Chapter IX: Ap Weights

CHAPTER

IX Weights

1. THE HARDY-LI'ITLEWOOD MAXIMAL THEOREM FOR REGULAR MEASURES Let p be a nonnegative Bore1 measure in R", finite on bounded sets. For this measure we pose the question of whether it differentiates p-locally integrable functions f ; in other words, if x E R" and Z is an open cube containing x, does the statement limp(r& 1 / p ( I))j r f ( y ) d p ( y ) = f ( x ) hold p a.e.? Here 111 denotes the measure of I. As in the case of the Lebesgue measure in Chapter IV we opt to approach this question by first considering a weak-type result for the corresponding maximal function. More precisely, if for x E R" and f locally in L,(R") we put

where the I's are open cubes containing x, is the mapping f + MJ of weak-type (1, l ) ? We can go about answering this as follows: let 0, = { M , f > A}; 0, is open since to each x in 0, there corresponds an open cube I, containing x such that

and consequently I, c OA.In fact

0,=

u I,.

xeU*

(1.3)

We want to estimate p ( 0 , ) in terms of the p-measure of the (complicated) set on the right-hand side of (1.3); it is apparent that we need some control 223

ZX. Ap Weights

224

over this set. So, suppose, in addition, that p is regular; in other words, if % is p-measurable, then

If K is a compact subset of OA now, there are finitely many Zx’s,Zx,,. .. ,I ,, say, so that K s U ,; Ixj; in fact, we may avoid unnecessary overlaps by discarding any cube Ix, such that Ix, c Uj+kI .,. However, even once this is done we may still be left with quite a bit of overlap. To handle this we proceed as follows. Since we are dealing with a finite number of cubes, there is one with largest sidelength (if there is more than one just pick any); separate it and rename it Zl.Now, if any of the remaining cubes, say I, intersects I,,since sidelength Z s sidelength of Zl,it follows that I c 311, the cube concentric with Zlwith sidelength three times that of Zl;all these cubes Z can be discarded as well. We are thus left with a finite collection of open cubes, each one disjoint with Il .Repeat for this family the procedure used to select Zl,that is select a cube with largest sidelength, call it I,, and discard all other cubes which intersect it. After a finite number of steps we are left with a collection {II,.. .,zk} of disjoint open cubes so that K E 31,. Thus

u;=,

Under what circumstances can we replace p ( 3 4 ) by p ( 4 ) in (1.5)? This can be done for the so called doubling measures, namely those measures p for which p(21) s cp(Z), all open cubes Z, c independent of I. In case p is doubling then from (1.5) it follows that

where c is the doubling constant of p. But the 4 ’ s are special cubes; in particular they all satisfy (1.2). Whence combining (1.6) and (1.2), and since the 4 ’ s are pairwise disjoint, we get k

Finally, on account of (1.4), (1.7) gives

225

2. The Hardy- Littlewood Maximal Function and the mapping is of weak-type (1,l) with norm =sc have proved

2

. Summing up, we

Theorem 1.1. Let p be a nonnegative Bore1 measure in R" which in addition is finite on bounded sets, doubling, and regular. Then the mapping f + MJ is of weak-type (1, I), with norm
IIqJllL;

CllfIlL;,

1 < P < 03.

=

cp

(1.9)

Proof. M, is of weak-type (1, l), and is bounded in L:. Thus the Marcinkiewicz interpolation theorem applies. W

Corollary1.3. Let p beasintheorem 1.1,andletfbelocallyin L:(R"). Then

Proof. Since p (K) < co for compact sets K , continuous functions are dense in L:(R"); the proof is therefore entirely analogous to Theorem 2.2 in Chapter IV. W 2. Ap WEIGHTS AND THE HARDY-LITILEWOOD MAXIMAL FUNCTION The boundedness of the Hardy-Littlewood maximal function is an essential ingredient in the consideration of the L P ( T )behavior of the various operators we have considered thus far. It is natural, then, to expect that the LP,(R")behavior of the Hardy-Littlewood maximal operator will have important applications in the study of weighted norm inequalities for similar, n-dimensional, operators. We open our discussion with the study of the necessary conditions; more precisely, suppose there is a constant k = k,,, independent off such that for some p, 1 s p < CO, and allf E LP,(R")and A>O h P P ( { M - > A))

kPllfll%;.

(2.1)

What can we, then, say about p ? Simple examples show, for instance, that p cannot have atoms, but in fact much more is true.

226

IX. Ap Weights

Theorem 2.1. Assume p is a nonnegative Bore1 measure, finite on bounded sets and assume that for some 1 S p < co (2.1) holds. Then (i) p is absolutely continuous with respect to Lebesgue measure, i.e., there is a nonnegative, locally integrable function w such that d p ( x ) = w (x) dx. (ii) for all open cubes I and f E LP,(R”)

where c = cp,k is independent of $ (iii) ( A pcondition) w satisfies Muckenhoupt’s A p ( R ” )= Ap condition, or w E Ap;i.e., there is a constant c = cp,k independent of the open cube I so that

or

I

The infimum over the constants on the right-hand side of (2.3) and (2.4) is called the Ap, or A l , constant of w ; moreover, the Al constant of w is less than or equal to ck and the Ap constant of w is less than or equal to ckP-’, p > 1. A statement which depends on the Ap constant of a weight w rather than on the weight itself is called “independent in Ap.” Theorem 3.1 below is an example of this. (iv) (Strong Doubling) For each open cube I and measurable subset E of I, s c(lll/lEI)PP(E),

where c

=

(2.5)

cp,k is independent of E and I.

Proof. (i) Suppose the Lebesgue measure IE( of E is 0. We show that also p ( E ) = 0. By the regularity of the measures involved we may assume that E is compact and that, given E > 0, there is an open set 0 3 E such that p(0\E) < E. Let f(y) = X B \ ~ ( ~ ) ;then JE LP,(R”) and l l f l l ~ ; = p(O\E) < E. Next observe that M f ( x ) = 1 for x E E : indeed, to each x E E there corresponds an open cube I c 0, x E I, and consequently 1/111 j r f ( y ) dy = I(O\E) n 11/1I1 = 1 since IEl = 0. By (2.1) it then follows that p ( E ) S p ( ( M f 2 f})C 2PkPllfllPL:s CE,which gives the desired result as E is arbitrary.

227

2. l%e Hardy- Littlewood Maximal Function

(ii) Fix an open cube I and consider for f~ LL the quantity which we may assume >O. Since (1/111) jrlf(y)l dy =

If], must be finite for each 1,for otherwise (2.1) cannot hold unless p is the 0 measure. Thus, if we put 0 = { M ( f X r )> lflr/2}, by (2.1) and (2.6) it follows that p ( I ) S p ( 0 ) s k ” ( l / l f l ~ ) ” l l f x ~ lwhich l ~ ~ : , is equivalent to (2.2). (iii) and (iv) Since there is no a priori reason why w cannot vanish on a set of positive measure, we introduce the measure d v ( y ) = d p ( y ) + E dx, E > 0, to avoid unnecessary technical difficulties. Clearly, v is also absolutely continuous with respect to Lebesgue measure, d v ( y ) = u ( y ) dy, u > 0, and, more importantly, (2.1) also holds with p replaced by v with constant independent of E. Assume p > 1 first. In order to estimate u(y)-”(”-’) dy we note that it equals Ill/ullpL’, l/p + l/p’ = 1, which by the converse to Holder’s inequality may be estimated by

I,

Now by (ii), which also holds for v, it follows that for such f’s

and consequently

Unraveling (2.7) gives (2.3),that is, the Ap condition for u. Next we show that (2.5) holds for u; indeed, note that

which by (2.7) does not exceed c k v ( E ) ’ / P ( I I I / v ( I ) ’ / PIn) . other words

v(I) s ck”(IZI/IEI)”v(E).

(2.8)

IX. Ap Weights

228

Since the constant in (2.8) is independent of E, we may let E + 0 there and obtain that p ( I ) 6 ckP(III/IEI)Pp(E)as well; i.e., (iv) holds. Moreover, if there is a set E with IEl > 0 so that p ( E ) = 0, then by (2.5) p is the 0 measure since it vanishes on any open cube I containing E. Thus d p ( x ) = w ( x ) dx, w > 0 a.e., and the above argument may be repeated with w in place of u; this completes the proof when p > 1 . The case p = 1 is rather simple since by putting f = xE, IEl > 0, in (ii), we see that (IEl/lIl) d c k ( p ( E ) / p ( I ) )or equivalently

P(O/IIl C k ( P ( E ) / I E l ) , (2.9) which is (iv). Furthermore, by choosing E to be a sequence of open cubes converging to a Lebesgue point x of w so that w ( x ) ess inf, w, (2.9) gives (2.4), and the proof is complete.

-

3. A , WEIGHTS As we have seen, A , is a necessary condition for the Hardy-Littlewood maximal operator M to map L,(R") into wk-L,(R"), but is it also sufficient? Theorem 3.1 (Muckenhoupt). Suppose w E A , . Then M maps L,(R") into wk-L,(R"), with norm independent in A l . Proof. First note that if w E A , , then p is doubling with doubling constant s c ( A , constant of w ) ; indeed, since (1/1211) d p ( y ) C c ess inf, w s c ( l / l I l ) d p ( y ) , it readily follows that p ( 2 I ) c c p ( I ) , c s 2 " ( A , constant of w ) . Moreover, since

I,,

I,

we also have that M f ( x ) C c M J ( x ) , c = A , constant of w. Thus {Mf> A} E {MJ>A/c}, and by Theorem 1.2 Ap({Mf>A})s 3 ( A / C ) P ( { M J > Ale)) =s c IlfllL,. Some observations concerning Al are obvious: for instance, A , is the limiting Ap condition as p + 1+ and an equivalent way of stating Al is M w ( x )s cw(x)

a.e.

(3.1)

3. A, Weights

229

But what are the A, weights? Can we give some examples or even characterize them? As a first step we consider powers of 1x1, i.e., 1x1". When n = 1 and 71 > 0, by letting I = ( 0 , 6 ) we note that (1/6) j(O,b)X" dx = l6"/(7 + 1) + 00 as 6 + 00, whereas inf, x" = 0. Thus positive powers of 1x1 are ruled out, but how about negative powers? We must have -n < 7 C 0 for otherwise 1x1" is not locally integrable, but this is essentially the only restriction. Indeed, we have

Proposition 3.2. Suppose -n < 7 C 0. Then 1x1" E A,; more precisely, there is a constant c independent of I such that

Proof. Fix a cube I and let I, denote the translate of I centered at 0; we consider two mutually exclusive cases, to wit (i) 21, n Z # 0 and (ii) 2 1 0 n I = 0. In case (i) we have that 610 2 I and ( l / ~ I ~ ) ~ , ~ xC ~ " d x ( l / ( I ( )j6r01xI"dx c ~(ll"'", where cis a (dimensional) constant, independent of I; clearly, (3.2) holds in this case. Case (ii) is easier, for then 1x1 lyl for x, y in I; indeed, we have 1x1 c Ix - yl + lyl G clIll'" + IyI clyl, and the opposite inequality follows by exchanging x and y above. Thus IyI" G cinfr)xlg,all y E I, and averaging over y in I, (3.2) holds for case (ii) as well. H Next we consider functions which behave like (XI", -n < 7 S 0, and show that they also are A, weights.

-

Proposition 3.3 (Coifman-Rochberg). Let p be a nonnegative Bore1 measure so that Mp(x) is not identically 00. Then for each 0 s E < 1, Mp(x)" E A,, with A, constant which depends only on E. Proof. Recall that Mp(x) = S u p x E ~ ( l / ~ Z ~ ) For p ( I )a. fixed open cube Z we estimate (l/lI() I ,Mp(x)" dx by A" = (inf, M p ) " as follows: for each x in I we divide those open cubes Q containing x into two families by setting $I = {Q: IQI C I2Zl) and $* = {Q: IQI > 1211). Thus

= A(x)

+ B(x),

say. The estimate for B ( x ) is readily obtained; since for Q 3 Q 2 Z, it follows that

(3.3) E

$2

we have

and B ( x ) 6 CA

(3.4)

IX. Ap Weights

230

with c independent of p. As for A(x),let p , denote the restriction of p to 61, i.e., d p , ( y ) = xsr(y) d p ( y ) , and note that

A ( x )d w h ( x ) .

(3.5)

Thus on account of (3.3), (3.4), and (3.5) we get that

and it suffices to prove the desired estimate with M p replaced by M p l . But by (a simple variant of the Lebesgue measure version of) Theorem 1.1 and 7.5 in Chapter IV, we readily see that [I

1 MP~(X dx) S~ -c(wk-L norm of M p I ) E ( I I 1 - "

with c depending only on

I[I

E,

and we have finished.

The interesting fact is that the converse to Proposition 3.3 also holds, name1y,

Theorem 3.4 (Coifman-Rochberg). Assume w E A,. Then there are functions b and f and 0 d E < 1 so that (i) 0 < A d b ( x ) d B < 00 a.e. (ii) J E L,,,(R"), M ~ ( Xis )finite ~ a.e. and w ( x ) = b(x)Mf(x)'.

The proof of this theorem relies on the so-called "reverse Holder" pr (perty of w ; this property is of independent interest and plays an importaat role in the theory of weights.

Theorem 3.5 (Reverse Holder). Suppose w E A,. Then there is a positive number q so that

where c = c,, is independent in A , and independent of 1,but not, of course, of q ; we abbreviate (3.7) by w E RH,,,,.

3. A, Weights

23 1

Now, suppose that Theorem 3.5 has been proved. Then by (3.7) also E A, ,M ( w'+")(x) s cw(x)'+" a.e. and Theorem 3.4 holds with b(x) = w ( x ) / M (W ' + " ) ( X ) ' ' ( ' + " ) , f(x) = w(x)'+", and E = 1 / ( 1 + q ) . It thus only remains to prove Theorem 3.5, which we do forthwith. In order to assure that the various integral expressions we consider are finite, we introduce the function v ( x ) = min(w(x), N); clearly, v E A,, (A, constant of v ) s ( A , constant of w), independently of N: indeed, for a given I let A = inf, w and consider two cases, namely, (i) A 3 N and (ii) A < N. We then have wl+"

in case (i),

N s inf, v, w ( y ) dy s c inf, w

s c inf, v, in case (ii),

thus proving the claim. We show (3.7) with w replaced by v first; let q > 0 and observe that

(3.8)

=A+B, say. Clearly, B

(3.9) which is the right estimate. The bound for A is not so readily obtained, and in the course of the proof we must keep track of the various constants appearing to be sure that they only depend on the A, constant of v, and 77 of course. First observe that with the notation 0,= { y E I: v ( y ) > t } we have A = (1 + q )

j

S V,+"IIl,

t"lo;l dt

Cv1,m)

lSSlds)' dt

IX. Ap Weights

232

which is also of the right order. Next we show that for an appropriate choice of 77, D is dominated by the (finite) quantity (3.12)

which may then be passed to the left-hand side of (3.8) to obtain the desired conclusion. We consider the innermost integral in D first. It equals (3.13)

Now since t > 0 1 , we may invoke the (n-dimensional version of the) Calder6n-Zygmund decomposition of u at level t, thus obtaining a collection of open, disjoint subcubes (4) of I with the following properties (i) v ( y ) s t a.e. in I\U 4 and (ii) t s (1/141) u ( y ) dy < 2"t, all j.

I,

From (i) it is clear that { u > t} c_ does not exceed

u4 and therefore the integral in (3.13) (3.14)

Furthermore, UIJ

c

since u E A,,

c inflJu,

since lj is a Calder6n-Zygmund cube.

{29,

Therefore by combining these bounds we get that ulJ=s(c infrJu)'-'(2"t)", all j , O < E < 1. Thus each summand in (3.14) is dominated by ct'lIjl(infIJ v)'-' s ct" jrJ u(y)'-" dy and (3.14) does not exceed (3.15)

We need one last observation: from the left-hand side inequality in (ii) it follows that 4 G {Mu > t}, and, since u E A, and M v ( x ) 6 c v ( x ) a.e., we also have U 4 G { u > t / c } . Consequently, (3.14) is bounded by

u

ct'

I

{V>t/C)

v(y)'-"dy,

0
4. A, Weights, p > 1

233

and the same is true of (3.13). Whence

(3.16)

First fix 0 < E < 1 and then choose 77 > 0 sufficiently small so that cv(1 + q ) / (+~7 ) < this is clearly possible. Thus D is dominated by (3.12) and (3.7) holds with v in place of w there. This is a minor inconvenience since by Fatou's lemma

t;

'/I+"

6 lim inf N+m

( I,i

'/I+"

v(y)l+? d y )

and the proof is complete. H 4. A,, WEIGHTS, p > 1

As we have seen, Ap is necessary for the Hardy-Littlewood maximal operator M to map LP,(R") into wk-LP,(R") and a simple argument similar to Theorem 3.1 shows it is also sufficient. However, a stronger result holds. Theorem 4.1 (Muckenhoupt). Suppose w E A,, 1 < p < 00. Then M maps LP,(R") continuously into itself, with norm independent in Ap.

Proof. For a nonnegative function f in LP,(R") and an integer -00 < k < 00, put A, = { y E R": 2, < M f ( y ) S 2k+'} and let %k be a compact subset of A,; we estimate Mf(y)'d p ( y ) by cllfll;;:, where c is independent of f and the %k's, and depends only on the Ap constant of w. A simple limiting argument then gives the desired result. To each y E A, we assign an open cube Iy containing y so that

Iuq,

f ( x ) dx

(<2,+').

(4.1)

IX. Ap Weights

234

Since %k E Ak, there are finitely many I , , ’ s , none of which is contained in the union of the others, { 4 , k } y i ? say, so that each cube verifies (4.1) and %k E U y5:) 4.k. Moreover, since

we must estimate the right-hand side of (4.2); a good estimate depends on our ability to avoid unnecessary overlaps of the $,k’s. This is achieved as fOllOwS: put E 1 . k = I 1 . k n % k , E 2 , k = ( I 2 , k \ I l , k ) n % k , and, in general, Ej,k

= (4,k\u

i
Ii,k)

%k,

j =

2, - *

* 3

n(k)

(4.3)

For each k the Eik’s are clearly disjoint, and by (4.3) it follows that (4.4)

Therefore, by (4.4) and (4.1) we also have

1

2kpp(E,k)

j,k

(4.5)

Let u ( x ) = W ( X ) - ” ( ~ - ’ ) , d v ( x ) = ~ ( xdx; ) then the right-hand side in (4.5) can be rewritten as

What we are attempting to do here is to bring a combination of the A, condition and Theorem 1.1 into play. Let ‘rn be the measure on Z + x Z given by

With this notation the expression (4.6) becomes I l { a j , k } l l h , where the sequence a j , k = ( I / V ( 4 . k ) ) j I j , k ( f ( x ) / u ( x d) )v ( x ) . Note that this expression also equals

235

4. Ap Weights, p > 1

where O,,= {(j,k ) E Z+x 2 : a j , k > A}. What we need then is a good estimate for m(O,,);we use the notation Z ( A ) = U ( j , k ) E O , Ij&. Observe that for each (j,k) in O,,by the Ap condition we have

(4.10)

Since the E j , k ’ s are pairwise disjoint we may replace the right-hand side of (4.10) by c I ,.M , ( ~ I ~ h ) / ~ ) dp(x) ( ~ ) P and ’ invoke Theorem 1.1 to estimate this quantity by (4.11)

Moreover, since W(X)-”‘dp(x) = w(x)’-”’ dx = dv(x) the expression in (4.11) is c v ( I ( h ) )and m(O,,) s cv(Z(.h)).

(4.12)

Now, the 4 , k ’ S whose union is I ( h ) are special cubes; in particular, by the definition of 4, if ( j , k) E OA, then a j , k > A. In other words, A < ( l / v ( I j , k ) )JIj,k(f(x)/u(x)) dv(x). n u s , each such Ij,k and also, consequently, I ( h ) is contained in { M , , ( f / u )> A} and by (4.12) we see that m(‘A) cv({Mu(f/u) > A}). (4.13) This is all we need to complete the proof. Indeed, on account of (4.8) J S c Ip3,m) v ( { M , , ( f / u > ) A}) dhp, which by Theorem 1.1 is dominated by c ~,~(f(x)/u(x))p dv(x) = c j,.f(x)” dp(x), since u(x)-” dv(x) = dp(x). This completes the proof. W Corollary 4.2. M maps LP,(R”)into itself if and only if M maps LP,(R“) into wk-LP,(R“), 1 < p < CO.

IX. Ap Weights

236

The question once again is, what are the Ap weights? Some properties, such as w E A, if and only if w-''(~-')E Ap,, l/p + l/p'= 1, 1 < p < 00, are readily verified, but we need some examples and if possible a characterization of these weights. Proposition 4.3. Assume w , , w, 00; then w E Ap.

E

A , and let w ( x ) = w l ( x ) w 2 ( x ) ' - p 1, < p <

hf. Holder's inequality. Corollary 4.4. lxlq E A p , 1 < p < 00, if and only if - n < 7 < n ( p - 1). In addition of being Apweights, the w's in Proposition 4.3 actually verify some additional properties. More precisely, Proposition 4.5. Assume w ( x ) = wl(x)wZ(x)'-p, w l , wz E A , , 1 < p < 00. Then

(i) (Open ended property) w E A,-,, some E > 0. (ii) (Reverse Holder) w E R H l + , , some 7 > 0. (iii) (Reverse doubling) There is 6 > 0 such that for all open cubes I and measurable subsets E of I, p ( E ) / p ( I )d c ( ~ E I / I I ~ ) where ', c is independent in A p , also independent of E, I. (iv) JRn(1 + Ixl)-" d p ( x ) c p ( I o ) , where I. denotes the unit cube in R" and c is independent in Ap.

Proof. (i) Since w, E A , , by Theorem 3.5, w, E RH1+,; put now E = (7/(1+7 ) ) ( p- 1)> Oandnotethatp - E - 1 = ( p - 1)/(1+7 ) .Itisthen readily seen that

(4.14) and also

Whence, by multiplying (4.14)and (4.15),it follows that w,w:-" we wanted to show.

E

A,-,, as

5. Factorization of Ap Weights

231

(ii) Since w2 E A, from Holder's inequality (applied to (1/1Z1) w2(y)'/"'/~ ~ ( y ) d' y/) ~it' follows that for some constant c > 0 independent of I

I,

(4.16) Let w1 E RH,,,. Then by (4.16)

as we wished to show. (iii) Let w E RH,,,. Then by Holder's inequality

= c( IEl/lIl) ? / l + ? p ( I ) .

(iv) By (i) and the doubling property (iv) of Theorem 2.1, k 3 1. p(2kZ0) < C 2 n k ( P - - E ) cL(Zo),

(4.17)

Thus

and we have finished.

5. FACTORIZATION OF A,, WEIGHTS This section is devoted to proving a remarkable fact, namely, the converse to Proposition 4.3. Before we proceed with the proof of this factorization result we need some preliminary observations; we start with a definition.

IX. Ap Weights

238

Definition 5.1. We say that an operator T is admissible provided it verifies the following four properties. (i) (ii) (iii) A > 0. (iv)

There exists r, 1 < r < 00, so that T is bounded in L'(R"). T is positive, i.e., T f ( x )2 0 for every f in L'(R"). T is positively homogeneous, i.e., T ( A f ) ( x )= ATf(x) a.e. for each

+

T is subadditive, i.e., T(f+ g ) ( x ) S T f ( x ) T g ( x ) a.e.

Some examples of admissible operators include If(x)l, M f ( x ) , and, more important for our purposes, ( ~ ( ~ f l " / v " ) ( x ) u ( x ) "for ) ' ~appropriate " 1< p < m , o < 7 d 1. We verify this last example and in the process we find the necessary conditions on u for this to hold. First, we must find r, the choice r = p / 7 being a natural one; in this case we have IITfll: = J R m M(lfl"/u")(x)'/"u(x)dx and, if we assume that u E All", then this expression may be estimated by c JRn ( l f ( x ) l " / " / u ( x ) ) u ( xdx ) = cllfll: and (i) holds; the verification of (ii) and (iii) are immediate. Thus it only remains to check (iv). Fix x and let I denote an open cube containing x; then by Minkowski's inequality we readily see that

=G

M(lfl"/u")(x)""+ M ( l g y / u " ) ( x ) ' / q

and since I is arbitrary it follows that M ( l f + glp/u")(x)l'pd M ( l f l p / u " ) ( x ) l /+ p M ( l g l P / u " ) ( x ) l / pand , (iv) follows at once. An important property of admissible mappings T is that they also are cT-subadditive; more precisely, we have

Proposition 5.2. Suppose T is admissible and r is the index in pro ert (i) in the definition of T. If {A},f are L'(R") functions with limN+, =f in L', then T f ( x ) ?s CJ:, TJ(x) a.e.

R Y

hf. Since f = (f lows that

zJzl&) + xJT,&,from properties (ii) and (iv) it folN

N j=l

239

5. Factorization of Ap Weights

Moreover, since 11 T(f-CJt,f;)llrG c l l f - C ~ , f i I l + . 0 as N + 00, there is a sequence Nk + 00 such that lirn,,,, T ( f - C z , f i ) ( x ) = 0 a.e. This is the sequence of N's we choose in (5.1), and by letting Nk + 00 there we get that T f ( x ) S ZT=, Tfi(x) a.e. We need one more definition. Definition 5.3. Given an admissible mapping T, we say that a nonnegative function w is in A , ( T ) if T w ( x )s c w ( x )

a.e.

(5.2)

We then have Proposition 5.4. Suppose that Tl and T2 are admissible mappings with the same r in (i). Then there exists a function 4 in L'(R") such that 4 is simultaneously in A,( T I )and A,( T2).

+

Proof. Put T = TI T2( T is also an admissible mapping) and let A 11 7'11, the norm of T i n L'(R"). For an arbitrary, nonnegative function g in L'(R") put 4 ( x ) = T j g ( x ) / A J We . show that this 4 will do. In the first place, 4 E L'(R") sinceCJz,((T'g((,/A's Tll/Ar)llgllr< 00. Moreover, by the a-additivity of TI we see that

(CJzo(II

This proves that 4 is in A,(T,) and a similar argument gives that 4 is in A1(T2)as well. W It is now a simple matter to prove the decomposition theorem for the Ap weights. Theorem 5.5 (Jones). Suppose w E Ap,1 < p < 00. Then there are weights w,,w 2 in A , so that w ( x ) = wl(x)w2(x)'-". Proof. Since w E Ap we also have that w - " ( ~ - ' )E Ap,,l/p + l/p' = 1. Let r = pp' and set T , f ( x )= ( M ( ~ f l p ' / ~ l / P ) ( ~ ) ~ ( ~and ) l / PT J) (l x/ )P=' ( M ( ~ ~ p w ' ~ p ) ( x ) w ( x ) - By l ~ the p ) l remarks ~p. at the beginning of this section it follows that TI and T2 are admissible and by Proposition 5.4 there is a

240

IX. Ap Weights

nonnegative, locally integrable function 4 simultaneously in A, ( TI) and A,( T,). This means that T , ~ ( x s )c ~ ( x ) or , M ( 4 p ’ w - 1 / p ) ( x )S c ~ ” ’ ( xW) ( X ) - ’ / ~

(5.3)

and T,+(x) s c ~ ( x ) or , M(+”w’l”)(x)c c+”(x)w(x)””. (5.4) In other words, c$pwl/pand +p‘w-l/pare A, weights. Put now w1 = 4pw1/p, w, = 4 p ‘ w - 1 / pand note that since p p’( 1 - p ) = 0 we have w1w i P p= 4P+P’(l-P)w1/Pw-(l-P)/P

+

=

Remark 5.6. It goes without saying that by Theorem 5.5, Ap weights satisfy properties (i)-(iv) in Proposition 4.5. 6. Ap AND BMO

As both the Ap condition and the definition of BMO deal with the, averaging of functions it is natural to consider whether there is any connection between these concepts. Proposition 6.1. Assume w is a nonnegative, locally integrable function. Then In w E BMO if and only if there is 7 > 0 such that w ” E A,.

Proof. We show the necessity first; by the John-Nirenberg inequality there are constants 7 ) s k/llln wll* and c, independent of I, such that

By removing the absolute values in (6.1) we also have

Whence multiplying the

+

and - estimates in (6.2) it follows that

that is, w“ is in A*. Conversely, assume such an

7)

exists and note that

6. Ap and BMO

241

say. Since both summands are handled in a similar fashion we only do A. By Jensen's inequality

s (A2

constant of

wq),

and we have finished. H A similar statement applies to Ap, namely, Corollary 6.2. Assume w is a nonnegative, locally integrable function, and for some 7 > 0, w" E Ap, 1 s p < 00. Then In w E BMO, Proof. If p s 2, then also w" E A 2 ,and the conclusion follows by Proposi, < 2, and again tion 6.1. If, on the other hand, p > 2, then w - " ' ( ~ - ' ) E A p rp' by Proposition 6.1 In( w - q ' ( p - l ) ) E BMO.

Proposition 6.3. Assume w is a nonnegative, locally integrable function. Then w E Ap if and only if

with c independent of I. As the proof should be obvious by now we omit it. Note however that by Jensen's inequality each factor in (6.4) is at least 1 and consequently the membership of w in Ap is equivalent to two separate conditions, to wit

and

An interesting application of these results is to evaluating the distance from BMO to L", more precisely, an estimate of the expression infgGL-ll4 gll* ,4 E BMO.What is relevant here is the quantity 7(4) defined as follows. Let 7 > 0, verify

IX. A, Weights

242

and put 7( 4 )= sup{7: (6.5) holds}. Two properties of this quantity are readily verified, namely, by the John-Nirenberg inequality 7( 4 )2 c/ )I4 (I*, c > 0, and ~ ( -4g ) = ~ ( 4for ) each bounded function g. We then have

Theorem 6.4 (Garnett-Jones). stants cl, c2 such that

There are absolute (dimensional) con-

c1/71(4) s infgsd14 - gll,

=Z

c2/7(4).

(6.6)

Proof. The left inequality in (6.6) follows at once from the comments preceding the statement of the theorem and we say no more. Next let 4 E BMO and pick 7 so that 7 ( 4 ) / 2< 7 < ~ ( 4 )by; Proposition 6.1 eT4 E A2 and consequently by Theorem 5.4 there are Al weights wl, w2 such that eT4(y) = wl(y)/ w2(y), or

74b)= In W l b )

- In

W2(Y).

(6.7)

Now since on account of Proposition 3.3 Mw,(x)" E A l , 0 < E < 1, with Al constant independent of wl, by Proposition 6.1 it follows that llln wlII* 6 absolute constant, and similarly for w2. Furthermore, since wl(y) zs Mw,(y) s cwl(y) a.e., the function g,(y) = ln(w,(y)/Mw,(y)) E L", and similarly for g 2 ( y ) = ln(w2(y)/Mw2(y)). Thus rewriting (6.7) as (In Mwdy) - In Mw2(y)) + (ln(w1(y)/Mwdy)) - 1n(w2(y)/Mw2(y)) = b ( y ) + g(y),sayweseeatoncethat4(y) = b ( y ) / +~ g ( ~ ) / r lIlg/71lms , 00, I(4 - g / 7 (I* s absolute const/ 7 s c/ 7(4 ) . Therefore the right inequality in (6.6) also holds and the proof is complete. 7. AN EXTRAPOLATION RESULT

This section is devoted to an important extrapolation property the A, weights verify; first we need some definitions and preliminary results. Definition 7.1. We say that the pair (w, v ) of nonnegative, locally integrable functions w, v satisfies the A, condition, 1 < p < 00, and we write (w,v ) E A,, if for all open cubes I and a constant c independent of I,

The infimum over the c's on the right-hand side of (7.1) is called the A, constant of (w, u ) , and a statement involving the pair (w, v ) is said to be independent in A, if it only depends on the A, constant of the pair, rather

243

7. An Extrapolation Result

than on the particular functions involved. Similarly, we say that ( w , u ) E A , provided that

where c is independent of I; the statement independent in A, has the obvious meaning. An example of a result independent in Ap is the following: let d p ( y ) = - w ( y ) dy, d v ( y ) = ~ ( y ) - ” ( ~ -dy, ’ ) v regular and doubling, then the weak type estimate (7.3) holds provided ( w , u ) E A p ,with the constant c in (7.3) independent in Ap. We do the case p > 1; first observe that, by Holder’s inequality and (7.1), for f b 0 we have

where c is the Ap constant of ( w , u ) . Whence by (7.4) it follows that, if

fi > A, then also

L

Let now K be an arbitrary compact subset of { M f > A}, by the estimate (7.9, and as in (1.6) we see that p ( K ) s ch-’ J R n f ( y ) ’ d v ( y )with c independent in A, and we have finished. Another result of interest to us is Proposition 7.2. Suppose 0 < 7 s 1, 1 < p < 00 and w E A p . Let g L”;”(R”) and consider G ( y ) = ( M ( g ’ / ” w ) ( y ) / w ( y ) ) ” .Then

(i) [IGl[L$’”ScIlgllL$/” and, (ii) (gw, G w ) E A”+p(l-”).

E

IX. A,, Weights

244

Furthermore, both the constant c in (i) and the Al)+p(l-l)) constant of the pair (gw, G w ) are independent in A,,. Proof. Statement (i) has essentially been proved in Section 5. As for (ii), let q = 7 +p(1 - 7 ) and note that q - 1 = ( p - 1)(1 - 7 ) 3 0, or q 3 1. If 7 = 1, then also q = 1and since G = M ( g w ) /w we have that (gw, G w ) E A , with A, constant 1. Let then 0 < 7 < 1, on account of (7.1) we must show that

x

(i

~ I ( M ( g ' i n w ) ( y ) / w ( y ) ) - q l o l w ( y ) - ' / qd-y)"' '

s c (7.6)

for a constant c independent of I and independent in Ap.Now, by Holder's inequality with indices l / and ~ its conjugate 1/1 - 7 we see at once that

I,

Also since for each y in I M ( g ' / " w ) ( y )3 (1/1I1) g ( x ) ' / " w ( x )dx and q - 1 = ( p - 1)(1- v), the other integral in (7.6) is dominated by

Whence, by multiplying (7.7) and (7.8), we get that the left-hand side of (7.6) is bounded by

s (Ap constant of w)'-?, thus (ii) holds and we are done.

Remark 7.3. Proposition 7.2 may be restated as follows: assume 1 S po < p and w E Ap; then to each nonnegative function g in L F / P ~ " ( R "there ) corresponds a function G 3 g such that 11 GI1L',pIp~)' s c 11 g I( J ~ ' , / ~ O " and

245

7. An Extrapolation Result

(gw, G w ) E A,, with both c and the A, constant of the pair (gw, G w ) independent in Ap.Actually a stronger result holds, namely, Proposition 7.4. Assume 1 s po < p , w E A,,;then to each nonnegative function g in L ~ / P o " ( R "we ) may assign G 2 g such that IIGllLF/po)'d c 11 g 11 L:Ipo" and G w E A,, with both c and the A, constant of Gw independent in Ap.

Proof. We proceed by induction. Let go = g and put g , in place of G in Remark 7.3. Here, g , verifies the estimate llg, 11 L$'po" S cllg 11 L$IPo", and by (7.3) the inequality

J

Ape {Mf>A)g o ( y ) w ( y )dv

k J R j f ( y ) l p o g l ( y ) w ( y )dy

holds for each f in Lp/po", A > 0 with constants c, k independent in A p . We can use g1 in place of go and so on; in general, given g,, we obtain gj+, 3 gj such that ~ ~ g , + , ~ ~ Lcllg,IILypo"d yP~)'~ ci+lllgollLypo" and the estimate

J

g j ( y ) w ( y ) dv

{Mf>At

k

J

R"

If(y)lpogj+l(y)w(y)dv

(7.9)

(7.10)

holds for every f in Lp/po",A > 0, with constants c and k independent in A,,. Now put G ( y ) ='C:o ( c + l)-jgj(y), where c is the constant in (7.9); since ( c + l ) - ' ~ ~ g j ~ ~ L $d~(mc /) c' + lyllgllLp'po" the series defining G converges in LjlpIPo)' and we readily see that G 2 g and IIGllLF'po)'d ( c + l ) ~ ~ g ~ ~ LMultiplying ~ ~ p o ) ' . each inequality (7.10) by ( c + l ) - j and summing over j we also get

i.e., the Hardy-Littlewood maximal function maps L2w(R") into wk-L2w(R")with norm independent in Ap. We may then invoke Theorem 2.1 part (iii) to infer that actually G w E A, with A, constant independent in Ap. Proposition 7.4 has a counterpart for the case p < po as well, namely, Proposition 7.5. Assume 1 < p < po and w E A p ;then to each nonnegative we may assign G 3 g such that 11 GI1Lf'po-p' s function g in LP,/'Po-P'(R") c ~ ~ g ~ ~ L $ ( pand o-pG ) - ' w E A,, with both c and the A , constant of G - ' w independent in Ap.

IX. Ap Weights

246

hf. We dualize Proposition 7.4; our assumptions are equivalent to 1 < p6 < p', u = w - ~ ' ( ~ - 'E) Ap,,and Ap.constant of 0 = Ap constant of w. Let d v ( x ) = u ( x ) dx.Then by Proposition 7.4 we conclude that to each nonnegative h in L$"lP;'(R") there corresponds H 2 h such that [ ~ H ~ ~ L s~ p ' ~ p ~ ) ' c ~ ~ h ~ ~ and L ~ Hu ' ~ Ep A:, ~ ) with ' c and the A,; constant of Hu independent in Ap.But (p'/p&)'= (po - l)p/( po - p ) so that h E L?'lp;)' is an equivalent way of saying h P o - l ~ - ( p ~ - P ) l (EPLpVl(po-p). -') Also Hu E Aio if and only if ( Hu )- I / ( P ; - ' ) = ( H P o - l W - ( P o - P ) / ( P - I ) ) - l W A . Thus given a nonnegative function g in L',/(po-P), we put g = hpo-lW-Ppp,-p)/(p-l) with h in L$P'/PO', obtain the corresponding function H from Proposition 7.4 and then define G = HPo-1 w-(Po-P)/(P-I).

We are now ready to prove the extrapolation theorem alluded to at the beginning of this section.

Theorem 7.6 (Rubio de Francia). Assume T is a sublinear operator which verifies the following property: there is a p o , 1 s po < 00, such that for every w E Am, d p ( x ) = w ( x ) dx,

IIrfllL2

cllfII~2,

(7.11)

where c is independent off and independent in Am. Then for every p with 1 < p < 00, and every w in Ap, T also satisfies the inequality

II

m.::

CllfII.:,

(7.12)

where c is independent o f f and independent in Ap. h f . We consider two cases; first suppose 1 s po < p , and let w E Ap and

f~ L',. As is readily seen

where the sup is taken over nonnegative g in L',"po" with IlgllLF'po"S 1. Fix such a function g and assign to it the function G 5 g of Proposition 7.4. Then by (7.11) and the properties of G given in that proposition we see that the integrals in (7.13) involving g are dominated by

where the constant c is independent of g and independent in Ap. Thus (7.12) holds and the discussion of this case is complete. Next suppose that

8. Notes; Further Results and Problems

247

1 < p < po and again let w E Ap and f E L:. Put g ( x ) = IlfI14Polf(x)lPo-P where f ( x ) # 0, and g ( x ) = 0 otherwise. Now note that

I

{f+-O)

If(x)l”og(x)-’ d P ( X ) = llfll2;

(7.14)

= 1. We are then in a position to invoke Proposition 7.5 and Ilg(lL;/(po-p) and obtain a function G 2 g with the properties given there. Observe that

where the constant c is independent in Ap.

8. NOTES; FURTHER RESULTS AND PROBLEMS

As expected, weighted inequalities are important in considering weighted mean convergence of orthogonal series, since the error terms can almost always be majorized by some version of the Hardy-Littlewood maximal function. In this context see Rosenblum [ 19621 and Muckenhoupt [ 19721. They are also important in the pointwise convergence of Fourier series as well: let s * ( f ;x ) = sup,Js,,(f; x)l, then Ils*(f)llL;S cllfllL;, 1 < p < 00, provided w E Ap (cf. Hunt and Young [1974]). Ap weights and their basic properties have been studied extensively; for instance, Feff erman and Muckenhoupt [ 19741 showed there are doubling measures which are not Ap weights for any p 3 1, and Stromberg [1979b] constructed examples to show that aside from the obvious implications, conditions such as doubling and reverse Holder as well as others we discuss in this section are independent of each other. The proof of Muckenhoupt’s Theorem 4.1 we present here is essentially due to Sawyer [19821 and Jawerth [ 19841 and it does not rely on the (difficult) implication “Ap implies Ap--.” as did the original proof. Sawyer’s idea is somehow related to the notion of Carleson measure which will be discussed in Chapter XV. The reader will note, however, that once the elements for the proof are set up, it very much looks like a Calder6n-Zygmund decomposition argument, especially relation (4.1). In fact Christ and R. Fefferman [1983] have shown that this is precisely the

ZX. Ap Weights

248

case; we prefer to give the more abstract proof since it applies to the very general context considered by Jawerth. The proof of the Jones Ap decomposition theorem given here is due to Rubio de Francia [1984]and that of Rubio de Francia’s extrapolation theorem is due to Garcia-Cuerva [1983].

Further Results and Problems

8.1 Suppose a nonnegative Borel measure p is doubling. Show that (R”

‘p(y) =

Let p be a regular, nonnegative Borel measure and let f~ L,(R”). Show that for some constant c, independent o f f , and A > 0, A p ( { M J > A}) S ~ j { ~ ~ , ~ J fd(p y( y)) I. (Hint: For a fixed A let 0 = { M J > A} and put f = fxo + f ~ ~ n = , ~f1+ f 2 , say. If we can show that 6 E { MJ1 > A}, then by Theorem 1.1 we are done; but this is easy since to each x in 6 there corresponds an open cube Z containing x such that (l/p(Z))j I l f ( y ) I d p ( y ) > A, which in turn implies that inf, M J > A and f = fl on I.) 8.3 The proof of Theorem 1.1 relied on a careful selection procedure of cubes out of an arbitrary family; results of this type are known as “covering lemmas.” That proof may also be obtained by making use of the following covering lemma, due to Wiener: Let E be a Borel measurable subset of R ” which is covered by the union of a family of open cubes {I,} of bounded sidelength; then from this family we can select a disjoint subsequence { r j } so that p ( E ) d c C p ( r j ) , where c is a constant that depends only on the doubling constant of p. Prove this lemma. (Hint: Choose I1essentially as large as possible, i.e., sidelength ZI2 i sup,(sidelength I,), discard any cube which intersects Zl,and so on.) 8.4 Maximal results, in turn, imply covering results; the following is an example: Assume that for a nonnegative, Borel measure p and some 1 < p < co the mappingf+ MJverifies IIMJllL: s cllfllL;, c independent o f f ; then given any finite collection of open cubes {Z,} it is possible to select a sequence { r j } so that (i) Z,) S c,p(U Zi) (that is the Zi’s cover a good portion of Z,) and (ii) jR.(C x1,(y))”’d p ( y ) S c 2 p ( u r j ) (that is the overlap of the Is’s is small when measured in L$( R ” )norm, l/p + l/p‘ = 1);the constants c l , c, depend only on the norm of the maximal operator and on p. (Hint: Since the Is’sare finitely many they may be ordered and we choose the first cube as I ] . For 1, we choose the first I, among the , n Zl)S $p( Z,). For Z3we choose the first I , remaining cubes so that p ( I among the cubes listed after Z2 so that p(Zp n (IIu I,)) S ip(Z,) and so on. Note that if an Z , was not selected then we have p(Z, n 4)) > ?p(Z,) 8.2

u

p(u

(u

8. Notes; Further Results and Problems

249

consequently p(uI s ) S p ( { M p ( x u I j> ) $1)c C ~ ~ I ) X U I ~=I I L ; c , p ( u $ ) and (i) holds. Next observe that if Ek = I k \ U j < k l j , then E L(&) 3

and

&.&(Ik).

We define now a linear operator T: L;(R") + LP,(R")as follows:

Clearly I Tf(x)l e M d ( x ) .Moreover, the adjoint T* :L$( R " ) + L$( R " ) can be explicitly written as

and consequently

and by taking p' norms we get (ii). this technique of proof is known as "linearization" and since at no point did we use the fact that the Is's were cubes the reader is invited to state and prove a general result in this direction. The substitute result for the case when the maximal operator is of weak-type ( 1 , l ) should also be considered. The proof above is from C6rdoba's work [ 19761 and the weak-type result was done independently by C6rdoba [ 19761 and Hayes [1976].) 8.5 Under very general conditions Corollary 1.3 admits the following converse (we only discuss the unweighted version here): a collection B = {B} of open, bounded subsets of R" is said to be a translation invariant Buseman-Feller, or B-F, basis if for each x in R" there is a subfamily B ( x ) of B such that (i) if B E B ( x ) , then x E B, (ii) each B ( x ) contains sets of arbitrarily small diameter; (iii) B ( x ) = x + B(0). Suppose that B differentiates L ( R " ) , that, is limlBI+o, B,sacx,(l/lBI) j B f ( y )dy = f ( x ) a.e.; then the mapping . f ( x ) M a f ( x ) = supBGB(x)(l/lBI)I B l f ( y ) l dy is Of weak-type ( 1 , l ) . (Hint: Suppose not and proceed exactly as in Chapter IV; the result is from de GuzmLn and Welland [1971].) 8.6 Although Ap is both necessary and sufficient for the ( p , p ) type and weak-type of the maximal operator, the same is not true for the restricted weak-type: the inequality A P d p ( x ) S c p ( E ) ,all A > 0 and measurable subsets E c R", is equivalent to the existence of a positive constant K such that for all cubes I and Lebesgue measurable E E I, lEl/lIl c K ( p ( E ) / p ( I ) ) ' / " (Hint: . Since M x E ( x )2 ( ~ E ~ / ~ I ~the ) ~condition ,(x) is necessary. Conversely, we readily see that M x E ( x ) K ( M d E ( x ) ) ' / P and that p is doubling. The result is Kerman's and the proof appears in Kerman and Torchinsky [ 19821.)

I{MxE,A)

IX. Ap Weights

250

For a nonnegative, locally integrable function w and an open cube = (tlf2)1/2 where tl = sup{t > O:I{x E I : w ( x ) s t}l s 1Z1/2} and tz = inf{t > 0: I{x E I : w ( x ) > t}l s lIl/2}. Show that for any real number a we have m,a(Z) = (mw(Z))"and ( w " ) a~$ ( m w ( Z ) ) aFurthermore, . we say that w satisfies condition A if w1 S cmw(Z),for a constant c independent of I. It is clear that, if 0 < a d 1 and w satisfies condition A, then ( w a ) , (m,( I ) ) a ,all I. Moreover, Apweights have the following interpretation in terms of the condition A : w E Ap if and only if w and w - ' / ( ~ - ' ) both satisfy condition A , 1 < p < 00. (Hint: One direction of the last The other follows statement is trivial since (m,( Z))-l/(p-l) = mW-m-1)(J). and ( W - l ' ( p - l ) ) r 2 from the inequalities W r 2 imw(I ) = $(r n , - m - l ) ( $ ( m w ( I ) ) - l / ( p - lThese ). results and those in the next remark are due to Stromberg and appear in Stromberg and Torchinsky [19801; they should be compared with Proposition 6.3.) 8.8 Let 1 < p , r < 00; then w E Ap n R H , if and only if w r and W - I ' ( ~ - ' ) satisfy condition A. (The statement is equivalent to w" satisfies condition A for all - l / ( p - 1) d a < r.) 8.9 Assume that a nonnegative function w verifies l{y E I ; w ( y ) < W r / A k } l s cvklIl,all I, where c is independent of Z and 0 < 7 < 1 < A < 00. Show that there is a p > 1 such that w E Ap. (Hint: Note that 8.7

Z put m , ( I )

-

< 1.) and choose p so large that 7A1'(p-1) We say that w verifies Muckenhoupt's A , condition, and write w E A,, if to each 0 < E < 1 there corresponds 0 < S < 1 so that for measurable subsets E of I we have p ( E ) < & p ( I whenever ) IEl < 8111. By Proposition 4.5(iii) each A, weight is an A, weight. Show that the converse is also true: if w E A , there is 1 < p < 00 so that w E Ap. (Hint: It suffices to show that for appropriate A and 7 the assumptions of 8.9 hold; A = 8 and 7 = (1 - d / 2 ) , where d is the 6 corresponding to E = $ will do in the one dimensional case, the n dimensional case requires minor adjustments. To see this fix Z and k, let E = {x E I : w ( x ) < ~ 1 / 8 ~and } , observe that p ( E ) < P(I)/< ~~ p ( 1 ) / 2 implies [El < (1 - d)lIl. Now since almost every x E E is a Lebesgue point of xE and Lebesgue measure is regular we may assume 8.10

8. Notes; Further Results and Problems

251

that E is compact and each point of E is a Lebesgue point of ,yE. To each x in E we may assign an open interval I, centered at x such that 11, n Z n El = (1 - d ) ( I , n ZI (this is possible since for I, large I, contains I and IEl < (1 - d)lIl and for I, small I, c I and 11, n . E ~ / ~+ I , l), ~ I, G cZ, c independent of x and I. Let S = UxGE I,, since E is compact we may assume that S is finite and choose I , as an I, in S of largest length. Then after 11,. . . ,I k have been chosen let s k be the family of the remaining 1,’s so that x fZ 4 and let I k + l be a largest interval in s k . Observe that each y in 4 belongs to, at most, two of the 4 ’ s and put El = Uj(4n I) E Z. Then ~ ( ~ 61 Cj1 d d y ) G 2 C j J E n i j n i d p ( y ) (since IE n IJ n 1 = implies p ( 4 n I ) 6 2 p ( n (I - d)l4 n 1 ~ 4 n I ) ) s 4jEuijuid p ( y ) (since each y belongs to at most two of the 4 ’ s ) “41, d p ( y ) s 4 ~ ( I ) / 8 ~ . How about the Lebesgue measure of El? Well,

u

uj”=l

IIjnr

’

Now, if k 3 2, it is possible to start with p ( E , ) < P ( E ) / ~ ~ - ’ or lE.11,31El/~. and repeat the above argument with E replaced by El. This gives E2 c I, p(E2) 6 and lE21 > IE1/q2; repeating the process k times we are done. This result is Muckenhoupt’s [1974] and insures that P A , = Uisp 0 independent of E and I, (ii) v is comparable to p, (iii) p is comparable to v, (iv) d v ( x ) = w ( x ) d p ( x ) where w E R H l + ” ( d p ) i.e., ,

for every cube I. Moreover, comparability is an equivalence relation. (The proof uses ideas similar to the ones discussed in this chapter, (iii) implies (iv) is the hardest implication. This observation is from CoifmanFefferman’s work [ 19741.) 8.12 A nonnegative weight w E A, if and only if

IX. A, Weights

252

I,

( ~ i n t : Since limp+m((l/lIl) w(x)-'/(p-')dx)p-' = e(('/lZl)J, In('/w(x)) h) the assertion here is that A, is obtained as the limiting A, condition, as p -* 00, much like A, is obtained as the limiting condition as p + 1. The proof is computational, and the necessity follows at once from 8.11 and Jensen's inequality. As for the sufficiency, let A denote the sup over I of the expression in question, A < 00. Then for each interval I and disjoint subsets of positive measure E, F of I, E u F = I, we have In A 2 In( w , ) ( I E l / l W n W ) E - (IFI/lIl)(ln W ) F , where (In W ) E = (1/lEl) In w ( x ) dx, and similarly for (In w ) ~Since . by Jensen's inequality (In w ) S~In( w E )we also have that

JE

Putting t = IFl/lIl and T = p ( F ) / p ( I ) we finally get 1nA 2 (1 - t ) ln((1 - t)/(l - 7))+ t ln(t/T); elementary considerations obtain now that for a constant c, which depends only on A, T S 1 - e-'/('-'o) = T~ < 1 provided that t < to, in other words w E A,. The proof is from HruSEev's paper [ 19841.) 8.13 There is yet another way of writing the A, condition, namely, the S, condition

Jz

M(Xrw'-P')(X)PW(X)dx s c

J,

w(x)'-"' dx,

all

I,

with c independent of I. (Hint: It is not hard to see that S, implies A,. Conversely, if w E S, and x E I c Io, then by A, we readily see that

=S CM,( X I o w-') (x)"".

Since for each f with s u p p f r I. and x in I. we have M f ( x ) = s ~ P ~ ~ ~ = ~j J~f(( ~ )/dlI ~I itI, follows ) M(XroW'-P')(X)P c ~ , ( x ~ , , ~ - ' ) ( x ) ~ ' , and S, follows from Corollary 1.2. This proof is due to Hunt and Kurtz and Neugebauer [1983]; an indirect proof follows from 8.14.) Assume that 1 < p S 8.14 (Sawyer's Two Weight Maximal Theorem) q < 00, and that u, w are nonnegative, locally integrable functions in R". Then the following two conditions are equivalent: (i) ((,. M f ( x ) % ( x ) dx)'/¶d C ( ~ , . ~ ~ ( X ) ~ ~ dx)'IP, U ( X ) with c independent of J: (ii) I, M ( ~ r u ' - P ' ) ( ~ dx ) 4s ~ ( ~ ) U(X)'-"' d x ) ¶ I P < 00, all I, with c independent of I.

.(Iz

8. Notes; Further Results and Problems

253

The proof of this interesting result is due to Sawyer [19821, and it follows along the lines of Theorem 4.1. An identical result holds for M replaced by the maximal function M,,of fractional order introduced in Chapter VI. 8.15 With the same notation and assumptions as in 7.14 of Chapter 111, 7.29 of Chapter IV, and 6.17 of chapter VIII, prove the following: For every r; < a,the infinite product real sequence {ik}, -1 < rk < 1, with (1 + rkfk(nkX)) converges for almost every x to a function w(x) in A, for 1 < p c 00. Moreover, w E Lp(T), p < co as well. (Hint: the convergence of the product is equivalent to that of the sum s(x) = CT='=,rkfk(ng); also for some constant c, Iln w(x) - s(x)l< c. The convergence of s(x) follows by classical arguments. Observe now that for some constants cl, c2 > 0, cles(x)c w(x) < c2es(x).This result is from Meyer's work [1979].) 8.16 Under the hypothesis of 8.15 and if p*(x) = SUPN,O~N(X), pN(x) = n;="=,l+ rkfk(ng)), denotes the maximal Riesz product, then p* E A,. (Hint: First, there is a constant c > 0 so that (1/27r) JTp*(x)dx s c whenever CT=, r i c E 2 (this follows since pN(x) S cesN@)and s* E BMO). Let I c T, if II( > 27r/n0 the estimate holds trivially sincep*(x) 5 1 - ro > 0 and (l/lIl) J r p * ( x )dx s (no/2r) JTp*(x) dx. Suppose next that 27r/nN+, < 111 < 2 r / n N and let x = center of I; then there exists a constant c > 0 such that p j ( t ) c cpj(x), pj(x) c c p j ( t ) , whenever 0 c j S N and t E I. The constant c only depends on 1 r;. To see this we majorize Ilnpj(t) - lnpj(x)l by

n:='=,

i k=O

Iln(1 + rkfk(nkt)- ln(1 + rkfk(n&)l i

d c

xlrklnzlx - fl

d c.

k=O

put Yk(t) = (1 + rN+lfN+l(nN+lf)+ * ' * + (1 + rN+kfN+k(nN+kt)) and note that p * ( t ) = sup(pl(t), * * ,PN(f), PN(f)Yl(f),. . . PN(t)Yk(f), ' * ') suP(pl(x), * . PN(X), PN(X)Yl(f), . . * , PN(X)'&(t), .) c sup(a, by*( t ) ) , where y*( t ) = SUpkal yk( t ) . Similarly, p*( t ) 3 c1 sup(a, by*( t ) ) 3 c,a. We will be finished once we show that (l/lIl) I, y * ( t ) dt < c, but this is not hard. The result is Meyer's.) 8.17 Assume w is a nonnegative function defined in a cube I. which verifies

-

-

9

3

( jzi

~ ( x ) dx)" "

c c1

-

ijz

W(X)dx

for all subcubes I c I,, some p > 1 and some constant c, independent of I. Show that there is 7 > 0 so that also

254

IX. A, Weights

for p S r < p + q, all I c I, and c2 = c,,,,, but independent of I. (Suppose I = [0,1], w ( x ) , d x = 1 and put EA = { x E I : w ( x ) > A}; the inequality follows at once from the estimate

I,

IEA

w ( x ) " dx d cAP-'

I,

w(x)dx,

A 3 1,

which in turn follows from an argument not unlike that of Theorem 3.5. This result, important both in applications and motivation, is due to Gehring [19731.) 8.18 For a locally integrable function 4 put p ( 4 ) = {infp: e b and e - @ belong to A,}. Note that p ( 4 ) can equal 00; also Holder's inequality shows that ed E Ap whenever p > ~ ( 4 ) Suppose . that p ( 4 ) = 00 and show that E BMO and p ( 4 ) - 1 = E ( + ) , where E ( + ) = inf{s > 0: supr(l/lIl)l{x E I : I+(x) > A}( =s e-"lE} whenever A > A. = AO(&, 4). (Hint: We must have p ( 4 ) =s2 (if p ( 4 ) > 2, then also e f b E A2, and this cannot be); then the A, estimates for e+' yield that

+

which in turn implies ~ ( 4=s ) p ( 4 ) - 1 since q ( 4 )= I/&(+). Conversely, for p - 1 > E ( + ) , again by the fact that q ( 4 )= l/&(+), we have that

and when p - 1 < 1, Holder's inequality shows that both eb and e-' The result is from Garnett-Jones [1978].) 8.19 Verify the following statements.

E A,.

(i) ( w , u ) E Ap if and only if (u'-~',w'-,') A,., 1 < p < 00. (ii) if (w, U ) E A,, O < S < 1 and ( q - l ) / ( p - 1) = S, then ( d ,u s ) E Aq. (iii) If (w, u ) E A,, 0 < S < 1, and d p s ( x ) = w s ( x )dx, d v 6 ( x ) = d ( x ) dx,then IIMfllL;, d ~ l l f l cl = ~ ~C6.p. (Hint: Since ( w ' , u s ) E A, and p > S ( p - 1) + 1 = q, we get that ({Mf > A}) s cllfll t:, and (iii) follows by the Marcinkiewin interpolation theorem. These observations and the next three res-ults are from Neugebauer's work [19831.) 8.20 Assume that

IlM-fIlL:

and IIMfllL!;-p. s CllfllLP;,-p,; then there are nonnegative functions w l , w2 such that w ( x ) l / p M w l ( x ) c l u ( x ) ' / p w l ( x )an ; identical inequality holdsfor w2, and W ( X ) ' / ~ U ( X )=' ~ ~ ' W ~ ( X ) W ~ ( X ) (Hint: ~ - ~ . Cf. Theorem 5.5 and consider T f x ) = w ( x ) ' / p M ( l f l u - ' / p ) ( x ) + u ( x ) ~ " p s M ( ~ f l w ' ~ p s ) (where x ) ' ~ ss = p / p ' . )

ClIflL:

255

8. Notes; Further Results and Problems

Let (w, u ) E Ap and 0 < 6 < 1. Then there exists a nonnegative function u = ug such that clw'(x) c u ( x ) s C ~ U ( X ) ~and u E Ap. (Hint: Choose 0 < E, 7) < 1, 6 = ET. From 8.19 (iii) we know that

8.21

~ v where ~ Mwj ~ ~ s c ~' ~ ( u / w ) j" = ' ~1, , 2. Note Thus, by 8.20, ~ ~ /= w,w;-" that uE = wl(u / W ) ' / ~ W ; - ~3 C M W , ( M W ~ )3' - cwI ~ w:-"( u/ w ) ~ ( ~ -=~ C) W' €~ and thus clws c (MW,)"(MW,)""-~'S c2us. Put now u= (Mw,)yMw2)""-p'. 1 8.22 Let ( w , u ) be a pair of nonnegative functions, then there exists u E A, with c1w ( x ) C u ( x )C c2u(x),if and only ifthere is T > 1such that (w', uT) E Ap.(Hint: Since u, u'-~'satisfy a reverse Holder inequality, there is T > 1 such that u' E A, and ( w T ,uT) E Ap. As for the converse, use 8.21 with 6 = l/T.) 8.23 Suppose that w is a nonnegative function and show that ( w , M w ) E Ap, 1 < p < 00. In particular M f ( x ) " w ( x )dx c c I R n l f ( x ) I p M w ( xdx ) in the same range of p's. (Hint: The proof is reminiscent of Proposition 7.2 (ii). If wI = 0 there is nothing to show. Otherwise, note that inf,Mw 5 w I ; thus ( w , M w ) E Ap with Ap constant 1. Consequently, if u ( x ) = M w ( x ) ,the maximal operator maps L: into wk-LP,for 1 < p < 00 and by interpolation also into Lc for the same p's. The result concerning the integral inequality was originally proved by Fefferman and Stein [1971].) 8.24 Given a nonnegative function w and 1 < p < 00, the following conditions are equivalent.

I,.

I,.

(i) There is a nonnegative, finite a.e. function u such that M f ( x ) " w ( x )dx S c J,nlf(x)l"v(x) dx, c independent of J: (ii) J,n w ( x ) / ( l + 1 ~ 1 dx~ <) 00. ~

(Hint: (i) implies (ii) follows by considering f = xA, measurable A. (ii) implies (i) requires a bit of work; by replacing w by max(1, w ) if necessary, we may assume that w 2 1. Let now u ( x ) = (1 + l ~ l " ) ' - ~note , that M ( u w ) ( x )< 00 a.e., and put u ( x ) = M ( u w ) ( x ) / w ( ~ )d~v (, x ) = v ( x ) dx. For k = 0, 1,. . . ,letfk(x) = f ( x ) ~ ( ~ k ~ l ~ l < then ~ k +by~ 8.23 ~ ( xit) follows ; that c

"

IX. Ap Weights

256

For 1x1 2 2k+Z,we have

where the last term can be estimated and summed. The above proof is from Young's paper [ 19821 and the result was obtained independently by Gatto and Gutierrez [1983]. The result followed this observation of Rubio de Francia [ 19811: given a nonnegative function u and 1 < p < 00 the following conditions are equivalent. (i) There is a nonnegative, finite a.e. function w such that

I,. M f ( x ) " w ( x )dx d c ~ R n l f ( x ) l p u (dx,x ) c independent of J: (ii) U ( X ) - ~ ' ( ~ - ' ) is locally integrable and lim S U P ~ + ~ I Qx~ I - ~ ' I, u( x) dx < co, where QR = {x E R": maxlSjSn lxjl 9 R}.) -l/(p-')

We say that a locally integrable function b has bounded lower oscillation, and we write b E BLO, if b, - inf, b s c, all I, where c is independent of I; BLO c BMO. Then b E BLO if and only if eqb E Al for some 7 > 0. (Hint: If eqb E A,, then ( e q b ) ,S c inf, eqb and the conclusion follows by Jensen's inequality. To prove the converse, note that the JohnNirenberg inequality gives ( e q b ) ,s c d q b ) 1 ,for 7 sufficiently small, and the conclusion follows easily from this. This means, in particular, that each b in BLO may be written as 7 l n f + h, where f s 0 is integrable and h is bounded. These results are Coifman and Rochberg's [19801.) 8.27 Theorem 7.6 has a weak-type version. More precisely, if T is a sublinear operator which verifies the following property; there is 1 S po < co, such that for every w E Ah, d p ( x ) = w ( x ) dx, and A > 0, A P o p ( { (TA > A}) s c llfll FLo, where c is independent off and independent in A,; then for every p with 1 < p < 00 and every w E Ap, T also verifies App({ITA> A}) d cllfll;;:, where c is independent o f f and independent in Ap. (Hint: In the first place if po < p , then for A > 0, 8.26

APOP({ITA > A } ) d P= APollX{,TJl>A)llL,/Po =

JR"

x{ITfl>A)(x)g(x) w ( x ) dx,

for some g 3 0 with IlglJLJp/po)s = 1. Associate with g a function G as in Proposition 7.4 and apply the weak-type assumption. If, on the other hand, 1 < p < p o , use Proposition 7.5 instead. The result is Garcia-Cuerva's [ 19831.)

257

8. Notes; Further Results and Problems 8.28

w

E

A nonnegative function w is said to satisfy the AP,¶condition, or AP,¶,if

where c is independent of I. The infimum over the c's above is called the AP,¶constant of w and a statement is said to be independent in AP,¶if it only depends on the Ap,4constant of the weights involved. Show that, if T is a sublinear operator which verifies

for some pair ( p o , qo), 1 < po s qo < 00, and all w E Ah,%, where c is independent in Apo,40, then it also satisfies the same inequality for any other pair ( p , q), 1 < p s q < 00 with l / p - l / q = l/po - l/qo for every w E AP,¶ with norm independent in AP,¶.(The proof, which is similar to that of Theorem 7.6, is in Harboure-Macias-Segovia [ 1984b1). 8.29 Let Mf,(x)

1

= sup -{)f(Y)l I

lIIl-7

dY,

where 0 < 77 < 1,

and the sup is taken over all open cubes I containing x. Then, if 0 < l / q = l / p - 77 < 00 and w E AP,¶,there exists a constant c, independent off and independent in AP,¶,such that

(I,"

M,f(x)'w(x)¶dx)liq

l/P

-

c( j R " I f ~ X ~ l P W ~ X ~ ~ d X )

Is the converse true? (Hint: The proof follows by a combination of the ideas discussed in this chapter; as an illustration we do the (easier) weak-type result. Assume that ~ , n l f ( x ) l p w ( x ) p d=x1 and note that by Holder's inequality and AP,¶we get that

the weak-type estimate follows without much difficulty from this. (Cf. the remarks in 8.14).)

258

IX. Ap Weights

8.30 For 0 < l / q = l / p - (Y < 1 the condition Ap,q is necessary and sufficient for the mapping

to verify

(cf. Muckenhoupt-Wheeden [19741). Welland [19751 observed that one may prove this inequality by using Theorem 2.4 in Chapter VI and 8.29. Results similar to 8.24 in this context have been established by Rubio de Francia [ 19811 and Harboure-Macias-Segovia [ 1984al.) 8.31 Liifstrom [1983] has shown there exist no nontrivial translation invariant operators on LP,(R”),if d p ( x ) = w ( x ) dx and w belongs to a class of rapidly varying weight functions, including for instance w ( x ) = e*lxla, (Y

> 1.