Chapter 1 Pointwise Convergence of a Sequence of Operators

1

CHAPTER

POINTWISE CONVERGENCE OF A SEQUENCE OF OPERATORS

There a r e q u i t e a number o f i m p o r t a n t problems i n F o u r i e r A n a l 1

s i s and i n o t h e r areas i n which a sequence ( o r g e n e r a l i z e d sequence ) o f o p e r a t o r s a r i s e s i n a n a t u r a l way. i)I f

f

E

L1([0,2x)),

f

periodic o f period

2 ~ r ,t h e p a r t i a l

sums

F o u r i e r s e r i e s can be i n t e r p r e t e d as t h e a p a l i c a t i o n o f t h e o D e r a t o r t o the function where

Dk(x)

f . E q u i v a l e n t l y S k f ( x ) = Dk

, the

*

f(x) =

D i r i c h l e t k e r n e l , i s d e f i n e d by

1

TI

Sk

f(x -y)Dk(y)dy

-Ti

s i n ( k t T1) x Dk(X) =

X

TI s i n 2

ii)I n an analogous way, t h e Cesaro sums

Okf(X)

=

S,f(x)

-t

Slf(X)

-t

.,. -t

Skf(X)

k + l

can be i n t e r p r e t e d as t h e a p p l i c a t i o n o f t h e o p e r a t o r i s t h e F e j 6 r k e r n e l , d e f i n e d by

Fk(x) =

1

2n( k + l )

1

ak

to

f. I f

fk

1. POINTWISE CONVERGENCE OF OPERATORS

2

We know t h a t

parameter

r

A r f ( x ) = Pr

,is

*

f(x)

Pr(x) =

i v ) If f E L1(Rn)

i s t h e mean v a l u e o f

f

over

where

1 -

2n

t h e Poisson k e r n e l of

Pr(x),

1-r2

1 - 2 r cos x + r 2

, B(0,r)

B(x,r)

= Cx

e Rn : 1x1

6

r

I

which i s c o n s i d e r e d i n t h e t h e o r y

o f differentiation o f integrals. v ) I n a more general way

\

,

k ( y ) d y = 1, k E ( x ) =

if

E

-n

k E L1(Rn)),

k),(

x

for

E

> 0, t h e n

a r e t h e o p e r a t o r s considered i n t h e s t u d y o f t h e a p p r o x i m a t i o n s o f t h e

id e n t it y

.

vi) If

f E L’(R”)

and

if

1x1

if

1x1 <

2 E

hE(x) = 0

,

E

1.0. INTRODUCTION

then

i s t h e truncated ( a t v i i ) If

k

E

) Hilbert transform.

i s a complex valued function defined on Rn--COI

such t h a t

then, f o r f 8 L 1 ( R n ) , KEf(x) = k E * f ( x ) i s t h e truncated ( a t E ) Calder6n-Zygmund transform of t h e function f , considered i n t h e theory o f singular integral operators. The most natural question in a l l t h e s e s i t u a t i o n s i s : ( A ) Tv dind vu.t W h d h a ,Oh u n d a which a d d i t i v n d nvn & v i a l c v n d i t i v a vn 5 v h an t h e vpehatvlrn Th ,the cornuponding bequence Tk6 ( x ) cvnuagen 6vh e v a y x O h denvot eVt?hy x and whdt ahe t h e p m ~ W e h ad the,& h L i 2 .

This i s , i n a l l the cases we have considered, a t the same time t h e most d i f f i c u l t question. In many of them, because of t h i s d i f f i c u l t y , t h e theory s t a r t e d w i t h a l e s s ambitious program: ( B ) Tv 5ind o u t w h e t h a , V R u n d a which a d d i t i v n d nvn .&ivial

cvnd&..LvnA vn 4 ah On t h e v p a d t v h n T k . t h e CvntLupanding oeyuence ~ u n C ; t i v a ~~6 cvnuenga i n Avme opace LP.

04

Question B has in many of t h e above presented s i t u a t i o n s a r a t h e r simple s o l u t i o n . Suppose t h a t we know t h a t T k i s l i n e a r , t h a t

4

1. POINTWISE CONVERGENCE OF OPERATORS

we can show, f o r example, t h a t , i f

11 T k f 11 6 c g E t o(Rn)

say,

Tg.

Ilf

p,q

]I f -

g

> ko,

.

c

w i l l converge i n

Hence, g i v e n

L2 (R')

with

independent o f

we a r e a b l e t o show t h a t

L e t us prove t h a t

and so T k f

2c

11

f E L2(Rn),

112

E

11 Tpg -

Tqg

L2(Rn).

T h i s would s o l v e q u e s t i o n

k

, and

and t h a t that for

L2(Rn)

Tkg i s t h e n a Cauchy sequence

{Tkfl

We can w r i t e , f o r

f E

tomn)

f

to, i n L2(Rn)

ego

(Rn),

such t h a t

g i s f i x e d , k o such t h a t , i f I T k f } i s a convergent sequence i n

and, once 6 4 2 . So

E/2

and

converges i n

> 0, we can f i r s t choose

6

f

Tk f E L2(Rn)

(B) and would l e a v e unanswered q u e s t i o n

( A ) . What can we do t o t h r o w some l i g h t on i t ? L e t us t a k e a c l o s e r l o o k a t i t s meaning. Assume, as b e f o r e , t h a t Tk i s l i n e a r . We wou d l i k e t o be a b l e t o prove, f o r example,that,

[A(f,X)I

=

for

f E L1(Rn)

( t x e R n : l i m sup I T f ( x ) P PY9

-

and

X

> 0,

Tqf(x)( >

X 1 = o

-+

T h i s would g i v e us t h e convergence o f Assume t h a t we know t h a t , f o r converges. Then, if h = f

- g,

g e

{Tkf(x)) 0(Rn)

and so t h e problem i s reduced t o prove t h a t if

h

fixed

i s o f small

X

L'

a t almost e v e r y

and f o r each

A(h,X)

x E Rn.

x E Rn , { T p g ( x ) l

i s o f s m a l l measure

- norm. Assume t h a t we can p r o v e t h a t , f o r each

> 0,

T h i s would s o l v e o u r prob em.

1.0. INTRODUCTION

5

However, t h e s e t A(f,A) has a r a t h e r unhandy s t r u c t u r e and so one can think of s u b s t i t u t i n g i t by some o t h e r e a s i e r t o handle. I t i s quite clear that lA(f,A)

defined by T*f(x) =

sup lTkf(x)[

has a r a t k k e r simple s t r u c t u r e . We may hope t h a t we w i l l be able t o prove now t h a t and t h a t t h e oper t o r T*

0 , and t h i s w i l l as well give us our desired IA*(fy A ) I 0 a s 11 f 1 1 1 almost everywhere convergence of {Tkf} +

-f

.

So we a r e led t o consider t h e operator

T*

defined by

.

I f {Tk} i s which i s c a l l e d t h e maxim& a p e h a t a h associated t o { T k } an ordinary sequence, k = 1,2,..., T*f i s c l e a r l y measurable. I f k i s not countable one has t o prove t h a t T*f i s anyway measurable o r e l s e t o deal with t h e o u t e r measure o f tT*f > A I . The operator T* i s such t h a t f o r each f and x, T*f(x) a 0 and, i f the Tk a r e l i n e a r , we can w r i t e

The relevance of t h e operator

T*

stems from t h e r o l e i t plays

in t h e pointwise convergence proofs, as i n d i c a t e d , and in t h e information i t furnishes about t h e l i m i t , when i t e x i s t s . Assume, f o r example, in t h e l a s t mentioned s i t u a t i o n , t h a t we can prove t h a t f o r each

f 6 L1(Rn)

with

f

c

independent of

6

L’(Rn)). Then we obtain f o r each

X

0,

1. POINTWISE CONVERGENCE OF OPERATORS

6 and so

ICT*f > X

11

-f

0

as

where convergence r e s u l t .

II Tf III

C IIT*f IIi

c

X

1) f ]I1

-f

0. Thus we o b t a i n t h e a l m o s t e v e r y

Furthermore i f t h e l i m i t i s

Tf,

I1 f I I 1 .

O f course,in o r d e r t o o b t a i n t h e almost everywhere convergence,

(*)

condition

i s somewhat s u p e r f l u o u s and sometimes f a l s e . I t i s good

enough t o know t h a t

f e L1@)

f o r each

.

X

and

O r even j u s t t o know t h a t f o r each

When

(**I

Condition at

o

X > 0 , with

and f o r each

from

L

to

(***) j u s t says t h a t

Observe t h a t c o n d i t i o n

(**)

i s o f weak t y p e

T*

?V .

independent of

f

31 > 0

T*

h o l d s one says t h a t

c

(1,l).

i s continuous i n measure

can be e q u i v a l e n t l y expressed by

saying t h a t

I n fact

11 f 11 1 >

(**) t r i v i a l l y i m p l i e s

( * * ) ' and, i f we have

(**I'

and

0, we can w r i t e

Our f i r s t t a s k w i l l be t o e s t a b l i s h some equivalences between a.e.

- convergence and p r o p e r t i e s of

the function

@(A)

T*

and t o c l e a r up a - l i t t l e th.e r o l e

p l a y s i n t h e whole business.

The general s e t t i n g i n which we w i l l p l a c e o u r s e l v e s i s t h e following: Genahae o e t t i n g . ( a ) We c o n s i d e r

(Q,F,p)

,

a measure space

t h a t w i l l be i n some cases o f f i n i t e measure and i n some o t h e r s + f i n i t e .

7

1.0. INTRODUCTION ( b ) We denote by a b l e f u n c t i o n s d e f i n e d on ( c ) With

Q to R

from

X

“I n ,

ing

k

X

u-a.e.

t ).

.

to

t h a t are f i n i t e

we denote a Banach space of measurable f u n c t i o n s

(ot to

( d ) The sequence t o r s from

t h e s e t of r e a l ( o r complex) valued measuc

w i l l be an o r d i n a r y sequence of opera-

{TkI

I n many cases t h e r e w i l l be no problem i n assum-

t o be a continuous parameter.

(e) Each w i l l be assumed t o be l i n e a r and i n some cases Tk j u s t t o s a t i s f y t h e f o l l o w i n g cond t i o n : f o r fl, f 2 e X , X 1 , 1 2 E lR we have

ITk(X1

fl

’

12

f2)

( f ) With and

x e fi

6 1x11

T*

we d e s i g n t h e maximal o p e r a t o r , i . e .

for

,

( 9 ) We denote by

T

t h e 1 imi t o p e r a t o r , i.e.

l i m Tkf k+m

Tf = when i t e x i s t s i n some sense.

(h) F i n a l l y f o r $(A)

X > 0, =

sup f ex

w i l l be

$(A)

u{

x

E

R

: T*f(x)

}

f E X

,

1. POINTWISE CONVERGENCE

8

1.1.

OF OPERATORS

AND CONTINUITY I N MEASURE OF THE MAXIMAL OPERATOR

FINITENESS A.E.

The f i r s t i m p o r t a n t r e s u l t we s h a l l s t u d y i s a general p r i n c i p l e

T h e com%ukty .in meanme ad each one 06 the ope ha to^ o a a @miey Y JRu a ~LnCteneohcmclump-tivn on t h e cahhanpond i n g maxim& opehatoh -impfie0 t h e continLLity .in meanwre at 0 0 6 t h e maL i m & o p e h a t u h .itnd4. T h i s statement, o f course, has a l l t h e f l a v o u r o f due t o Banach. Roughly s t a t e d :

a u n i f o r m boundedness p r i n c i p l e , and so i t i s .

I t can be o b t a i n e d by a

s i m p l e a p p l i c a t i o n o f t h e general u n i f o r m boundedness theorem and t h i s i s t h e way we f o l l o w here.

[1970

, pages

F o r an a l t e r n a t i v e p r o o f one can see A.Garsia

1-4 1 .

I n o r d e r t o p r e s e n t t h e theorem as a p a r t i c u l a r case o f t h e u n i f o r m boundedness p r i n c i p l e , we endow t h e l i n e a r space p-measurable p-a.e.

p-a.e.

m

and f o r

f E ??I (R)

i n t h e sense o f Yosida [1965 only i f

{ f n IE d(f

- fn)

and -f

More s p e c i f i c a l l y , l e t

l e t us s e t

I t i s an easy e x e r c i s e t o check t h a t

a sequence

of a l l

w i t h a d i s t a n c e t h a t w i l l d e f i n e i n "I ( 0 )

a r e t h e same)

t h e topology o f t h e convergence i n measure. p(Q) <

'"I(a)

f i n i t e f u n c t i o n s (where f u n c t i o n s t h a t c o i n c i d e

d:/)n(R)

, p.30 I . "m , we

f e

[ 0,m)

+

i s a quasi-norm

A l s o i t i s easy t o show t h a t f o r have

fn

-f

f

(p-measure)

i f and

0.

For a s u b l i n e a r o p e r a t o r

T

f r o m a normed space

to

X

'hl (R)

one a l s o shows e a s i l y t h a t 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 .

-' M(R) i s

(a)

The o p e r a t o r

T : X

(b)

The f u n c t i o n

4 : (0,m)

-f

[O,m)

continuous a t d e f i n e d by

0

E

X

9

1.1. FINITENESS A.E. AND CONTINUITY

tends to 0 as X tends to

a.

Of course, if T is linear, then continuity of T on X .

(a) is equivalent to the

Likewise, let (TcOaeA be a family of sublinear operators from X to W (R). Then one easily shows that the follow ng statements are a1 so equi Val ent

I > h } + O

asXtm

For the theorem that follows we shall use the following form of the uniform boundedness principle, that can be seen in Yosida [1965, p.681: LeI

k%mu~

(X,il

A ~ U C ~ L.

11

~ A

dying doh each f, g e X

d(Ta(Af)) 7 6 the

(Y,d) a q u a i - n a m e d be a ~amieqo p e h a t a u &horn X t o Y A& -

) be a Banach npace and

=

d(ATaf)

{Taf : a e A} c Y d(Taf) = 0 u n i ~ a m L qi n a Ilfll-. 0 We recall here that the fact that a each neighborhood U of the origin lim

A&

bounded doh each f

E

X,

,then

e A.

set ScY is bounded means that for there is an E > 0 such that E S c U.

With these preliminaries the proof of the following theorem is straightforward.

1. POINTWISE CONVERGENCE OF OPERATORS

10

lei {TkIF=l be a bequence o d hubfineah ope l d u r n dhom X, a Banach hpace, t o "@l (a) w i t h u(Q) < m. h w n e t h a t each Tk LA conLLnuvlLs and that t h e maxim& opeha2vh T* dedined doh f E X and x E R an 1.1.1.

A huch thud T * f T*

A d

6

A ~.ivLite u-a.e. Then

%doh I each f, i .e. T*f at 0 , and thenedahe

o continuouA

Phaod. imal operator

Clearly T i 0 & T;f(x) each

TtlEOREM.

c

For

Ti

..

n = 1,2,3,.

we d e f i n e t h e t r u n c a t e d

i n t h e f o l l o w i n g way. For

f

E

i s s u b l i n e a r , continuous f r o m X t o T*f(x) f o r each x e a , we have

X

and

x e

( a t n) max-

n

and s i n c e d(T*,f) c d ( T * f )

for

f.

Therefore t h e uniform boundedness p r i n c i p l e a p p l i e s and continuous a t

~(6) <

-

0,

{T*,)

i.e.

Observe t h a t , i f

p(R) =

,

m

we can s e l e c t

c R

and, i f we d e f i n e

&i)

i s equi-

=

sup

Ilf I1 6 1

p

1 x

E

6 : T*f(x)

> X 1

t h e n w i t h t h e same h y p o t h e s i s of t h e theorem we a l s o o b t a i n

with

1.2. CONTINUITY AND A.E.

0

1.2. CONTINUITY I N MEASURE AT CONVERGENCE

E

X

CONVERGENCE

OF THE MAXIMAL OPERATOR AND A.E.

We a l r e a d y know t h a t t h e f i n i t e n e s s

a.e.

a t o r i m p l i e s i t s c o n t i n u i t y i n measure a t

0 e X

t h i s c o n t i n u i t y i m p l i e s t h e closedness i n

X

of

i n which t h e sequence

X

Tkf

11

.

o f t h e maximal operWe s h a l l now see t h a t

o f t h e s e t o f elements

converges

f

I n most i n t e r e s t i n g

a.e.

cases i t i s easy t o e s t a b l i s h such a convergence f o r some s e t dense i n and so we o b t a i n t h e

a.e.

convergence f o r a l l f u n c t i o n s i n

theorem t h a t f o l l o w s i t i s n o t necessary t o assume t h a t

For t h e

X.

X

X

i s complete.

be a sequence 06 &hean. opehatom 6hom X, a named Apace, t o 'hl (n). h b w n e t h a t t h e ma&& opehdtoh T* h cow%nuouh in memwte &am X t o at 0 e X . Then t h e be;t E ad d e m e h f 06 x dolt which {Tkf} CUnvmgtA at a.e. x e R iA doseed i n X.

we have

1.2.1.

THEOREM.

Phood. -

Let

PIX e =

R :

f

Since f e E.

X

l i m sup m,n -+a

PIX E R ; l i m sup m,n

G

E

LeA

211 I x 8

@

ITk)

and c o n s i d e r

ITmf(x)

-

g

E

E.

Tnf(x)[ >

\Tm(f -g)(x)

-

a > 0

Then f o r any

CL

1

=

Tn(f -g)(x)J >

~1

}

G

f

E

+a

R :

[h) 0 J-

as

COROLLARY.

I]

f -g

11 J-

we see t h a t i f

then

t h e p h e c e h g Theohem .h a nomed subspace oh %'t and id doh each g e E Me have T k g ( x ) - + g ( x ) at a . e . x e R , then we &o have doh each f E , T k f ( x ) f ( x ) at 1.2.3.

16 ,the space

0

X

06

-+

1. POINTWISE CONVERGENCE OF OPERATORS

LI C X E fi :

lirn sup k + -

=

FIX E R : I i r n sup k +-

<

~ C 6X 0.: T * ( f - g ) ( x )

b u t t h i s tends t o c e r o as

g

>

-f

a )+ 7

f ( h ),

V{X

e n : I(f-g)(x)l

>

a 1 2