The Carleson-Hunt Theorem on Fourier Series, Lecture Notes in Mathematics , Vol. 911

Lecture Notes in Mathematics Edited by A. Dold and B. Eckmann

911 Ole G. Jersboe Leif Mejlbro

The Carleson-Hunt Theorem on Fourier Series

Springer-Verlag Berlin Heidelberg New York 1982

Authors

Ole Greth Jersboe Leif Mejlbro Department of Mathematics, Technical University of Denmark DK-2800 Lyngby, Denmark

AMS Subject Classifications (1980): 43A 50 ISBN 3-540-11198-0 Springer-Verlag Berlin Heidelberg New York ISBN 0-38?-11198-0 Springer-Verlag New York Heidelberg Berlin This work is subject to copyright. All rights are reserved, whether the whole or part of the material is concerned, specifically those of translation, reprinting, re-use of illustrations, broadcasting, reproduction by photocopying machine or similar means, and storage in data banks. Under w 54 of the German Copyright Law where copies are made for other than private use. a fee is payable to "Verwertungsgesellschaft Wort", Munich. 9 by Springer-Verlag Berrin Heiderberg 1982" Printed in Germany Printing and binding: Beltz Offsetdruck, Hemsbach/Bergstr. t2141/3140-543210

CONTENTS.

PREFACE

3

CHAPTER I

4

i.

Interpolation theorems.

2.

The Hardy-Littlewood maximal operator.

10

3.

The Steln-Welss theorem.

15

4.

Carleson-Hunt's theorem.

17

23

CHAPTER II

Py

and

Qy .

24

5.

The operators

6.

Existence of the Hilbert transform and estimates for the Hilbert transform and the maximal Hilbert transform.

7.

33

Exponential estimates for the Hilbert transform

40

and the maximal Hilbert transform.

45

CHAPTER III 8.

The dyadic intervals and the modified Hilbert transforms.

46

9.

Generalized Fourier coefficients.

51

lO.

The functions

S~(x;f;m*)

and the operator

M* .

69

CHAPTER IV II.

Construction of the sets

12.

Construction of the sets

13.

60

Gk

and

Estimates of the index set

Y*

S*

and

70

~L "

Pk(X;~)-functions and the and

Pk(X;W)

~

and

74

X* .

and introduction of 80

G~ .

14.

Construction of the splitting

15.

Construction of the sets

16.

Estimation for elements

T*

~(p*,r) and

p* ~ G* rL "

V*

of

~* .

87

and

EN .

91 101

IV

17.

Final estimate of

e . o. Sn(X,XF,~_I)

18.

Proof of theorem 4.2.

9

II0 118

REFERENCES

122

INDEX

123

CHAPTER I.

This chapter is composed of four sections. In w 1 we introduce the concept of (weak and strong) type of an operator, and we prove an interpolation theorem, which is a special case of a theorem due to Marcinklewicz for the general formulation). mal operator

0

(ef. [9]

In w 2 we introduce the Hardy-Littlewood maxi-

and prove that

@

is of type

p

for all

p ~ ] I,+~[

.

In w 3 another classical interpolation theorem is proved, namely the SteinWeiss theorem, and finally, in w 4 , we prove .the Carleson-Hunt theorem under the assumption that some operator all

p~ ] i,+~[

M

defined below is of type

p

for

.

For technical reasons we shall always consider real-valued functions defined on a finite interval, although their Fourier expansions will be written by means of the complex exponential functions. This assumption will save us for a lot of trouble in the estimates in the following chapters, and we do not loose any generality, since for a complex-valued function we may consider the two real-valued functions shall further assume that

f

and more complicated, full generality.)

and

is integrable , f eLl(1)

finite interval mentioned above. this chapter also hold for

Re f

, where

I

is the

(We may note that most of the results in

f ~LI(R)

f

Im f instead. We

, but their proofs may be different

so we have avoided to prove the theorems in their

w i. InterDola_lion theorems, Let

f

be a real-valued function defined on an interval

pose that

f ~LI([-A,A])

.

The Lebesgue-measure on

R

is denoted by

depending on the function (i.i)

Ey = {xE

understand the function

(1.2)

and sup-

m . We introduce the sets

E

Y

under consideration by [-A,A] I [f(x) l > y} , y e R+

By the distribution function

Definition I.i.

%f : R++ [0,2A]

~f(y) , y c R+ , we shall

defined by

~f(y) = m(Ey) = m({x~ [-A,A] I [f(x) l > y})

Clearly, we have and

f

[-A,A]

%f(y)

0 =y}) y}) y} < ( ~ ) P ~

IIf[l~

and

llTfll < A llfll

,

then IITfllp < p " ~ =

Proof.

9 A ~

9 -P-Po

llfll

9

We use the same notation as in the proof of theorem 1.9. We choose

the constant

A

as

~

. Then

IlfYll~ =< ~ y

, IITfYII~ =< y , and conse-

.co

quently

;~Tfy(y) = 0 .

This gives

the estimate

~Tf (2y) ~ ~Tf (y) ~ A o Y

y_Pol

fy(X) Ip~ I dx ,

and (exactly as in the proof of theorem 1.9) 2P

=

~

po-p

1 P-Po

9

\A /

proving the theorem.

j

D

w 2. The Hardy-L~ttlewood maximal operatoK. In this section we shall consider the Hardy-Littlewood maximal operator and derive estimates for this operator using theorem i.i0. Let

f ~LI(R)

. We define the maximal operator

G

by

I Ix+t 8f(x) = sup ~--~ If(y) Idy , x E R ts + ~x-t

(2.1)

Clearly, the function

Of

is measurable for each

f e LI(R) , because

@f

is lower semicontinuous as a supremum of continuous functions, and the operator

0

is a sublinear operator, defined on

Theorem 2.1.

The operator

precisely,

satisfies the following estimates

O

0

is of type

+~

LI(R)

and of weak type

I . More

11

(2.~)

llofll. ~ llfll , 4

(~.s)

xoffy) = max IOf(x) >y}) 2 ~ llflll

for all

y e R+

From theorem 2.1 and theorem i.i0 we infer at once that the operator of type

p

for all

p ~ ]I, +~[

g

is

. In the applications later on we only need

functions of compact support~ so let us assume for the proof that

f

has

compact support. It is obvious that (2.2) is satisfied. In the proof of (2.3) we may as well assume that xs

I Of(t) >y}

f

is non-negative. Let

there exists an interval

I

y s R+ . To each

(with center at

x

x ) such

that I

f(t)dt > y .m(l x) . I x

As

f

has compact support, the set

pose that all the intervals

I

x

{Of >y}

are contained in an interval

We claim the existence of a sequence intervals

I

x

is bounded, and we may sup-

(In )

[-B,B]

(extracted from the class of

above) of pairwise disjoint intervals, such that

m

In

U Ix

> [m

n I

x

Assume for the moment that this has been proved. Then the proof is finished as follows:

m({x I Of(x) >y}) __<m

4 =< --Y

U Ix x

< 4m

4 I f(y)dy = ~ n

;

U In n=i

= 4 I m(l n) n= I

4

U I n n

f(y)dy < ~llfll I =

9

Let us prove the assertion above (this is a theorem of Besieovitch-type; results in that direction can be found in e.g. [3]) Let

SI

be the class of all intervals

a I = sup{m(l x) II x r S I} m(ll ) > 3 a I sect

. Let

I I , let

S2

I

considered above. Let x and choose an interval I I from S I with

be the class of intervals

a 2 = sup{m(l x) II x ~ S 2}

Ix

and choose

that do not inter12

from

S2

with

m(l 2) > ~ a 2 . In this way we continue. If the process stops after a finite number of steps, the result is obvious, so we may assume that we obtain a

12

sequence

(a k)

of real numbers

disjoint intervals

(Ik)

Let us consider an interval denote

the first

ak + 0

with I

and a corresponding

m(l k) + 0

for

and let

x

index for which

sequence of

k -+ +

k

I x ~ Sk ; Ik. I

in that case

Ix n l k _ I ~ @

i

and

m(Ik_ I) ~ ~ m ( l x)

. Then we have

where

Ik_ I

Jk-i

and

ter and where

i

i

i

i

Ix c J k _ l

have the same cen-

m(J k i ) = 4m(l k i ) . Using , and the theorem is proved. It

s h o u l d be n o t e d t h a t

possible one.

Note that

the constant

4

appearing in (2.3) is not the best

D 9

is not of type i (if we e.g. let

e

f(x) = X]0,1[(x)

, then

e

is not even integrable')

Corollary 2.2.

The operator

e

is of type

p

for all

p ~ ]i, +|

. More

precisely we have (2.4)

llefllPp __< 4p. 2p . ~

Iifiip

for all

p E ]1, +~[

o

As mentioned above this follows from theorem 2.1 using theorem i.I0 with Po =I

If

, A ~ =4

g(x)

and

A

is a real-valued

g+(x) = max{g(x),0}

Theorem 2.3.

(2.S)

=i

.

~ .

function, we define another function

. A well-known example is

The operator

e

log +

satisfies

[ (of(x)- 2)+~ ~= 8 [ if(x)llog+if(x)j~ J

J

g+(x)

by

13 Proof. We may of course assume that f ~ 0 and that f log+f grable. We define functions fY and r by Y fYtx~/ = ~ f(x)

L

0

0

if f(x) ~y fy(X) =

otherwise

is inte-

if f(x)~y

f ( x ) otherwise

Then clearly Ofy ~ y and so m({x J@f(x) >2y}) ~ m({x JOfy(X) >y}) Using (2.3) on the function f we get Y (2.6) m({x 1Of(x) >2y}) ~ 4 ] fy(x)dx . An integration of (2.6) from I

to

+~

J 0f(x) >2y})dy _
i}

I +~ m({x jef(x) >t})dt = ~If (0f(x) -2)+dx . The left hand side equals 2S2 and thus the result follows.

Later on we shall also need the following two simple lemmata:

Lemma 2.4. Let F

x

on

R

by

f~LI(R)

. For any fixed

we define a function

Fx(t) = S~f(x+y)dy 9 Then

JFx(t)[ ~ 2JtJ@f(x)

Proof.

xER

for all

x~R

, a~l

t ~R

The len~na follows immediately from

irx+t (x+t [x+,t, JFx(t) J = Jx f(y)dy < Ix f(y) dy < Jx-JtJ Jf(y) Jdy ~ 21tJ "Of(x) .

Le~ma 2.5. If

feLl(R)

, then

if(x) I ~ t

If(y) Idy

for all

t9

Jx-t and

[x+t

I -2t ~ - t

If(y) Idy -> If(x)]

where we have used Lebesgue's (cf. e.g. [3]).

for

t~0 +

differentiation theorem for

x 9R ,

Ll-functions

D

From lenm~a 2.5 and theorem 2.1 follows that if get

for almost every

f 9

nL=(R)

we even

[[fll= = ll@fll~

Finally we shall prove an exponential estimate for f(x)

Of , when

f ~L ~

and

is zero outside a compact set.

Theorem 2.6.

Let

c 9 R+

be any positive constant and let

f

be any es-

sentially bounded function, the support of which is contained in an interval

I

of length

A . Then for

y 9

(2.7)

XOf(y) = m({x Igf(x) >y}) y)

and

dist(x,l) = d > 0 , then an

application of (2.1) gives

y < Of(x) < ~

If(T) IdT __y}) j A + 2

Let

t

=

9

A llfIl~ (llfll~) Y A. i + - -y -

~-F77F-l Y e ]0,I[ . Then it is enough to prove that

15

1 1 i + ~ =< 2exp c. ~ 9 exp(-ct)

,

tE ]0,I[

,

which is equivalent to the trivial estimate (l+t) exp(ct) ~ 2. expc and the theorem is proved.

for

t E ]0,I[

,

D

w 3, The Stein-Weiss theorem. We shall now introduce an operator

T~

associated with the operator

T .

In the following we assume that all functions considered are defined on a fixed interval p , where

[-A,A]

p e ]I, +=[

. Let

Let

q

E ~[-A,A]

be the number conjugate to

A p ~ R+

such that

,

p , i.e.

. We define a set function

contained in

[-A,A]

y

! + ! = 1 , and let P q on the class of Borel sets

by

(3.2)

y(E) = I (T•

H~Ider's inequality implies that that

operator of restricted type

IITXEIIp =< AplIXEII p = Ap[m(E)] I/p

f eLq([-A,A]) E

linear

be a

, i.e. there exists a constant

for every measurable set (3.1)

T

X

y

is well-defined,

and it is easy to see

is a countably additive set function which is absolutely continu-

ous with respect to Lebesgue measure. Using Radon-Nikodym's theorem we then get an (up to nullsets) uniquely determined function

(3.3)

~(E) = ~ h ( x ) d x

for all Borel sets

E S[-A,A]

Because of this relation we

of

such that

= IXE. hdx = ITXE" fdx ,

.

define

(3.4) Clearly,

h

an operator

T*

on

Lq([-A,A]

by

T*f = h . T*

is linear, and

T . We have e.g. that if

(3.5)

formally g

Tg 9 f dx = A

T~

behaves as the adjoint operator

is a simple function, then

Iig9 T * f d x

9

16

Lemma 3.1.

Assume

p ~ ]I, +~[

where

Proof.

that the linear operator

. Then

T~

is of restricted q , where

By assumption there exists a constant

(3.6)

type

P ,

~ + ~ = I . P q

A oR+ , such that P

][TXEH p =< ApI[XEI[p = Ap[m(E)] I/p

We shall prove the existence of a constant

(3.7)

XT, f(Y) ~ and let

[Ifll

Let

f EL q

E

(x I lh(x) l >y} 9 We put

Y

T

is of weak type

h=T*f

for all

and E

Bq~ R+ , such that

f

and all

y~R+

.

%(y) = %h(y) = m(Ey) , where as usual,

= E + uE- , where Y Y Y

E + = (x lh(x) >y} , Y

E- = {x [h(x) < - y } Y

Finally, we define X+(y) = m(E~) , Clearly,

A-(y) = m(Ey) .

X(y) = X+(y) +X-(y) , and for

%+(y)

we get the following esti-

mate

Y

Y

ffi I TEE+y" f dx

Y

< IIT• p y

= Ap[m(E~)] I/p

tlfllq

where we have used the definition of

IIfllq ~ ApIIXE~IIp [IfI[q

= Aq[X+(y)] l-(I/q)

B

= 2 I/q .A q

X+(y) = 0

X (y) , and hence

X(y) = X+(y) + X-(y) ~ 2 which is (3.7) with

0 P

Hft[q

,

T* , H~Ider's inequality and (3.6).

From the inequality above we deduce that either [X+(y)] I/q ~ (Ap/y) 9 Ilf]lq We have a similar estimate for

Y

IlfI[

or

17

If the operator

Theorem 3.2. (Stein-Weiss).

ed weak type

Po

p

p c ]po,Pl [ .

for every

Proof.

Let

Pl

p c ]po,Pl [

stricted type T*

and

p'

is of weak type

q'

for all

Tf = T**f T

for

T .

" then

T

T*

T** = (T*)*

" Then

T*

is of weak type

ql'

is a bounded linear operator on is bounded on

T**

T

is of re-

are linear and

T**

and and

T*

Pl' qo' '

is of type

L q , which

L p . From (3.5) we conclude that

for all simple functions, i.e. on a dense set in and

is of type

Po'

q' r ]ql,qo [ . We choose

theorem (theorem 1.9) we infer that

q , and especially that implies that

i8 linear and of restrict-

p' e ]po,Pl [ , and then lemma 3.1 tells us that

Po < p o < p Ek/P 1 9 Then

m(E , k) < ~p Ek

be the poly-

20 Proof.

From lenm~a 4.1 we get

IIf-Pe,k[l~F < e

k

I) k


X}) ~ ~ IlfllI ,

(6.5) for all

X ~ R+

and all

f e LI(R)

with compact support (cf. a remark in

w 2 and definition 1.4). In general, (6.6)

f=f+-f-

, where

f+, f-~ 0

Ifl =f+ +f- 9 Furthermore,

m((x IH*f(x) >X}) ~ m({x I H*(f+)(x) >~}) + m({x ] H*(f-)(x) >~}) ,

so we may in the following assume that Let

and

X ~ R+

be given. For each

e 9 R+

E + = {x I supNyf(X) >X}

c

Y~

[Note that even if

f

'

is non-negative,

f>= 0 . we define the two sets E- = {x I sup (-Hy f(x)) > X}

s

Y~

H f need not be non-negative; Y E + and prove that

cf. (5.1).] We shall only consider the set

:

m(E ) ~ ~-~

f(t)dt , as the proof for

E~

is analogous.

35

For any finite interval of

I , and

Ic

Let us consider

I

of the real axis let

the complement

of

c(1)

denote the center

I .

the family of open intervals

I , for which

I f i ~ J I c f(t) 9 c(1)_------~dt > X .

(6.7)

By the definition

E + these intervals I cover E + , and as E + is c c E (here we use that f has compact support) already a finite number

bounded

of

of these intervals cover

E + . Using the same reasoning of Besicovitch type E as in the proof of theorem 2.1 we can find disjoint intervals I I, . . . , I n such that (6.8)

m(E~)

n !im(lj)

~ 4 J

and

(6.9)

1 I&l~ f(t) 9 )--------~dt c(lj1 > X ,

j = 1,2 . . . . . n

.

J The function tends to

0

gx(t) = ~ as

I

Itl + + ~

is uniformly . Thus,

to any

continuous

for

~s ]0,i[

Ix-tl ~ E , and it

we can find a decompo-

sition of the real axis into a finite number of small intervals

J

infinite

and

each

intervals

such that for each of the small intervals

j =1,2, ... , n

~i I c. f(t).

(6.10)

J

and two for

,

d t - ~ ji

n .=@ J f(t)dt, c(lj)-c(J)

J

< 6 9I .

J

We can even choose

the intervals J in such a way that for each I. either J j c I. or J A I. = @ . In the following we shall suppose that this has = j J been done. Due to the facts that f has compact support and gx(t) tends to

0

as

Itl ->+~

vals mentioned

we shall never need to consider

the two infinite inter-

above.

We define

g(x) = ~I

J f(t)dt 9 x-c(J) I

and

gj (x) = ~I J

. = j

The function

J f(t)dt 9 x-c(J)

36 g(x) - gj(x) ffi i

f(t)dt 9 x-c(J) Jn

J

is clearly decreasing in the interval get for

x=c(lj)

I. , and from (6.9) and (6.10) we J

,

g(c(lj)) - gj(c(lj)) > (I-6)X , for

x

in the left half of

I.. J

so

g(x) - gj(x) > (I-~)X

Thus we deduce that

n i .I ~m(lj) < m({x I g(x) > 89 ~=i =

+

n I m({x ] gj(x) < j=l

-

~(I-~)}) 2

.

An application of len~na 6.1 then gives !llm(lj)< __ 2 11 j =~(--(T~-6)J ~

2 ~i fJ f(t)dt+ j=l~~ X(I-~) J I. = J

f(t)dt~}) < 2 c A 9 exp - I0--6~ = II proving theorem 7.1.

D

)

42 Theorem 7.2. f

There exist positive constants

cI

and

c2

such that, if

is any essentially bounded function, the support of which i8 contained

in an interval of length

A , then

(?.$)

m({x I H~f(x) >X}) ~ c I A---~--exp~- C2]T~T~~ ] .

Proof.

We remark that it suffices to prove (7.3) for large values of

We may assume that

f ~0

(write

f= f+-f- )

E + = {x I sup Hy f(x) > X}

and that

E~ = {x I sup (-Hy f(x)) > X}

y~ where

mate

X >0 for

and

m(E )

X .

IlfIlm = I . Let

y~

g >0 . We shall give an estimate for

m(E~) , the e s t i -

being similar.

~s in the proof of theorem 6.2 we can find disjoint intervals

II, 12 , ..., In ,

such that n m(m~) X ,

j=l, 2 .... ,n .

J Let

fj(x) = f(x) Xl.(X) , j= i .... ,n , and consider the function

I[

f(t) dt x-t

gj(x) = H f(x) - H fj(x) = ~'I~ J The function

gj

by (7.5), hence

is decreasing in gj(x) >X

lj

(because

in the left half of

f ~0 ) and

gj(c(lj)) >

lj . Thus we infer that

n

I

X ~m(lj) < m({x I Hf(x) > m({x I H fj(x) < - { } ) j=l = ~ }) + j=l From theorem 7.1 we deduce that n

2

12 J~I. = m(lj) 6} tha t

(8.8) As

IIH

llf~

H f(x) ~ [Nf~

f~

~ Cp Hf~

and

IIH* f~

l + H * f~ ~ c*I1f~ P

= 41/Pllfllp , we get from (8.8) for IIH flip

lIH f~

+ llH* f~

, say, and

p e ]I,+~[

,

~ 4 I/p (Cp + C;) llfllp

Finally, we shall introduce the modified maximal Hilbert transform

A H

sub-

ordinated the dyadic intervals introduced above. Consider a given

m~j v

and let

x

be an interior point of

~ jv , x E intw~j v . r (Ox)r=~,~+l,.. " of inter-

Then there exists a uniquely determined sequence vals, such that each

o r is a smoothing interval from level r , o~ c~* x belongs to the middle half of each o r . Note that x depends on x and that ~ ~v . A The modified maximal Hilbert transform H. with respect to ~J~ is dejv and such that

x

fined by (8.9)

where

Hj~ f(x) = sup [ I (pv) f~ r_>~ or x-t x f ~LI(] -7, 7])

and where

fo

dt I

'

x ~ int m

,

as usual denotes the periodic exten-

49 sion of

f

to

]-4~, 4~] . We shall sometimes for short write (8.9) in the

form f(~)

=

I I sup l;(pv) o x

A H

We shall prove that

f~x-t dtl q

also i s of t y p e

x

p

for every

normally is a skew interval with respect to

p E ] 1 , + ~ [ . As r x x , we shall first prove the

f o l l o w i n g lemma.

Lemma 8.2. f

to

Let

f eL1(]-~, ~])

1- 4~, 4~] . By

A=

]-a, b[ , - ~ < - a

by

~*

~

and let

~

be the periodic extension of

we shall denote the family of intervals I a , satisfying the condition ~ < ~ X}

l >-~} U {x~

With trivial modifications

l + H* f~

+ 3@ f~

is contained in

[H*f~

> ~}

this is also true for

U {x~*

I ef~

> ~}.

w* -W[l

9 Thus, if

f ~ L~

f o l l o w s i m m e d i a t e l y from theorem 2 . 6 ,

we also have an exponential

Theorem 8.4.

t h e o r e m 7 . 1 and t h e o r e m 7 . 2 t h a t ^ estimate for Hw, . In fact,

There exist positive constants

is any essentially bounded function and transform with respect to (s.13)

~

cI

and

c2

such that, if f

is the modified maximal Hilbert

~* , then

m ( { x e ~ * [~f(x) > x}) -! ci" m(~*) 9 T

" exp -c a

A We note that m ( { x ~ * I H f(x) > X}) < m(m*) , and as the function i ~(t) = ~ e x p ( - c 2 t ) , t E R + , is decreasing and ~(t) ~exp(-e2t) for we can omit the factor

1

llfll.

t

1

in front of the exponential

the right hand side of (8.13), provided have the following corollary.

that

c I ~ exp c 2

t~l

function on

. Thus we also

,

51

There exist positive constants

Corollary 8.5.

f

is any essentially bounded function and

bert transform with respect to

The constonts

cI

and

c2

and

c2

such that, if

is the modified maximal Hil-

~* , then

I~f(x)> X} ~

m({xe~*

cI

H

clm(~*)exp

-c 2 9 ~

do not depend on the choice of

~* .

w 9 . Gene_ra_alized Fourier coefficients. In the following

~

will always denote a dyadic interval as introduced in

w 8, and

m.jv will denote one of the dyadic intervals of length 2~. 2 -v contained in ]-2~, 2~] . Similarly, ~* will denote a smoothing interval (including

m~-i = ]- 4~, 4~])

intervals of length For each

n eN

o

m~. jv

4~. 2 -v .

[n~ 2 "~r

equal to ~jv

denotes the greatest non-negative

is any smoothing interval from level

(9.2)

~*(n; ~ m* =m~l

Remark 9.1. ~(n; ~jv)

jv

integer less than or

v c No ' let

mk,v+l

n*[~v]

for

~*(n; ~;v)

n[~jv]

for

. This has been done in order to

For the same reason we have avoided the use of

the Lebesgue measure,

.

~*(n; m*l ) = n , which is consistent with (9.2).

Here we have changed the usual notation, which is and

be any

(8.4) . We define

n . ) = ~(n; ~k,v+l ) = [n * 2 -v-l] = [~-~m(mjv)]

we define

avoid confusion.

nlml

'

n 9 2 -v

of the four dyadic intervals satisfying

and

m. we define j.v

n m(~jv)] = in " 2-v] ~(n; ~-'~)3 = [ ,2~_

where

For

will denote one of the smoothing

and each dyadic interval

(9.1)

If

and

since it may be difficult

. In our notation they are written

I'I

for

to distinguish between

~(n; m)

and

n .m(m)

n[~]

. In

chapter IV they will both occur in formu]m, which are very much alike.

52

For an arbitrary function

f ELI(] -~, ~])

the periodic extension of

f

We define, for

e E R , e = e . 3~

Fourier coefficient

e =n E Z

we let (as before)

]-4~, 4~]

and

ca(u;f) f~

We note that if

to

with period

f E LI(] - ~, ~]) , the

~'th generalized

exp (- i 2V~x)dx = ~

f~

exp

- im--~/ax 9

we have the ordinary Fourier coefficients

of

f

~ ) , but in the following it will be neces-

sary also to deal with the generalized coefficient, ~ = ~ , p EZ

denote

by

(with respect to the interval

case where

fo

2~ .

especially with the

e

We first remark that we of course have the estimate

(9.4)

lca(co;f) l 0

Iml + max

[0,2~]

Im"l} ,

[0,2w]

in the len~na is due to the

change of variable and the fact that we differentiate

Lemma 9.5.

I~"I

twice.

such that we ~or

D

n ~ NO and

have IOn.2_v(~;f) I < c 3 9 C~(n;~) (~;f) 9

Proof.

Let

8 =n-~(n;m)

9 2 v = n_2VLn.2-Vjrl

Using lemma 9.4 on the function t E ~ . jv

k0(t) = e -iBt

. Then we have and

~=m.

jv

0 &~8 < 2

~

we get for

'

~ Z Y~ exp

~

~!Z ~p exp

with (l+~2) Iyp[ ~ c2~ {maxl~] + 2 -2v maxl~"]} ~ c 2. {i+2 -2v 82 } ~ c2{i+2-2v'2 2v} = 2e 2 9

57 Then we have _~(m;f) = ~ I

e

Ir176

I I e-iBt exp[-i~(n;m)2vt] 9 f~ (t)dt = m(m)

~EZ U ~(n;~)+~ and hence ICn.2_~(m;f)[ < ~ Iy~['Ic

:~EZ

~(n;~)+~

(w;f) l < 20c2.~0

=

~ Ic

p~Z ~(n;~)+~

(m;f) l. ~ I

I+p2

= 20c 2. C~(n;w) (~;f) proving the lemma.

D

Lemma 9.6. Let n E Z and feL2(~) and M ~ 2 be constants such that

be given where

~=~j~ . Let A, B c R +

I If(t)[2dt ~ A 2m(w) and [Cm(~;f) I < B

for

In-ml < M .

Then we have

Cn(oJ;f)89

iCn+21 3

[

9

1 ~

I

~

I~[> y 12-~;~-nl

I+~2

. In w 4

S~(x;f;~*)

2 2~.2 -~

-

I

2m(P)

.

m(e)=2~.2 As this is true for all

~ eN

we finally get for

~§

that

O

I Ian(m) I2m(m) ~ 2m(F) . (n,m) E Gk

~ m(~) ~ 22k+ly-pm(F) (n,o) ~ Gk

Corollary 12.4.

Proof.

If

(n,~) ~G k , then

Jan(m) I ~ 2-kyp/2

D

.

and so i ~ 22ky-Plan(~)I2 .

Prom lenma 12.3 follows that m(m) ~ 22k 9 y-P (n,~) E G k

~ lan(e) 12m(~) ~ 22k+l y-Pm(F) . (n,m) e Gk

We shall now introduce the exceptional sets Gk

we define for each

(12.9)

ken

a function

X*

Ak(X)

and on

lan(~) 12XJX)

Ak(X ) =

Y* . Using the sets ]- 2~, 2~]

by

,

(n,m) E Gk and we define the set (12.10) If

Xk

by

X k = {xe ]-2~, 2~] IAk(X) > 2kyp}

x e X k , one can find a finite subset lan(m) 12X~(X) > 2k y p

(n,~)

I of the index set Gk, such that

, and as the left hand side is a step func-

~ I

tion, there exists a dyadic interval for all

~' such that x~ m' and Ak(Z)> 2ky p

z ~ m' . Hence we have proved the existence of a dyadic interval

79

m' , such that

xs

=c Xk,

so

is a union of dyadic intervals. Using

Xk

the notation from (ii.i) we define

(12.11)

x~ =

u g

and

X~ =

~X k

m(~)

Theorem 12.5.

Proof.

U ~. ~L k=l

o

2-10km(w')

(~s.?)

m(~)

It should be noted that we write Furthermore, if

~=~3~"

and

~/(~';c0) (= ['~(~'; m ' ) " m ~

~' =~s

at most

10k possible choices of the

m' . On the other hand, for given

i.e. for given

~'

and

2 9 210k- i

w'

]) in (13.6)

it follows from (13.7) that

220k

(n,m) ~ G~~ , condition (13.10) can-

I~'-~"I" m(m) --i-~- < 2-10k

for any two exponents

~'

or

and

~"

occurring in

Q~(x;~) . But I~'-~''I" re(m) 2~

I~ , m(m) _ ~,,. m(m) I

=

"

2~

-f#-

~ I~(~';~) -~(~";~)I

In-~(%';m) l + In-~(%'';m) I +I < 2 .210k+l

+ 1

< 220k ,

so the latter possibility is ruled out, and we get I~'- X"I" m(m) --i~--< 2-10k

(13.14)

We may of course write I j=l where

il.x J

'

j=l

I ~i6 j = ~ 9 2-v we get 6,

~

I00 .

belongs to the middle half

= 2 .2~ .2 -~ " ~ =< N - ]

and assume that there exists an

rEN

. Let

, such that

~ G*rL and , o 2 - r y =mo>ml

> ...>mj

0,

0 < n j = ? . I f

nj ~0

, then

m(~) J

> 8 9 2~ 9 2 - N

[ log(l+2 -i) < [ 2 -i = i , we have i=l i=l

N >= 7

, then the conditions

kj+ I < kj

and

so if

.

J)nj

imply for

8~

9

~*(nj,mj)" ~ ~ ~I , and as

=
120 9 2

o , then choose

~

according to

O

theorem 16.1 and

n ,~* ,r

according to theorem 16.2. E s p e c i a l l y ,

r ~k ~

Using theorem 15.7 we get (17.12)

S~(x;• I n ~

o

= i00 , we get , S~o (x;XF;mo)o

(17.14)

* o 1~0 .2

2k o

. Then

-k n1 ~ ~ ~ (1+~

If

Proof.

n