t h e s e two B a n a c h

spaces :

n

444

IV. WEIGHTED NORM INEQUALITIES

B.M.O.

=

( L ~+ H L ~ ) / R

the norm in the right hand side being the quotient norm corresponding to the seminorm inf tII9II,

+

INI,

: f = 9

+

W I

Now the space (Lm(R) + HLm(R))/R is naturally identified with the dual of H1(R), by doing exactly the same that we did for the torus at the beginning of section 9 in Chapter I. Thus, corollary 5.12 gives yet another proof (the third one!) of the duality theorem (H')* = B.M.O. Observe that the proof of the decomposition in coroi lary 5 . 1 2 is a constructive one, whereas the two previous proofs of the duality theorem (sections 1.9 and 111.5), do not give an explicit construction o f the decomposition. COROLLARY 5.13.(C. Fefferman's duality theorem). (H1(R))* B.M.O.(R).

=

=

In Chapter I we have proved the Helson-Szep8 theorem only for the torus. However, the theorem also works for the real line substituting the Hilbert transform for the conjugate function. The proof is not essentially different, but some changes have to be made due to the unboundedness of the domain. By using this Helson-SzegB theorem, we can prove a converse of theorem 5 . 1 9 , namely: if IIvllm IT/^ then eHv(x) = k(x)w(x), where k > 0 i s bounded away from 0 and m and w E W,(H). Indeed, the Helson-Szeg8 theorem im plies that eHv E A Z , so that i t has the right integrability (condition a) in definition 5 . 8 ) . Besides, eHv-iv = eHV(cos v - i sin v) is the boundary function of a holomorphic function F in the upper half plane, and it can be shown that F(x+it) = Pt*(e Hv-iv1 (XI * Thus, F(x+it) = Pt*(w+i$)(x) where w(x) = eHv(x) cos v(x) and $(x) = -eHv(x) sin v(x). But, since IIvllrn < ~ / 2 , we have: 0 < cos 5 cos v(x) 5 1 , s o that if we write k(x) = l/cos v(x), k is bounded away from 0 and a . Also I@(x) I 2 eHv(x) 2 c w(x) with C = l/cos IIvII,. We have obtained eHv(x) = k(x)w(x) with k > 0 bounded away from 0 and m and w f: W1(H).

IIv~/~

What we have is a characterization of W1(H). Using the Helson-SzegG theorem once more, we can write: exp(Lm).W1(H) = = exp(Lm) .A2. In other words, the functions belonging to W 1 (H) coincide with those in A2 up to multiplication by functions bounded away from 0 and m . This can also be written as W1(H) - A2.

IV.5.

445

FACTORIZATION AND EXTRAPOLATION

A s a f i n a l a p p l i c a t i o n o f t h e f a c t o r i z a t i o n t h e o r e m we s h a l l g i v e a theorem t h a t r e l a t e s t h e f a c t t h a t w belongs t o A2 w i t h the fact that

log w

THEOREM 5 . 1 4 . @

t a n t s C,

a) e

'

e A2

@

C2,

is close to

Lm

i n t h e m e t r i c of B.M.O.

b e a f u n c t i o n on

Rn.

Then, t h e r e e x i s t cons-

d e p e n d i n g onZy o n t h e d i m e n s i o n

provided

n,

such t h a t :

W

i n f { ( l @ - g ( I * : g e L 1 5 C1

b ) inf{ll@-gll* : g e Lm} 5 C 2

e@ e A2.

whenever

P r o o f : a ) I f i n f {ll@-gllx : g e Lm) < C , we s h a l l be a b l e t o f i n d g e L m , s u c h t h a t @ = g+h w i t h llhll* < C . Now i f C i s small enough, i t f o l l o w s from t h e J o h n - N i r e n b e r g t h e o r e m ( c o r o l l a r y 3 . 1 0 i n C h a p t e r 11) p l u s t h e c h a r a c t e r i z a t i o n o f A 2 w e i g h t s i n c o r o l l a r y 2 . 1 8 , t h a t e h e A 2 . Then e' = e g e h e A 2 . I f n = 1 , p a r t b ) c a n b e d e r i v e d from t h e Helson-Szegijtheorem. The weak v e r s i o n c o n t a i n e d i n t h e o r e m 5 . 1 1 s u f f i c e s . I n d e e d , i f

e A 2 , we s h a l l b e a b l e t o w r i t e e' = e'+Hv w i t h )Iullm < IIvllm < TI. Thus I$-ul/, = llHvll* 2 C IIVI, < Ca, s o t h a t m i n f t11$-g1!, : g E L 1 6 CTI = c 2 .

e'

w

and

I n t h e g e n e r a l c a s e , p a r t b ) c a n be d e r i v e d from c o r o l l a r y 5.4. ( a ) . I n d e e d , i f e0 E A 2 we h a v e : '(x) = b ( x ) + h ( x ) , where b(x)

=

log k(x)

i s a bounded f u n c t i o n , and h ( x ) = c1 l o g Mf(x) Ihll* 6 C i n d e p e n d e n t o f

- 6 l o g MP,(X) i s a B.M.O. f u n c t i o n w i t h

@. Therefore inf {ll@-gll* : R e L

W

1 5 IIhll* 5 C

=

C2.

IJ

A n o t h e r way t o f o r m u l a t e t h e o r e m 5 . 1 4 i s t h e f o l l o w i n g : COROLLARY 5 . 1 5 .

(Garnett-Jones theorem).

A(f) = sup{h > 0 : e A f e Az}.

For

f

f

B.M.O.,

e Lw}

-

A(f)-l.

we w r i t e

Then

d i s t ( f , L m ) = inf{I(f-gl(, : g B.M.O.

P r o o f : C, and C 2 w i l l have t h e same meaning a s i n t h e o r e m 5 . 1 4 , and " d i s t " w i l l mean t h e d i s t a n c e i n B . M . O . =

C1,

a ) L e t X = C l / d i s t ( f , L m ) . Then d i s t ( X f , L m ) = X d i s t ( f , L m ) = s o t h a t , a c c o r d i n g t o p a r t a ) o f t h e t h e o r e m , e Af e A 2 o r ,

i n the notation introduced i n the statement: -1 t h e same a s s a y i n g : A ( f ) - ' 5 C, dist(f,Lw).

X 5 A(f),

which i s

446

I V . W E I G H T E D NORM INEQUALITIES

b ) Let

0 < A <

e Af e A 2 ,

Then

A(f).

so t h a t according t o

p a r t b) o f t h e theorem: A d i s t (f,Lm) = d i s t ( h f , L m ) 5 C2.

d i s t (f,Lm) 5 C2 A ( f ) - l .

It f o l l o w s t h a t

0

We s h a l l f i n i s h t h e s e c t i o n by p r e s e n t i n g a n e x t r a p o l a t i c n

The k e y t o t h i s e x t r a p o l a t i o n t h e o r e m theoren for the classes A P' i s the following. LEMMA 5 . 1 6 . S u p p o s e w E A t o r a g i v e n 1 < p < m , and l e t

<=

0 < t

P

1.

Consider t h e operator

S ( u ) = (M(lu1 ' I t w) . w - ' ) ~ Then: -

a) S

LPtIt(w)

i s b o u n d ed i n

t h e A - c o n s t a n t for w . P b) For e v e r y f u n c t i o n

u

2

w i t h norm d e p e n d i n g o n l y u p o n

LP'lt(w),

belonging t o

0

the pair

(uw, S ( u ) w ) b e l o n g s t o t h e c l a s s A r of p a i r s of w e i g h t s for r = ( 1 - t ) p + t , w i t h an Ar-constant f o r t h e p a i r which depends exA - c o n s t a n t for P

c l u s i v e l y upon t h e

Proof: a)

w'-~'

j

~ ( u ) ~ ' / t =w E A

=

A -constant for P

b ) The c a s e S ( u ) w = M(uw).

w. t

=

P

j

M(lull/tw)p'w1-~'

, with 1

A

P'

Q,

5 c j l u l ~ ' / t w since

- c o n s t a n t d e p e n d i n g o n l y upon t h e

i s c l e a r , since then

Now l e t cube

w.

0 < t < 1. Observe t h a t we h a v e t o e s t i m a t e :

r-1

=

r

=

1

and

(1-t)(p-1).

For a

On t h e s e c o n d f a c t o r we u s e t h e f a c t t h a t

On t h e f i r s t , a f t e r w r i t i n g

w

=

w ~ w ' - ~ , we u s e I-IBlder's i n e q u a l i t y

IV.5. FACTORIZATION AND EXTRAPOLATION

with exponents

if

C

l/t

and

(l/t)l

=

l/(l-t).

447

We get:

A -constant f o r w. P Combining lemma 5.16 with lemma 5.1, we obtain is the

LEMMA 5.17.

ant

Let W, p

b e a s i n lemma 5 . 1 6 .

t

Then, f o r every

u e LP’/~(~), t h e r e e x i s t s a n o n - n e g a t i v e

non-ne a t i v e

such t h a t :

a) u(x)

<= v(x)

for a.e.

lI~llp‘,t,w

b) IIVllpl/t,w 2 2 V.W e A,

C)

for

d e p e n d i n g o n Z y on t h e

x

r = (1-t)p+t, w i t h a n A - c o n s t a n t f o r w. P

Ar-constant

Proof: It is enough to apply lemma 5.1 to the operator S defined in lemma 5.16. Given u 2 0 belonging to LU’It(w), we have v satisfying a) and b) above and, besides, Sv(x) 5 C v(x) with C = 2 USll. We know that (vw, S(v)w) e A, with an A,-constant that depends only upon the A -constant for w. Combining this P with the fact that Sv <= Cv, with C depending only upon the A P -constant for w , we obtain v.w E A r with an A,-constant depending only on the A -constant for w. P The extrapolation theorem will follow immediately from the next lemma, which is little more than a reformulation of lemma 5.17. 1 < p < w , 1 2 r < m , r # p. D e n o t e b y s -1 -- 11 - -r1 . Let w e Ap. Then, for S P i n L’(w), there exists a v 2 o L’(w) such

LEMMA 5.18. S u p p o s e

t h e exponent g i v e n by

u

every that: -

2 o

a) u(x)

v(x)

b ) Nvlls,w

2 c lI~NS,w9 __ with C

c)

If

A,.-constants

for

~

w.

r < p,

f o r a.e.

vw e A,,

x

and i f

an a b s o l u t e c o n s t a n t p < r,

v-?

E

t h a t , i n b o t h c a s e s , d e p e n d onZy o n t h e

with

AT,

A -constant P

I V . WEIGHTED NORM INEQUALITIES

448

P r o o f : 1 ) I f r < p , t h e n r = ( 1 - t ) p + t f o r some 0 < t <= 1 , and l / s = 1 - ( r / p ) = 1 - (1 - t + ( t / p ) ) = t / p ' , so t h a t s = p ' / t . T h e r e f o r e , i n t h i s c a s e , t h e r e s u l t c o i n c i d e s w i t h lemma 5 . 1 7 . 2 ) L e t p < r . Then l / s = ( r / p ) - 1 = ( r - p ) / p , that is: s = p/(r-p). Now r ' c p ' , and i f we w r i t e l / s * = 11 ; ( r ' / p ' ) / ,

we s e e t h a t s * = s ( r - 1 ) - S i n c e w e A we have wimp e A P' P' We a r e i n a p o s i t i o n t o a p p l y t h e f i r s t p a r t o f thelemma a l r e a d y p r o v e d , w i t h p ' i n p l a c e o f p , r ' i n p l a c e o f r , and w l - p ' i n p l a c e of w. I f u 2 0 b e l o n g s t o Ls(w), we s h a l l have s/s* wp'/s* E L'*(W'-~'). Therefore, there w i l l e x i s t uo = u

-

e L'*(W'-~')

e Arl.

w i t h uo 2 v o , I f we w r i t e vo

IvoII = vs / s

5

C Iu,II

wP'/S*,

and, besides, t h a t i s , i f we

= v s * / s , - P ' / S , we s e e t h a t define 0 and b e s i d e s : v - l w = v 0- s * / s w l + ( p ' / s ) -- vo1 - r w ( r = (vow - W P - 1 ) ) 1 - r = (vow1 -P

Here i s , f i n a l l y , t h e e x t r a p o l a t i o n theorem

Let T b e a s u b Z i n e a r o p e r a t o r . L e t 1 5 r < m, 1 < p < m . S u p p o s e t h a t T i s bounded i n Lr(w) ( r e s p e c t i v e Z y of weak t y p e ( r , r ) w i t h r e s p e c t t o w) f o r e v e r y w e i g h t w e A r , w i t h a norm t h a t d e p e n d s onZy upon t h e A r - c o n s t a n t f o r w. Then, f o r every w E A T i s bounded i n Lp(w) f r e s p e c t i v e Z y , T 2 P' u t weak-type (p,p) w i t h r e s p e c t t o wl, w i t h a norm t h a t d e p e n d s o n l y upon t h e A - c o n s t a n t for w . P THEOREM 5 . 1 9 .

P r o o f : We s t a r t by d e a l i n g w i t h t h e c a s e i n which be o f s t r o n g t y p e . T h e r e a r e two p o s s i b i l i t i e s .

T

i s assumed t o

1 ) L e t r < p . Take w e A We a r e g o i n g t o a p p l y lemma 518, P' o b s e r v i n g t h a t , w i t h t h e n o t a t i o n u s e d i n t h a t lemma, l / s = 1 - ( J p ) , s o t h a t s = ( p / r ) ' , We have = j / r f ( x ) Iru(x)w(x)dx

Ls(w), s u c h t h a t u 2 0 a n d I I U ~ ~ =, 1~ . I f we t a k e a s s o c i a t e d t o u a s i n lemma 5 . 1 8 , we s h a l l h a v e : pw(x)dx) r / p 5- J I T f ( x ) l r v ( x ) w ( x ) d x 5

I V . 5 . FACTORIZATION AND EXTRAPOLATION

5

C Ilf(x)l'

v(x)w(x)dx

449

<=

5 C ( j ( l f ( x ) I r 1 p / r w ( x ) d x ) r / P ( j v ( ~ ) ( ~ / ~ ) ' w ( x ) 11 d x( )p / r ) ' =

c

IIvIIs,w

I I f l l p ,rw

I I f l l p ,rw

5 C'

Observe t h a t t h e c o n s t a n t

=

d e p e n d s o n l y on t h e

C'

A -constant f o r

P

W.

2 ) Now l e t p < r . Take from H 8 l d e r ' s i n e q u a l i t y t h a t

I f 0 < a < 1 , it follows P' (Ifla 5 (/Iflu-')"(\ ua'('-") 1 ,

w e A

and we know t h a t , a c t u a l l y , t h e l e f t hand s i d e e q u a l s t h e minimum o f t h e r i g h t hand s i d e o v e r a l l p o s s i b l e c h o i c e s o f

u,

that is: u-1

w ( x ) dx

Observe t h a t , w i t h t h e n o t a t i o n o f lemma 5.18 l / s = ( r / p ) - 1 = = (r-p)/p, s o t h a t s = p / ( r - p ) . T h u s , g i v e n f e Lp(w), w e h a v e

I ll p ,w

j l f ( x ) l r u ( x ) - ' w(x)dx

=

f

f o r some u 2 0 w i t h I ( U I I ~ , ~ = 1 . Now c o n s i d e r t h e f u n c t i o n a s s o c i a t e d t o u a s i n lemma 5 . 1 8 . We c a n w r i t e : ( j i w x ) I P ~ ( X ) ~ X )=~ II/ iPT f i r n p / r , w

5 ~ ~ T f ( x ) I r v ( x ) - l w ( x ) d x . ~ ~ v [ s2, wC 5

C I I f ( x ) Iru(x)-'w(x)dx

Observe t h a t t h e c o n s t a n t

=

I

v

2 If(x)lrv(x)-'w(x)dx

5

CIIfll'P , W A -constant for w. P i s o f weak t y p e , w e o n l y n e e d a

d e p e n d s o n l y on t h e

C

When we j u s t assume t h a t

T

minor m o d i f i c a t i o n i n t h e two a r g u m e n t s g i v e n a b o v e . For e x a m p l e , let

r < p

and

w e A

Given

P'

f E Lp(w),

E X = { x e Rn

Then

( W ( E ~ ) ) ' / =~ IIXE

/Ir

P,W

=

1

write,

: ITf(x)I > A )

XEX(x)u(x)w(x)dx

f o r each

X > 0:

I V . WEIGHTED NORM INEQUALITIES

450

f o r some u E Ls(w) s u c h t h a t c o r r e s p o n d i n g v , we h a v e : W ( E , ) ~ ' 5~

1

xE

x

u

2

0

(x)v(x)w(x)dx

6

I I U I I ~ , ~=

and

1.

Taking t h e

C A - ~J l f ( x ) I r v ( x ) w ( x ) d x

Thus w(EA) 5 C A-p ~ ~ f ~ and ~ ~ ,T w i s, o f weak-type ( p , p ) r e s p e c t t o w a s we wanted t o p r o v e . The same m o d i f i c a t i o n a p p l i e s when

p

c

2

with

0

r.

W e may s t r e n g t h e n t h e o r e m 5.19 by combining i t w i t h t h e M a r c i n k i e w i c z i n t e r p o l a t i o n theorem ( t h e o r e m 2 . 1 1 i n C h a p t e r 1 1 ) . We g e t t h e following COROLLARY 5 . 2 0 . Let T b e a s u b l i n e a r o p e r a t o r . Let 1 5 r < m, 1 < p < S u p p o s e t h a t T i s of w e a k - t y p e (r,r) with respect t o

-.

w f o r e v e r y w e i g h t w e A,, w i t h a norm t h a t d e p e n d s o n l y u p o n t h e A r - c o n s t a n t f o r w. T h e n , for e v e r y w e A T i s bounded i n P' Lp(w), w i t h a norm t h a t d e p e n d s o n l y u p o n t h e A - c o n s t a n t f o r w .

P

Proof: Let w e A Then we know ( t h e o r e m 2 . 6 ) t h a t w e A for P' P-E some E > 0 . The E and t h e A - c o n s t a n t f o r w depend o n l y P-E f o r e v e r y q > p . Now, upon t h e A - c o n s t a n t f o r w . A l s o w E A 9 P a p p l y i n g theorem 5 . 1 9 , we g e t t h a t T i s o f w e a k - t y p e ( P - E , P - E ) w i t h r e s p e c t t o w and a l s o o f weak t y p e ( q , q ) w i t h r e s p e c t t o w f o r e a c h q > p . The norms depend o n l y upon t h e A - c o n s t a n t P f o r w . Then, by a p p l y i n g t h e M a r c i n k i e w i c z i n t e r p o l a t i o n t h e o r e m , we c o n c l u d e t h a t T i s o f s t r o n g t y p e ( p , p ) w i t h r e s p e c t t o w ( t h a t i s , bounded i n L p ( w ) ) , w i t h norm d e p e n d i n g o n l y upon t h e A -constant f o r w. 0 P 6 . WEIGHTS IN PRODUCT SPACES

The o p e r a t o r s c o n s i d e r e d s o f a r i n t h i s c h a p t e r , a s w e l l a s i n p r e v i o u s c h a p t e r s , were a s s o c i a t e d t o t h e o n e - p a r a m e t e r f a m i l y o f dilations: x w t x , t > 0 , i n the sense t h a t the conditions sat i s f i e d by them w e r e n o t a f f e c t e d b y s u c h a change o f s c a l e i n B n . I n some c a s e s , t h e o p e r a t o r s t h e m s e l v e s were i n v a r i a n t u n d e r t h e s e dilations, i.e. T(ft)

=

(T f ) t

where

ft(x) = f(tx),

t > 0

( e . g . , t h e H a r d y - L i t t l e w o o d maximal f u n c t i o n o r a homogeneous s i n -

IV.6. WEIGHTS IN PRODUCT SPACES

451

gular integral operator). The classes of weights associated to such operators were, accordingly, dilation invariant: If w E A then P' also wt E A with the same Ap constant as w. P Now, we shall study operators and weights which are invariant under the n-parameter dilations in Rn, defined by X

6(x)

h

( 6 1 x1,62 X2,"',6,

=

Xn)

Here are the simplest examples of

with 6 = (61,62,...,6n) e (R,)". such operators:

i) The strong maximal function

where R denotes the family of bounded n-dimensional intervals R = [al,bl] x [a2,b21 x...x [an,bnl. ii) The multiple Hilbert transform, defined by

H f(x)

I'

1im

=

f(Y)dY

n

n or, in terms of Fourier transforms: = (-iIn s i p ( S ~ E ~ . . . ~ ," ~, (! s J .

xn-Yn) (ff f)^(S)

=

iii) The partial sum operators

sI

f(x)

=

J,~ ( S Ie21rix.S dS

where 1 is a (not necessarily bounded) n-dimensional interval. These are the analogues in Rn of the partial sum operators in R studied in 3.12. For an arbitrary function f and 6 = ( 6 1 ,62,...,6n) E ( R , ) n , we shall denote by f6 (x)

=

f6

the function

f(6(x))

=

f(6,X1,62X2 ,..., 6,Xn)

Then the following identities hold for the operators just described: M,(f6)

=

ff(f6)

(Msf)6;

=

(H f)&;

S6(Il(f 6)

=

(SIf) 6

The boundedness of these operators in Lp(Rn) follows by writing them as products o f one-dimensional operators. Given an operator T acting on functions in R, we denote by Tj, j=1,2 ,..., n, the operator defined on functions in Rn by letting T act on the j-th variable while keeping the remaining variables fixed, namely Tjf(x)

=

T(f(xl ,x2,. . . , X ~ - ~ , . , ,. X ~. .+,xn))(xj) ~

IV. WEIGHTED NORM INEQUALITIES

452

If T = Tm is a multiplier operator, with to verify that (TJf)A(S1,S2,...,Sn)

=

p(S1,S2,

m

E

Lm(R),

it is easy

...,c;,)m(Sj)

In particular, we have Hf(x) SIf(X)

H 1O H2

=

0 . .

.oHnf(x)

SI 1 0SIZ0.. 2

s;

f(x) n where H is the (ordinary) Hilbert transform and I = I 1XI 2x...xIn, with I intervals in R . Also, if M denotes the one-dimensional j Hardy-Littlewood operator, then =

. O

1

J

and a moment's thought shows that MS f(x) 5 M 1

0

M2

0 . .

.o

Mnf(x)

On the other hand, we make the following simple observation based on Fubini's theorem: If T is a bounded operator in Lp(R) with norm

pdx and similarly for T2,. ..,Tn. From the identities for ff, SI and the inequality for MS previously displayed, it is now evidentthat, f o r all 1 < p < 00 and f E Lp(Rn), we have IIMSf

(Ip

5

cp I1 f l,

cp u f Ilp llSIfllp 5 cp Ilfllp IIHfllp 5

(Cp

independent of

I)

There is no weak type (1,l) inequality for these operators. For 1 , we have example, letting f(x1,x2) = P1(x1)P2(x2) = IT (Xl+l)(x;+l) x1 x2 , which is not in weak-L1 (R 2 ) . By using Hf(x,,x2) = 2 2 77 (x,+l) (x2+1) the estimates near L (see Chapter 11, 2 . 3 )

'

for the Hardy-Littlewood maximal function one can prove that MS is bounded from

IV.6. WEIGHTS IN PRODUCT SPACES

453

L(1og L)"-l (R") to weak-L1(R") . However, we are rather interested in Lp estimates, and our next objective will be the analogues in Lp(w), for suitable weights w(x) in Rn, o f the above inequalities. Our previous experience with A weights suggets that the P condition similar t o Ap with n-dimensional intervals instead o f cubes must be necessary, while the method used in the unweighted case indicates that we can easily deal with weights w(x) such that w(x1,x2 ,..., X ~ - ~ , . ,,,..., uniformly in X ~ + xn) e A (B) P to give a precise meaning (x1,x2,. . . , X ~ - ~ , X ~ + ~ , . . . , XIn ~ ) order . to this, we establish the following: NOTATION: G i v e n

& Rn, n 2 1

v(x) '> 0

the A -constant f o r -

P

v

(which can be

we d e n o t e by

+a),

I.[

( R")

namely

The last definition is, of course, motivated by (2.15). As usual, Q stands for an arbitrary cube (interval if n = 1) in Rn. DEFINITION 6.1. Let s a y t h a t w e A* i-f (R+)n,

i.e.9

P

sup 6

we

be a w e i g h t i n Rn. For 1 5 p 5 0, w6 E A u n i f o r m l y i n 6 = (61,62,...,6n) e

W(X)

P

[w6]Ap(Rn)

<

m'

Since dilations by 6 = (1S~,6~,...,6,) e (R+)n transform cubes in arbitrary n-dimensional intervals, a change of variables proves that, for 1 < p < m, w E A* if and only if P

and similarly for Thus, A* is the P nal intervals.

p = 1,

Ap

p = m (see theorems 6.5 and 6.7 below). class naturally associated with n-dimensio-

THEOREM 6 . 2 . G i v e n a w e i g h t

W(X)

following statements are equivalent:

a) w e A* P

Rn

1 < p <

m,

the

IV. WEIGHTED NORM INEQUALITIES

454

b) T h e r e e x i s t s

C > 0

j = 1,2,

s u c h t h a t , for e a c h

...,n

have -

[ w ( x ~ ~ x ~ ~ ~ ~ ~ ~ x j - ~ Y ~ #< x cj + ~ ~ ~ . ~ ~ ~ ~ ~ ] ~ f o r almost every C)

MS

(x1,x2,.. . , X ~ - ~ , X ~.+.,xn) ~ , . e Rn- 1

i s a bounded o p e r a t o r i n

Lp(w)

d) The p a r t i a l sum o p e r a t o r 4 (SI) a r e u n i f o r m l y bounded i n LP(W) e) tl

is a bounded o p e r a t o r i n

Lp(w)

Proof: For notational simplicity, we shall prove the theorem f o r n = 2 , since this case contains all the essential difficulties of the general situation. The mutual implications of statements a)-e) will be established according to the following diagram

(a) implies (b): Given R we define

w(x1,x2)

Now, given another interval apply Hijlder's inequality

where R = I x J I = (xl-6, x1+6)

and and

6

+

P'

for each interval

mJ(xl)

is the A*-constant for P 0, we obtain for

a.e.

x1

over

w. E

I

J

in

and

Letting

B

sup mJ(xl)P, and it is enough to consider J the countable family of intervals J with rational endpoints. This proves [w(xl, .)IA 5 C (a.e. x 1 e R), and similarly, [w(xl

'.)lAp(R)

A*

I, we average

5

mJ(xl)P

But

C

in

[W(*,X2)]Ap(R)

5

=

P

(a.e* x2 e R).

IV.6. WEIGHTS IN PRODUCT SPACES

455

(b) implies (c): By Muckenhoupt’s theorem 2.8 applied to the maximal operator M in R, the hypothesis implies

and

for almost every x, and x2 in R. Now, both inequalities consecutively with g = MLf, that MSf 5 M1(M2f).

f2 0

(c) implies [a): For every function > (=

1

(

f)X,.

taking into account

R e R,

and

MSf

Thus, from (c) we get

Taking f = w - l/(p-l)xR ( o r an increasing sequence of bounded fun2 tions in R converging to w -l’(p-l)) it follows that w E A* P’

(b) implies ( d ) : It is proved exactly as (b) -4 2 f), and I = I1 x 12, we know that SIf = S 1I (SI 1 2 w together with Corollary 3.12 imply that both bounded in Lp(w) with norms independent of 1 1 ,

I

(d) implies (e): Take Then Hf and therefore

=

=

-(S1f

[O,m)

x [O,a)

S-If)

+

(Sif

+

IIHfllp,w 5 C Ilfllp,w

?

and +

s

for all

=

(c). Given the conditions on S;l

s:,

and

are

12. [O,m)

x

(-m,O].

if) f

f

LP(W).

(e) implies (a): Given R = [a,a+h] x [b, b+l], consider the rectangle fi = [a-h,a] x [b-L,b] which is symmetric to R with respect to the common vertex (a,b). Take f > 0 supported in R. Then, for every x = (xl,x,) e

Using the fact that (6.3)

H

is bounded in

(&i, flp 1-

Now, we choose first

1?

f(x)

=

w 5

c

1

Lp(w)

R

w(x) - ” ( p - ’ )

lflP w in

(6.3).

This gives

IV. WEIGHTED NORM INEQUALITIES

456

But we can also interchange the roles of R and then take f(x) = XR(X). This gives w(R) 5 C w(E) in (6.4), proves the condition A* for w(x). 0

in ( h . 3 ) , and which, inserted

P

One of the consequences of the preceding theorem is the equivalence between A; and the one-dimensional A conditions on each P variable (uniformly in the remaining variables). It is natural to ask whether such an equivalence continues to hold in the extremal cases: p = 1, p = m . The next two theorems will give affirmative answers in both cases, while yielding some other equivalent characte rizations of At and Acf, (remember that A; and were already defined in (6.1)).

At

THEOREM 6.5. The f o Z Z o w i n g c o n d i t i o n s f o r a w e i g h t

& Rn

W(X)

are equivalent:

i) w 'A;

c 2 c

ii) M~ w(x) 5 iii) ~j w(x)

f o r a.e.

x

w(x)

f o r a.e.

x e R"

iv) T h e o p e r a t o r s t o weak-Ll (w) .

Mj,

Observe that iii) means for almost every j = 1,2 ,...,n.

R"

w(x)

j

=

1 ,2,.

E

. . ,n

[w(xl,x2

and

j

a r e bounded f r o m

1

L1 (w)

,...,X~-~,.,X~+,,...,X n)] A,(W n- 1

~ + ~ xn) e R (x1,x2,..., X ~ - , , X ,...,

Proof: By a change o f variables, the definition of reformulated as

1 -

I,z,...,n

=


and

A;

can be

5 C (ess. inf. w(x))

for all R e R x e P The equivalence between this and ii) can be seen exactly as in the case of A 1 weights (see the argument after (1.8) and ( 1 . 9 ) ) . IRI

R

w(x)dx

To show that ii) and iii) are equivalent, we s i m p l y use the ma jorizations ~jw(x) 5 ~ ~ w ( x )5 hi1

0

~2

0 ,

. . o

~"w(x)

(a.e. x

R")

E

The second one is already known, and the first one follows by a step by step repetition o f the argument we used to prove (a)

(b)

IV.6. WEIGHTS IN PRODUCT SPACES

in theorem 6.2, where we take now

p

=

457

1.

Let us now prove the equivalence of iii) and iv). We apply the inequality (2.14) of Chapter I 1 to arbitrary functions f(x1,x2, ...,xn) and weights w(x1,x2, . . . ,xn) considered as functions of x, only, with the remaining variables fixed. By integrating then with respect to x2,x3, ...,xn, we get

and similarly for M2,M3,. . . ,Mn. This shows that iii) implies iv) . Conversely, if iv) holds, we apply the weak type inequality for M1 to a function of the form f(x) = g(xl)XE(F), with g 0 and E a measurable subset of Rn-’ (we are denoting x = (xl,x) E Rn, with x1 e R, x E: Rn-l). Then, M1f(x) = XE(?) Mg(x,), and we get

,

-

eR :Mg (x ) > A 1

and

w(xl ,x)dxlI dx

I

Since E is arbitrary, this implies that, f o r each X > 0, the inequality

g 2 0

holds f o r a.e. 2 e Rn-l. Let A be the set of points ? e Rn-l for which (6.6) holds for every function g = X I , where I is any interval with rational endpoints, and every A rational. Let B

=

{x of

=

(xl,x)

e

-

Rn : x

e

A

and

x1

is a Lebesgue point

-

w(.,x)l. -

Then, clearly, IRn\BI = 0. Now, if (xl,x) e B and x 1 is an interior point of an interval I with rational endpoints, we take intervals { I ~ I ~ = with , rational endpoints such that x 1 E I k C I and lIkl + 0. Then we apply (6.6) to gk(yl) = XI (yl). Since k > IkI 111 for all y 1 e I, we obtain -

/

1 -

III

the supremum f o r all intervals such as

I

I V . WEIGHTED NORM INEQUALITIES

458

-

-

M 1w(xl,x) 5 C w(xl,x) The same argument works for

for all

M2,M3, ...,Mn,

-

(x,,x) e B and iii) is proved.

0

Next, we shall establish several equivalent formulations o f the condition A: THEOREM 6 . 7 . The f o l l o w i n g c o n d i t i o n s f o r a w e i g h t

i n Rn -

w(x)

are equivaZent:

i ) w e c

R e R

iii) For e v e r y

and e v e r y m e a s u r a b l e s e t

f o r some f i x e d c o n s t a n t s

iv) T h e r e e z i s t a whenever we have A C R V) T h e r e e z i s t

I IX

E:

R :

C,

E

> 0

A C R

depending o n l y on

w.

and f.

R

CY

W(X)

a 5-

I RI

s a t i s f i e s a reverse Hzlder's inequality over a l l n - d i m e n s i o n a l i n t e r v a Z s R, i.e., f o r some c o n s t a n t s C,

vi) w

E

> 0

Proof: By a change o f variables, each one of the statements ii)-vi) is equivalent to saying that the weights Iw6 16e(R+)n satisfy the corresponding condition with cubes Q instead of intervals R E R , uniformly in 6. Taking into account the different characterizations of Am (see 2.13 and 2 . 1 5 ) these statements are equivalent to [w 6]Am(Rn) 5 Constant (for all 6 e (R+)")

IV.6. WEIGHTS IN PRODUCT SPACES

459

which is the definitiorsof w e A.: Thus, all the conditions i)-vi) are equivalent. On the other hand, we know that 6 < C implies [w ]Ap(Rnl 5 B for some p = p (C,n) < m [w A,(R"I and B = B(C,n). Therefore, i) implies vii), and the converse is obvious because [ 2 [ . 1 ( ~ n ) by Jensen's inequality. Finally, we prove the equivalence between vii) and viii) by observing that each of them is equivalent to the following statement:

.

C

(*) There e x i s t

p

such t h a t

i m

Indeed, vii) \-a ( * ) by theorem 6.2, while ( * ) lary 2 . 1 3 applied t o weights in R. 0

viii) by corol-

A consequerice of the characterization of :A provided by theorem 6.7.(vi) is that, if 1 < p 5 m, w e A* implies P W e A* for some E > 0. This follows also from the correspondP ing property for A weights and definition 6.1. We shall now use P this fact in order to describe the classes A* in terms of maximal P functions. Given u(x) > 0 and v(x) > 0, we shall write: u(x) v(x) to indicate that both u ( x ) and v(x) are L" functions.

'+'

-

m

THEOREM 6.8. Let a) w e A; g(x)

function

W(X)

b) For A* P

-

W(X)

<

a.e.

m

Then

i f a n d o n l y i f t h e r e e x i s t a ZocaZZy i n t e g r a b l e and a p o s i t i v e number 6 < 1 s u c h t h a t

-

(M'g(x))'

1 < p <

- ... -

(PPg

XI l 6

we h a v e

00

= A;(A;)'-~

(Mzg(x ) 6

=

w, e A:}

{w0 Wi-p

1 locaZZy i n t e g r a b l e f u n c t i o n s

M1g(x) M'h(x)

-

g(x)

M2g(x) M2h(x)

- ... ... -

h(x) Mng(x) Mnh(x)

and some

such t h a t

460

IV. WEIGHTED NORM INEQUALITIES

o n l y and P r o o f : a ) By c o n s i d e r i n g ( M j g ( x ) ) & a s a f u n c t i o n o f x j u s i n g ( 3 . 4 ) i n C h a p t e r 11, i t f o l l o w s t h a t Mj((Mjg)')(x) 5 < C , ( M ~ ~ ( X ) a) .~e . Thus, i f w(x) (Mjg(~))~ for a l l j=1,2 n , t h e n Mjw(x) 5 C w(x) a . e . ( j = l , 2 , . . . ,n ) , which i s

-

,...,

e q u i v a l e n t t o w e A;. w l + € e A: f o r some

C o n v e r s e l y , assume t h a t w e A;. Then which means Mj(wl+')(x) - w ( x ) ~ + ~ 1+E 1 and 6 = we n . L e t t i n g g ( x ) = w(x) for a l l j=1,2 1+ E (MJg(x))' ( j = 1 , 2 , ...,n ) a s d e s i r e d . have w(x)

,...,

-

E

> 0,

b) We j u s t n e e d t o p r o v e t h e f a c t o r i z a t i o n r e s u l t : A* = P A;(A;)l-P. But t h i s i s a p a r t i c u l a r c a s e o f theorem 5 . 2 , s i n c e A Yt a r e t h e c l a s s e s o f w e i g h t s a s s o c i a t e d t o t h e s t r o n g maximal o p e r a t o r , i . e . A; = Wp (MS), 1 5 p a, ( b y 6 . 2 a n d 6 . 5 ) and A* = P' =

The r e a d e r c a n e a s i l y c h e c k t h a t t h e s t a t e m e n t : W(X)

-

(M1gl(x))6

0 < 6 < 1

f o r some

-

(M2g,(x)l6

- ... -

(Mngn(x)) 6

and some ( d i f f e r e n t ) l o c a l l y i n t e g r a b l e f u n c -

which i s a p p a r e n t l y weaker t h a n t h e one i n tions gl,g2, ...,g,, However, we p a r t a ) of t h e o r e m 6 . 8 , i s a l s o e q u i v a l e n t t o w E: A;. do n o t know i f

(M,f)&

i s , f o r every

f e Lloc(Rn)

and

0 < 6 < 1 , a w e i g h t i n A: ( t h e converse i s t r u e due t o t h e r e v e r s e H i j l d e r ' s i n e q u a l i t y : w e A; i m p l i e s w(x) (MSf(x)) 6 w i t h f = w l + E , 6 = - and E > 0 ) . T h u s , t h e p r e c e d i n g theorem 1+ E d o e s n o t g i v e an e a s y method t o p r o d u c e "enough" A; weights i n

-

Lp(Rn), b u t we c a n f i n d s u c h a method by i m i t a t i n g t h e argument used f o r t h e f a c t o r i z a t i o n t h e o r e m , namely: Given

where

Mg

1 < p <

u e Lp(Rn),

denotes, as usual, the

e q u a l s t w i c e t h e norm o f in

Lp

MS

in

we t a k e

m,

q =

k-th i t e r a t e of Lq.

6

and d e f i n e

MS,

and

C

Then, t h e s e r i e s converges

and u(x)

U e A;

with

5

U(X)

A;-constant

d e p e n d i n g o n l y on

p

L i k e w i s e , t h e whole p r o c e s s t h a t l e d t o t h e e x t r a p o l a t i o n t h e o r e m

IV.6. WEIGHTS IN PRODUCT SPACES

.

applies mutatis mutandis to weights in state the final result

461

We limit ourselves to

THEOREM 6.9. S u p p o s e t h a t we r e p l a c e

statements

Ap & A*P e v e r y w h e r e i n t h e o f t h e o r e m 5 . 1 9 and c o r o l l a r y 5 . 2 0 . T h e n , t h e r e s u l t i n g

statements are a l s o true.

The fact that, not only the extrapolation theorem, but also the lemmas we used in order to establish it can be extended to the product setting, will be important for our next result. What we shall need is the particular case of Lemma 5 . 1 8 corresponding to r = 1, which we now recall: LEMMA 6 . 1 0 . Let w e A * 1 < p < m. Then, f o r every P' Lpl(w), t h e r e e x i s t s v 2 0 Lpl(w) s u c h t h a t : u(x) 5 v(x) with

a.e.9

IIvllpl,w 5

AT-constant independent o f

c

u

lIullp,,w a n d vw

>0

&

E A;

u.

The last objective in this section will be the study of the strong maximal function with respect to a regular Borel measure in Rn, which is defined by

This is the analogue of the operator Mu studied in Chapter 11, where only cubes (instead of arbitrary n-dimensional intervals) were considered. In that case, Mp was shown to be bounded in Lp(p), 1 c p 5 m, provided that p is doubling: p ( Q 2 ) 5 C p ( Q ) f o r every cube Q . In our present situation, a trivial result is that is bounded in Lp(p) f o r 1 < p 5 m when p is a product MS,!J measure: p = p l @ p 2 @ . . . P pn for some regular Borel measures p 1 , p 2 , ...,pn in R. In fact, in this case we can repeat the argument given for Lebesgue measure, since f(x) 5 MA

M 2 o...O Mn f(x) 1 p2 % and M p , is bounded in Lp(R;pj), 1 < p < m, for each j=l,Z,.?.,n. (This has been proved in Chapter 11, theorem 2.6 when the p . ' s are doubling measures, but the doubling condition is not 1 necessary for one-dimensional measures; see 7.4 in Chapter 11). MS,u

A more significant result will be obtained for measures

which are not necessarily products of measures in

R, namely:

p

IV. WEIGHTED NORM INEQUALITIES

462

*

dp(x) = w(x)dx with w e Am. This will be based on covering arguments for the family R. Since this technique is new for us, we shall establish first the general result which allows to pass from covering lemmas to boundedness properties of maximal functions: LEMMA 6 . 1 1 . G i v e n a b a s i s 8 ( i . e . , a c o l l e c t i o n of o p e n s e t s ) i n Rn, and a measure p s u c h t h a t 0 < p ( B ) < m f o r a l l B e 8 , suppose t h a t , from any f i n i t e sequence s u b s e q u e n c e {iikj s u c h t h a t

f o r some f i x e d c o n s t a n t s

C,

{BjIC 8

we c a n s e l e c t a

1 < q <

C' > 0

m.

Then, the

maximal o p e r a t o r

is of weak t y p e q' < P ' " .

(q',q')

with respect t o

p

and bounded i n

Lp(p)

for

Proof: Given

f(x) 2 0 EX

Every and

x

EX

u

E

EX

in

= {X e

Lq' (p)

and

X > 0, set

Bn : M g Y P f(x)

belongs to some

B

e

> XI

8 such that

will b e covered by a countable family

EY = l
satisfying ( 6 . 1 2 )

and select from

.

IBjIlcjLN

I,

f

dv > A P ( B ) ,

{B.}.

3 JZ1*

a subsequence

Let

fi.1 3

Then

letting N -+ m, the weak type inequality is proved. The strong type (p,p) for p > q' follows from Marcinkiewicz interpolation theorem.

0

It is remarkable that the converse of this lemma is also true (it was proved by CBrdoba [ Z ] ) , but we shall have no need of this fact. For results of this kind and generalizations we refer to M. de Guzm5n [Z] , Ch. 6 .

IV.6. WEIGHTS IN PRODUCT SPACES

THEOREM 6.13. Let

dP(x)

=

for all

w e A.:

w(x)dx

maximal o p e r a t o r w i t h r e s p e c t t o

463

~.r,

MS,p,

Then, the strong

is bounded i n

Lp(p)

p 5".

1

Proof: We know that w e A: for some r < m , and it will be enough to prove the inequalities (6.12) for r 5 q < m, since this will for q ' arbitrarily close imply the weak type (q',q') o f MS,Ll to 1 . Given IRjIlcjiN in R , we take R 1 = R1 and, once ." R 1 , R 2 , * - * s R k - 1 have been selected, we choose Rk to be the first interval in the given sequence (if any) such that at least half of it (in the sense of Lebesgue measure) is not covered by the intervals already selected, i.e.

-

- -

Now, we claim that

f

Indeed, let x -

E

U Rj,

there are two possibilites: either

J

uk

be the characteristic function of x

6,.

If

belongs to some

Rk, in which case MSf(x) = 1 , or x e R. for some R . which has 3 J been discarded in our selection process, in which case we must have 1 1 I R j n ( U 6,) 1 lRjl, and therefore biSf(x) NOW, since k we know that MS is bounded in Lr(w)

z.

which is the second inequality in (6.12). To prove the first inequa lity we use duality

o f unit norm. Since w e A ~ A:,C we apply for some u f: ~:'(w) lemma 6.10 to find v(x) u ( x ) such that 5 C 1 and vw e A; with A;-constant 5 C 2 (independent of u ) . We denote E k = B,\ (ilu . . . a,-,), so that { E L } is a disjoint scquence and lk,l i 1 l E k l . Then

\IVI~~,,~,

fi2u

< 2 C, -

C

&li

u

\ E l i [ ess.inf. (v(x)w(x))

xeRk

5 2 C2

C

li

1

Ek

v(x')w(x)dx

=

I V . WEIGHTED NORM INEQUALITIES

464

and t h i s completes t h e proof.

0

The p r o o f we h a v e g i v e n o f t h e l a s t t h e o r e m i s b a s e d on t h e i n Lp(w), w e A* For an a l t e r n a b o u n d e d n e s s p r o p e r t i e s o f MS P‘ t i v e ( a c t u a l l y , t h e o r i g i n a l ) a p p r o a c h w h i c h d o e s n o t u s e A*-weights P for p < m, s e e R . F e f f e r m a n [ Z ] . We o b s e r v e t h a t , c o n v e r s e l y , t h e f a c t t h a t MS i s b o u n d e d i n Lp(w) f o r w e A* 1 < p < m, P ’ i s a consequence o f theorem 6 . 1 3 t o g e t h e r w i t h t h e i m p l i c a t i o n :

w E A* P o f Ap imply

-

w E A* f o r some q < p . I n d e e d , j u s t as i n t h e c a s e 9 w e i g h t s , t h e c o n d i t i o n w E A* a n d H E l d e r ’ s i n e q u a l i t y 9

t h a t i s , MSf(x) 5 C {M (lf]q)(x)31’q, and t h e r i g h t hand s i d e S,W i s , a c c o r d i n g t o 6 . 1 3 , a bounded o p e r a t o r i n Lp(w), p > q .

7 . NOTES A N D FURTHER RESULTS

-

Weighted i n e q u a l i t i e s f o r o p e r a t o r s of t h e w i t h w e i g h t s o f t h e f o r m 1x1‘ were o b t a i n e d i n Aljart f r o m t h a t , t h e c l o s e s t a n t e c e d e n t s o f A P p r o b a b l y Rosenblum [l] a n d t h e F e f f e r m a n - S t e i n 7.1.

t y p e considered h e r e E.M. S t e i n [3]. weights are inequality 2.12

o f c h a p t e r 11. The b o u n d e d n e s s o f t h e maximal f u n c t i o n i n Lp(w) for w e A i s d u e t o Muckenhoupt [ l ] . The c o r r e s p o n d i n g r e s u l t P f o r t h e H i l b e r t t r a n s f o r m was e s t a b l i s h e d s h o r t l y a f t e r w a r d s i n

.

f l u n t , Muchenhoupt a n d Wheeden [ l ] The more s y s t e m a t i c a p p r o a c h o f Coifman a n d C . F e f f e r m a n [ l ] a l l o w e d them t o d e a l w i t h r e g u l a r s i n g u l a r i n t e g r a l o p e r a t o r s T i n Rn b y means o f t h e i n e q u a l i t y :

465

IV.7. NOTES AND FURTHER RESULTS

(w e Am, 1 < p < m). This approach is different from the one followed in section 3, which relies upon the sharp maximal function. More general weights for which ( * ) is true are found in E.T. Sawyer

121

*

The fact that

Am

=

u

n>1

Ap

is due independently

to

Muckenhoupt

[3] and Coifman and C. Feiferman [l].

All other characterizations of Am given in 2.13 and 2.15 are more o r less implicit in both papers except for 2.15 (iii) (the equivalence of arithmetic and geometric means over cubes) which was found by the authors while writing this monograph and, independently, by HrusEev [ l ] . The characterization of A 1 given in 2.16 is due to Coifman (see Coifman and Rochberg [ l ] where it appears in connection with B.L.O.); CBrdoba and C. Fefferman had previously proved that (Mf)& E Am if 6 < 1 , a result which they used to obtain inequalities like corollary 3 . 8 (see CBrdoba and C. Fefferman [ l ] ) . 7.2.- The aforementioned papers of Muckenhoupt and Coifman-Feffennan prompted a long series of results about weighted norm inequalities by many different authors. We cannot even attempt to give here a short account of these developments. As a sample of them, let us merely mention the following:

a) The l a s t part of section 3 (specially theorem 3.9) is taken from Kurtz and Wheeden [l], where other related results are also obtained. Weighted multiplier inequalities beyond the A classes P have been investigated in Muckenhoupt, Wheeden and W.-S. Young [l], 121 * b) Let mu(x) and Au(x) denote, respectively, the nontange; tial maximal function and the area function (defined in 7.14 of Chap ter 11) of a harmonic function u(x,t) in the upper half-space Ign+1 . The equivalence between the norms in LP(Rn) of both func+

tions, 0 < p < m , which was stated in chapter 111, 8.5, has the following extension: Let w B Am and 0 < p < a ; if Au E Lp(w) and u(x,t) is suitably normalized, then

See Gundy

and Wheeden [l].

IV. WEIGHTED NORM INEQUALITIES

466

c) The following weighted version of the Carleson-Hunt theorem on pointwise convergence of Fourier series (see Carleson [ 3 ] , Hunt [l]) was obtained in Hunt and W.-S. Young [l]: For a funtion f in R, let S*f(x) = sup ISIf(x) I ; then S* is a bounded operator on I Lp(w) if w E Ap and 1 < p < m. This is clearly a strong improvement of corollary 3.12 (a). 7.3.- Analogues of the contexts, such u s :

AP

theory have been developed in different

a) Spaces of homogeneous type, as defined in chapter 11, 7.11. F o r the maximal operator in this setting see A.P. CalderBn [6] (from which the proof we have given of the R.H.I., lemma 2.5, was taken)

.

For the Bergman projection in the unit ball of (cn and Jawerth [l] and other domains, weighted inequalities are obtained in Bekoll6 [ l ] .

b) Martingales. For regular martingales (an abstract generalization of the process of dividing each dyadic cube of R” into 2” dvadic subcubes) the R.H.I. can be obtained by essentially repeating the proof given for Rn, and a complete analogy was found by several authors (e.g. Izumisawa and Kazamaki [ l ] ) . For general martingales, the R.H.I. may fail (Uchiyama [l]), but the boundedness of the maximal operator in Lp(w) for w e A can nevertheless be proved by using P the ideas of section 4 ; see Jawerth [l]. c) Ergodic theory. See Atencia and de la Torre [l], and also Martin Reyes [l]. 7.4.- We shall state here some simple general facts about weighted inequalities for an arbitrary operator T. To simplify matters, suppose T is defined in L”(Rn), 1 < p < m , and is linear and (essentially) self-adjoint. Consider the class Wp(T) consisting of 0 such that T is a bounded operator in those weights w(x) Lp(w), and denote by Np(w) the least constant such that (/lTf(x)

Ip

for all (good enough)

w(x)dx)l’P f.

5 Np(w) (jIf(x)

Ip

w(x)dx)”P

Then:

a) Wp is a cone, and Np(.) satisfies the ultrametrical inequality: N (u+v) 5 max (N ( u ) , Np(w)). Also, N (Au) = ND(u) if P P P

A > 0.

b) Wpl(T)

=

W (T)’-’’, P

i.e.,

w

E

W

(T) P’

if and only if

IV.7.

u E W (T) P

w = u ' - ~ ' with Np

I

(w)

= Np

(u)

467

NOTES AND FURTHER RESULTS

and, i f t h i s is the case, then

*

c ) Wp(T) i s a l a t t i c e : u , v E W (T) i m p l i e s m a x ( u , v ) E W (T) P P a n d min ( u , v ) E W ( T ) . M o r e o v e r P N (max ( u , v ) ) 5 , ' I p m a x ( N p ( u ) , N p ( v ) ) P To p r o v e t h i s , u s e ( a ) t o g e t h e r w i t h u + v 5 max ( u , v ) 5 u + v . A similar i n e q u a l i t y holds f o r

Z'Ip

min ( u , v )

with

Z1/p'

instead of

(use b ) ) . d) I f

T

i s t r a n s l a t i o n i n v a r i a n t , so a r e

W

P

and

Np(.).

As

a c o n s e q u e n c e , t h e y a r e a l s o i n v a r i a n t by c o n v o l u t i o n w i t h a f u n c -

t i o n k(x) 2 0 u E W and k P < Np(U).

*

( o r a measure p 2 0) u(x) < m a . e . , then

i n the following sense: I f * u E WP a n d N p ( k * u ) 5

k

The same p r o p e r t i e s h o l d f o r p a i r s o f w e i g h t s . A s a n a p p l i c a t i o n o f p r o p e r t y ( c ) , when p r o v i n g a n i n e q u a l i t y f o r Ap w e i g h t s , i t i s e n o u g h t o c o n s i d e r w e i g h t s w w h i c h a r e b o u n d e d away f r o m 0 and m ( i . e . w and w-l belong t o L m ) , p r o v i d e d t h a t t h e c o n s t a n t i n t h e r e s u l t i n g i n e q u a l i t y depends o n l y upon t h e constant Ap for w, a n d n o t upon llwll, o r IIw- 1 \loo. T h i s o b s e r v a t i o n c a n b e used f o r i n s t a n c e i n theorem 3 . 1 . A p p l i c a t i o n s o f p r o p e r t y (d) w i l l be given below. 7.5.-

We showed i n c h a p t e r I t h a t i f t h e c o n j u g a t e f u n c t i o n o p e r a -

must b e a b s o l u t e l y t o r i s bounded i n L 2 ( p ) , t h e n t h e measure c o n t i n u o u s w i t h r e s p e c t t o Lebesgue measure. I n t h i s c h a p t e r , i n e q u a lities in = w(x)dx,

have been considered only f o r measures dp(x) = s o t h a t t h e f o l l o w i n g q u e s t i o n a r i s e s n a t u r a l l y : Let Lp(p)

R. j = 1 , 2 , . . . ,n d e n o t e t h e R i e s z t r a n s f o r m s i n 1' r e g u l a r m e a s u r e s p i n Rn do t h e i n e q u a l i t i e s

R";

f o r which

( o r t h e i r weak t y p e a n a l o g u e s ) h o l d ? The a n s w e r i s t h a t ( * ) i m p l i e s t h a t p i s a b s o l u t e l y continuous (and t h e n , of course, " AP a s we know f r o m s e c t i o n 3 ) . The a r g u m e n t s k e t c h e d b e l o w was i n d i c a t e d

=

t o u s by B . J a w e r t h : I f (*I h o l d s , we c a n a p p l y t h e a n a l o g u e o f p r o p e r t y ( d ) i n 7 . 4 t o show t h a t t h e same i n e q u a l i t y i s s a t i s f i e d b y t h e w e i g h t f u n c t i o n s

468

IV. WEIGHTED NORM INEQUALITIES

u g * p = w6,

where

u

is

2 0 c o n t i n u o u s w i t h compact s u p p o r t ,

I u = 1 , and u 6 ( x ) = 6 - " u ( t ) . By theorem 3 . 7 , w 6 E A C Am, P w i t h A m - c o n s t a n t i n d e p e n d e n t o f 6 . Now, g i v e n a compact s e t E C R " w i t h [ E l = 0 , we f i x a cube Q c o n t a i n i n g E i n i t s i n t e r i o r , and and E C V j C Q . S i n c e t a k e open s e t s V . s u c h t h a t l V j l < J w6 + p ( 6 + 0) i n t h e weak-* t o p o l o g y o f m e a s u r e s , U r y s h o n ' s lemma g i v e s

with

C,

E >

0

fixed; therefore,

p(E) = 0 .

A n o t h e r r e l a t e d q u e s t i o n i s : Can t h e H a r d y - L i t t l e w o o d maximal operator o r a s i n g u l a r i n t e g r a l operator v e r i f y a s t r o n g type ( 1 , l ) i n e q u a l i t y w i t h r e s p e c t t o a p a i r o f r e g u l a r m e a s u r e s i n Rn? F o r Lebesgue m e a s u r e , we know t h a t o n l y t h e weak t y p e ( 1 , l ) i n e q u a l i t y h o l d s , and n o t h i n g b e t t e r c a n h o l d f o r any o t h e r m e a s u r e s i f t h e o p e r a t o r b e i n g c o n s i d e r e d i s t r a n s l a t i o n and d i l a t i o n i n v a r i a n t . I n f a c t , i f II Rfll 1 < c llfll 1 (where R i s some R i e s z t r a n s L [wl L ful f o r m ) , we c a n assume t h a t d u ( x ) = u ( x ) d x and d v ( x ) = v ( x ) d x w i t h u and v c o n t i n u o u s and e v e r y w h e r e p o s i t i v e ( j u s t c o n v o l v e w i t h a p o s i t i v e i n t e g r a b l e f u n c t i o n ) . By d i l a t i o n i n v a r i a n c e , t h e same i n e q u a l i t y h o l d s f o r t h e w e i g h t s u 6 ( x ) = u ( 6 x ) and v 6 (x) = v ( 6 x ) , and l e t t i n g 6 + 0 we o b t a i n t h a t R i s a bounded o p e r a t o r i n L 1 ( R n ) , b u t we know t h a t t h i s i s a b s u r d . T h i s s o r t o f argument w i l l a p p e a r a g a i n i n c h a p t e r V I , 2 . 8 and 2 . 9 . 7 . 6 . - The a n a l o g u e o f Problem 2 ( i n s e c t i o n 1 ) f o r t h e F o u r i e r t r a n s f o r m i s p r o b a b l y o n e o f t h e most i n t e r e s t i n g and h a r d p r o b l e m s concerning w e i g h t e d i n e q u a l i t i e s . The o b s e r v a t i o n made i n t h e p r e c e d i n g n o t e a b o u t t h e R i e s z t r a n s f o r m s no l o n g e r a p p l i e s i n t h i s c a s e , and o n e l o o k s f o r i n e q u a l i t i e s of t h e form

f o r some p , q 2 1 and r e g u l a r Bore1 m e a s u r e s p,v 2 0 . The b e s t p o s s i b l e i n e q u a l i t i e s o f t h i s t y p e w i t h m e a s u r e s o f t h e form d v ( x ) = ( x 1 8 d x a r e known ( s e e S t e i n [5] a n d , f o r du(x) = (xl'dx, the p a r t i c u l a r case p = q , c o r o l l a r y 7 . 4 i n chapter VI). Extensions

469

IV.7. NOTES AND FURTHER RESULTS

t o more g e n e r a l a b s o l u t e l y c o n t i n u o u s m e a s u r e s d p ( x ) = u ( x ) d x , d v ( x ) = v ( x ) d x a r e i n Muckenhoupt [4] ( s e e a l s o t h e r e f e r e n c e s c i t e d t h e r e ) , where a s u f f i c i e n t c o n d i t i o n f o r i n the case q = p ' , i s s p e c i a l l y simple:

i s given which,

(*)

(j

o(x)dx) 5 C for a l l r > 0 u(x)dx) ( l { u ( x ) > B Yl Io(x)
r. F o r m e a s u r e s which a r e n o t a b s o l u t e l y c o n t i n u o u s , t h e c a s e more e x t e n s i v e l y s t u d i e d c o r r e s p o n d s t o d v ( x ) = dx and d p ( x ) = r o t a t i o n i n v a r i a n t m e a s u r e s u p p o r t e d i n t h e u n i t s p h e r e En-,. If (*) h o l d s i n t h i s c a s e , we s a y t h a t t h e (Lp,Lq) r e s t r i c t i o n ( t o t h e u n i t s p h e r e ) holds f o r t h e F o u r i e r t r a n s f o r m . Observe t h a t t h i s means, i n p a r t i c u l a r , t h a t i t makes s e n s e t o d e f i n e which i s a s e t o f a measure 0 , f o r e v e r y t h a t t h i s a c t u a l l y happens f o r some p > 1 E.M.

h

f

in

f t- Lp(Rn). The f a c t was f i r s t n o t i c e d by

S t e i n , and an argument o f A . Knapp ( b a s e d on t h e d i l a t i o n p r o -

p e r t i e s o f t h e F o u r i e r t r a n s f o r m ) shows t h a t n e c e s s a r y c o n d i t i o n s f o r the (Lp,Lq) r e s t r i c t i o n i n Rn a r e

The c o n j e c t u r e i s t h a t t h i s i s a c t u a l l y t h e b e s t p o s s i b l e r a n g e . I n 2 R , t h e c o n j e c t u r e was c o m p l e t e l y p r o v e d by Zygmund [3]. For n L 3 , the conjecture is t r u e i n the case q = 2 , i.e., (Lp,L2) r e s t r i c n+ 1 t i o n h o l d s i f and o n l y i f 1 5 p 5 2 (3); s e e Tomas [l] . T h i s r e s u l t i s c o n n e c t e d w i t h t h e s p h e r i c a l summation m u l t i p l i e r s d i s c u s s e d i n 7 . 1 5 o f c h a p t e r 11; s e e 7.7.-

c.

F e f f e r m a n [4].

The c h a r a c t e r i z a t i o n o f p a i r s o f w e i g h t s f o r which t h e Hardy-

(theorem 4 . 9 ) L i t t l e w o o d maximal o p e r a t o r i s o f s t r o n g t y p e ( p , p ) T h i s c a n b e use'd t o s o l v e , was f i r s t o b t a i n e d i n E . T . Sawyer [ l ] . t h e f o l l o w i n g weak v e r s i o n o f t h e two f o r t h e maximal o p e r a t o r M , w e i g h t s p r o b l e m : F i n d t h o s e u ( x ) > 0 ( r e s p . w(x) > 0 ) f o r which

Lp(w) t o L p ( u ) f o r some w(x) > 0 ( r e s p . S a v y e r [3]. The p r e c i s e s t a t e m e n t s c a n be f o u n d i n c h a n t e r V I , where a d i f f e r e n t and s y s t e m a t i c a p p r o a c h t o t h i s k i n d o f p r o b l e m s w i l l be f o l l o w e d .

M

i s bounded from

u(x) > 0 ) ;

see

E.T.

For s i n g u l a r i n t e g r a l o p e r a t o r s , and even f o r t h e H i l b e r t t r a n s f o r m , t h e two w e i g h t s p r o b l e m ( i . e . , Problems 2 and 3 o f s e c t i o n 1 )

I V . WEIGHTED NORM INEQUALITIES

470

r e m a i n s c o m p l e t e l y o p e n ; s e e , however C o t l a r a n d S a d o s k y [2] f o r some p a r t i a l r e s u l t s i n t h e s p i r i t o f t h e H e l s o n - S z e g t i t h e o r e m . The i d e a o f r e l a t i n g C a r l e s o n m e a s u r e s a n d

w e i g h t s , which

A1

appears i n s e c t i o n 2 o f c h a p t e r I1 ( f o l l o w i n g C . Fefferman and S t e i n [ l ] ) can be f u r t h e r e x p l o i t e d , giving a u n i f i e d approach t o Carleson or S c o n d i t i o n s f o r weights. Consider t h e measures and t h e A P P maximal o p e r a t o r M f ( x , t ) d e f i n e d i n c h a p t e r 11, r i g h t b e f o r e

t h e o r e m 2 . 1 5 , w h i c h maps f u n c t i o n s f ( x ) i n Rn i n t o f u n c t i o n s d e f i n e d i n R Y c l = Rn x [ O , m ) . We t r y t o f i n d p a i r s (w,p), where w(x)

is a weight i n

that

M

Rn

i s bounded from

Lp(R:+',

p).

(a)

j6

and

The s t r o n g t y p e IM!a

a p o s i t i v e measure i n

p

Lp(Rn; w ( x ) d x )

(p,p)

XQ)(x,t))Pdu(x,t)

w h i l e , f o r t h e weak t y p e dition is

(p,p),

to

LP(R:+',

R:", p)

such or

h o l d s i f and o n l y i f

5

C a(Q)

t h e n e c e s s a r y and s u f f i c i e n t

con-

0

w h e r e , as u s u a l , a ( x ) = w(x) - l ' ( p - l ) and denotes t h e cube 1 R n + Q x [0, l ( Q ) ] i n . When w(x) : 1 , b o t h (a) a n d ( b ) a r e e q u i +

i s supp o r t e d i n Rn x { O } a n d i s g i v e n b y d p ( x ) = u ( x ) d x , t h e n ( a ) a n d ( b ) become, r e s p e c t i v e l y , t h e S and A conditions for the pair P S e e R u i z [ l ] a n d R u i z and T o r r e a 711. (u,w).

v a l e n t t o t h e C a r l e s o n measure c o n d i t i o n f o r

7.8.-

p.

When

p

This note is intended t o i l l u s t r a t e t h e second motivation f o r

t h e study of weighted i n e q u a l i t i e s given a t t h e i n t r o d u c t i o n t o t h i s c h a p t e r . Let IP,(x) d e n o t e t h e o r t h o n o r m a l s e q u e n c e o f Legend r e P o l y n o m i a l s . We want t o know f o r w h i c h v a l u e s o f p i s i t t r u e that

(1

N

1 -1

f

pk) pk(x)

~

* f(x)

(in

LP

norm)

for every f E Lp([-l,l]). The N - t h p a r t i a l sum, T f i s an i n t e N 'N g r a l o p e r a t o r g i v e n by t h e D i r i c h l e t k e r n e l D N ( x , y ) = Pk(x)Pk(y) w h i c h , by t h e C h r i s t o f f e l - D a r b o u x f o r m u l a , c a n b e w r i t t e n a s DN(X9Y)

N+l

=

- -

X-Y

('N+l(')

'N(Y)

(KN(x)LN(y)

-

- pN(x) ' N + , ( y ) )

KN(y)LN(x))

+

=

t r i v i a l e r r o r terms

IV. 7

NOTES AND FUTHER RESULTS

where IKN(t) I 5 C(1-t2)- /4 -1 5 x 5 1, and denoting by TNf(x)

=

and

H

ILN(t)I

5

471

C(l-t2)1/4.

Thus, for

the I-iilbert transform

KN(x) H(fLN(x))

- LN(x)H(fKN)(x)

+

error

and a sufficient condition to have TN uniformly bounded in Lp 4 < p < 4. is (l-t2)’P/4 e A P (in the interval [ - l , l ] ) , i.e. 3 This can easily be shown to be the exact range of p ’ s (see Pallard [ I ] , Newman and Rudin [ l ] ) . By using the same argument together with the Hunt - W.S. Young result stated in 7.2 (c), i t a l s o follows that T,f(x) = s u p ITNf(x)l is bounded in L p ( [ - l , l ] ) if

4

< p <

N> 0

4, and therefore, the Legendre series of each 4 p > 3, converges a.e. to f(x).

f e LP,

For similar results concergining other orthogonal expansions we refer to Muckenhoupt and Stein [ l ] , Askey and Wainger [ l ] , Benedek, Murphy and Panzone [1] . 7.9.- The first proof of the Ganett-Jones theorem (in LJ. Garnett and P. Jones [ l ] ) was long and complicated. Further elaboration of the ideas in that proof led to a still more complicated stopping time argument, by means of which, the factorization theorem for A P weights was originally proved (in P. Jones [ l ] ) . This result had been previously conjectured by bluckenhoupt. The approach followed here, based on the reiteration method of lemma 5 . 1 , originates in Rubio de Francia [ S ] , where the extrapolation theorem was also found. That approach was later clarified in Coifman, Jones and Kubio de Francia [ l ] (which contains also the application to the weak version of the Ilelson - S z e g i j theorem and the constructive decomposition of BMO) and Garcia-Cuerva [ 2 ] . B. Jawerth [ l ] further exploited these ideas. obtaining general versions of the factorization and extrapolation theorems. The result about restrictions of .4 weights and P B.M.O. functions to arbitrary sets (5.6 and 5.71 are due to ‘l’, Wolff [ l ] . Another application of lemma 5.1 appears i n Neugebauer [ l ] : “ G i v e n p o s i t i v e function u ( x ) , v ( x ) - R”, t l z e nc,c17sstiY’ld i 7 i l ; i sufficient c o n d i t i o r i f o r t k P e x i s t i ~ i ? o sof w t‘ A szAch Lhz:, !’ u ( x ) 5 w ( x ) 5 v(x) 7:s that ( u ’ + € , v l C E ) E. .I for. s c m c E > 0“. I’ This result is interesting for the two weights problem for maximal and singular integral operators.

472

7.10.-

IV. WEIGHTED NORM INEQUALITIES

The extension of the AD theory to the product setting her straightforward, but we can mention heretwonon-trivial If A;

1 f E Lloc(Rn) and MSf(x) for every 6 < I ? ( + )

<

m

a.e., is it true that

Does the analogue of the Fefferman-Stein inequality (2.12 in chapter 11) hold for the strong maximal function, i.e.

("1 for arbitrary

j(MSf(x)ip

w(x)dx 5 jlf(x) IpMsw(x) dx

( 1 < PI

w(x) 2 O?

The best we know concerning the second problem is that ( * ) holds for all p > 1 if w e ;:A see Lin [ l ] . Weights of product type can also be defined in Rn+m = Rn x Rm. The family of intervals to be considered in this case is p = {Q(') x Q(2) : Q(') and Q(21 cubes in Rn and Rm respectively), the classes Ap(P) are defined as expected, and the whole section 6 applies with the obvious modifications. In particular, if T(l) and T(2) are regular singular integral operators in Rn and (defined as the comuosiR ~ ,respec ively, then, T = T(l) P T(2) tion of T ( acting on the first n variables and T (2) acting for every w e Ap(P), on the last m variables) is bounded in Lp(w) l < p < m . There is, however, a more interesting extension of the theory of s ngular integrals to product domains, which was initiated in R. Fefferman and E.M. Stein [l]: One can easily write the cancellation and size conditions satisfied by the kernel o f T = T(l) 0 T(2), and taking these as the starting hypothesis, without assuming that T is of product type, the LP-boundedness of T as well as A -weighted inequalities can be proved. A far-reaching P theory of singular integrals in product spaces (which we cannot describe here) has been built from this, including Hp and BMO spaces and non translation invariant operators. We refer to the survey by A. Chang and R. Fefferman [ I ] .

The close connection between B.M.O. functions and A weights, P as well as important consequences of it, became apparent in sections 2 and 5 of this chapter. Some other results emphasizing this connection are collected in the last four notes: (+) This question has recently been answered in the negative by

F. Soria.

IV.7. NOTES AND FURTHER RESULTS

473

7.11.- Let h : R -+ R be a homeomorphism of the real line. Suppose, for instance, that h is increasing. Then, h preserves B.M.O. (i.e. f(h-l(x)) belongs to B.M.O. whenever f(x) does) if and only if h' e A,. If this is the case, then also (h-')' E A, (by theorem 2.11) and f f0h-l is actually an isomorphism of B.M.O. onto itself. See P. Jones [2]. The corresponding problem in Rn was solved by H.M. Reimann, who showed that h preserves B.M.O. (R") if and only if h is quasiconformal (Reimann [l]).

+

7.12.- Given a measurable function b(x) in B, we denote by Mb the operator of pointwise multiplication by b(x). The commutator of this operator and the Hilbert transform H is [Mb,H]f(x)

=

b(x)Hf(x)

- H(bf)(x)

=

P.V. TI

lm

b(x)-b(y) X-Y

-m

f(y)dy

The following result was proved by Coifman, Rochberg and Weiss [ l ] "FOP any

p,

and o n l y if

1 < p < m, [Mb, H] b e B.FI.O.(R)".

i s bounde d i n

Lp(R)

3

We shall sketch the proof of the "if" part in the case p = 2. We suppose that b(x) is real, and consider the power series in the complex variable z = r e ia

By Cauchy's formula, the term corresponding to

n

=

1

is

But IITzll 5 llT,ll if I z I = r (11.11 stands for the norm as an operator in L2) and, for r > 0, Tr i s bounded in L 2 if and only if e 2rb e A2, which is certainly the case for small enough r, since b e B.M.O. Therefore 11 rMb, 1 5 1 llTrll < O1

The same proof applies to higher order commutators and to regular singular integral operators in R". The boundedness of [Mb, Rj] for b e B.M.O.(Rn) where R. J are the Riesz transforms is then used by Coifman, Rochberg and Weiss to give a weak factorization theorem for H1(Rn), See note 8.10 i n

IV. WEIGHTED NORM INEQUALITIES

474

chapter 111. 7.13.- The following characterization of the closure of Lm in B.M.O. can be obtained from the Garnett-Jones theorem. Given a weight w(x), we write w e R.H.I.(q) to indicate that w satisfies a reverse Hb'lder's inequality with exponent q (i.e., lemma 2.5 holds with 1+F. = q) . Then, for a real function f E B.M.O.(Rn), setting w = e f , the following assertions are equivalent: belongs to the closure of

a) f b) w

E

R.H.I.(q)

c) w e A d) w

E

A

P

P

e) wq e Am

and

w-'

for all

p > 1

and

E

w-'

A

P and w - E~ Am

E

Lm

in

R.H.I.(q)

and

w

for all for all

E

B.M.O. for all

R.H.I.(q)

q <

m

for all

q <

m

p > 1

q <

Observe that every f E V.M.O.(Rn) (defined as in the case of the torus, see 11.18 in chapter I) satisfies a). This is used by Guadalupe and Rezola (11 to prove A,-weighted estimates for the conjugate function operators in certain curves of the plane. 7.14,- The typical example of a function in B.M.O.(Rn) which is not bounded is log 1x1. The following extension of this fact has been noticed by E . M . Stein [ l l ] : "Let P(x) be a poZinomiaZ of degree k Rn; then log \ P ( x ) ) B.M.0. & \ \ l o g l P \ \ \ * 5 < ck,n ( c o n s t a n t d e p e n d i n g onZU o n k a n d n)". Here is the simple proof of this result. It is enough to prove a uniform R.H.I., for instance

for all cubes Q and all po ynomials of degree k. By translation and dilation invariance of the polynomials, it suffices to prove i t for the unit cube Q = [0,1]". But then, both sides of ( * ) define norms on the finite-dimensional space of polynomials of degree k in Rn, and therefore, they are equivalent.