On the Pointwise Convergence of Fourier Series

Lecture Notes in Mathematics A collection of informal reports and seminars Edited by A. Dold, Heidelberg and B. Eckmann, Z0rich

199 Charles J. Mozzochi Yale University, New Haven, CT/USA

On the Pointwise Convergence of Fourier Series

Springer-Verlag Berlin Heidelbera - New York 1971

A M S S u b j e c t C l a s s i f i c a t i o n s (1970): 43 A 50

I S B N 3-540-05475-8 S p r i n g e r - V e r l a g Berlin • H e i d e l b e r g • N e w Y o r k I S B N 0-387-05475-8 S p r i n g e r - V e r l a g N e w Y o r k • H e i d e l b e r g . Berlin

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Dedicated to the memory of my father and mother

Foreword

This monograph is a detailed (essentially) self-contained treatment of the work of Carleson and Hunt and others needed to establish the Main Theorem:

If

f

e Lp (-~,~) l
L (-4z,4~).

In Chapter 6 a large number of statements are presented without proof. Every proof that is omitted in Chapter 6 can be found in Appendix B.

The reader who is not familiar with the concept of Cauchy principal value (denoted:

P.V.) or the concept of the Hilbert transform is

referred to Appendix A for a summary of the results needed in the sequel. Appendix C contains the recent results of Kahane and Katznelson on divergence sets which in a sense are opposite to those of Carleson and Hunt. I would like to thank Professor S.A. Gaal, Professor R.E. Edwards, and Professor S. Kakutani for their encouragement during the preliminary stages of the writing of this monograph, and Professor R.A. Hunt for his very generous help during the spring and summer of 1969 at which time he explained in detail a portion of his original papers to me and made a number of suggestions for improving a few of the proofs contained in them.

VI Without his offer of assistance in the spring of 1969 I would not have seriously considered writing this monograph.

Also, I would like to thank

Professor Y. Katznelson for his permission to reproduce in Appendix C a portion of his text:

An Introduction to Harmonic Analysis.

An outline of the proof of Theorem ~2.2) based on Theorem (1.18) and Theorem [3.6) was communicated to me by Professor E.M. Stein and my appreciation for his doing so is to be noted here. This paper was completed while the author was conducting a seminar on the contents of a preliminary draft at Yale University during the fall of 1969.

I would like to further acknowledge my indebtedness to the men

who participated in this seminar:

James Arthur, Eugene J. Boyer,

G.I. Gaudry, Michael Keane, S.J. Sidney, Joseph E. Sommese, Charles Stanton, A. Figa-Talamanca and Professors S. Kakutani and C.E. Rickart.

September, 1970

C . J . Mozzochi

TABLE OF CONTENTS

Chapter i;

A Theorem of Stein and Weiss

Chapter 2:

The Main Theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

Chapter 3:

A Proof of Theorem

(2.2) . . . . . . . . . . . . . . . . . . . . . .

ii

Chapter 4:

A Proof of Theorem

(3.6) . . . . . . . . . . . . . . . . . . . . . .

19

Chapter 5:

A Proof of Theorem

(4.2) . . . . . . . . . . . . . . . . . . . . . .

20

Chapter 6:

A Proof of Theorem

(5.2) . . . . . . . . . . . . . . . . . . . . . .

24

Appendix A: The Hilbert Transform . . . . . . . . . . . . . . . . . . . . . . . .

44

Appendix B: P r o o f o f Unproved S t a t e m e n t s in C h a p t e r 6

53

...................

.......

1 8

Appendix C: The Results of Kahane and Katznelson . . . . . . . . . . . .

75

Bibliography

86

.........................................

I.

A THEOREM OF STEIN AND WEISS

Throughout this monograph we assume each function f is real-valued and in LI(-~,~) and hence finite almost everywhere in (-v,~).

We say f

and g are equivalent iff f(x) = g(x) for almost every x in {-~,~). we may assume, when necessary,

Hence

that f is finite everywhere in (-~,~).

The proofs in this chapter have been taken directly with only slight modification

from [14].

m or ~ denotes the Lebesgue measure on (-~,~). (i.i)

Definition.

For each y > O the function

Xf(y) = m{x ¢(-~,v) is called the distribution (1.2)

I If(x) l > y}

function of f.

Remark. Since kf(y) <

everywhere Limit %f(y) = O. y-~

co for each y > 0 and f is finite almost

Clearly, %f is non-negative

and non-increasing.

Using the fact co

U

n=l

(x c

(-~,~) 1 Ifcx)1 > Yo

we have that Xf is continuous

+

n1 } : ~ × ~

(_~,~) I If(x)l > yo }

from the right.

has a countable number of discontinuities;

Since Xf is monotonic,

it

so that it is measurable.

Let T be a mapping from a subset of the integrable real-valued functions defined on (-~,~) that contains the simple functions into the set of measurable real-valued functions defined on (-~,~).

In this

chapter we assume 1 < p < ~, 1 < q < (1.3)

Definition.

T is of type (p,q) iff there exists A > O such

that ;l~f~lq~ "" "" A--I~f[~p for every simple function f.

(1.4)

Definition. T is of weak type (p,q) iff there exists A > 0

such that for each simple function f and y > O

~Tf(Y) <

[y

['f~(P~ q

(1.5) Definition. T is of restricted type (p,q) iff there exists A > 0

such that for each measurable set E ~ (-~,~)

llv~"EI! q -

A IJXEU p

, where ~ E

is the characteristic function

of E. (1.6)

Definition. T is of restricted weak type (p,q) iff there

exists A > O such that for each measurable set E ~

(-~,~)

q X E (1.7) restricted Proof. measurable

Lemma. weak t y p e

If T is of restricted type (p,q), then T is of (p,q).

Let Ey = ( x E ( - ~ , ~ ) set contained

1 ITXE(X) I >

y}.

Let E be any

in (-~,~).

-Tf

(1.8)

Lemma. For 1 < p <

-

and

f ~ LI(-~,~)

we h a v e

Proof.

[f[Pdp:

pyp-ldy

. ~T

d~=

.'IT

so that by Fubini's theorem since the set If(x)]

j

>

y}

]f]Pdu=

~T

[o If (x)])

o .TT

'

{(x,y)

(-Tr,~) and

I x

is product measurable

Jo-(

pyp-I

X

(y)

~lu

.~T [o, f(x)|)

~[ (y) = %E (x) [o, lfCx) l ) Y

)

dy.

But

where E : {x e xIifCx) I > y}. Y

This completes the proof of (1.8). In the rest of this chapter we assume: 1 < p £ k

1

q

<

~

(k = 0,1), p 4: P , o 1

k

:

(l-t)

Pt

+

t

Po

;

Pl

1

qt

=

q ~'q o 1

(l-t)

qo

+

and for 0 < t < i

t

ql

If s ~ i, then s' is that number (including ~

)

satisfying ( 1i + _1 ) = S

(1.91

S v

Lemma.

Let T be of restricted weak type (Po' qo )

(pl,ql), then it is of restricted type (pt,qt) for 0 < t < i. Proof. Suppose E ~

Let p = p , q = qt for a fixed t between 0 and i. t (-~,z) is a measurable set, ~ E its

characteristic function, h = T ~ of h.

and l(y) the distribution function E We can assume, without loss of generality, that q < q . Then, o

using the restricted weak type relations (with constants

1

A

o

and AI),

4

we obtain by (1.8) for any C > 0

[h[qd~= q

yq-lx(y)dy = q

yq-lX(y)dy + q

.~

yq-lx(y)dy JC

< q

fC

yq-I

( Ao ]l/Po)qo l;yq- ~ l( 7- [~(E) dy + q

1/Pl) q 1 [u(E)] 1 dy

0

:["q

Aq° 7

(q-q°)-[ [v(E)]q°/P° cq-q°

[q Alql ] (~l---q)'] [v(E) ]ql/Plc q-qt

+

Letting C = [~(E)]s , where

(q-qo) we have [~(E)

]

qo/Po

P

(q-ql)

(q-qo)

C

= [~(E)]

P

q/p

Pl

= [~(E)] .ql/pl cq-ql

Thus we have shown that

.•

Ihl q d~< Aq

[~(E)] q/p ,

where

q )

A=

q-q

_

+

o

1/q

1

q -q 1

This completes the proof of (1.9) Suppose T is linear and of restricted type (p,q) and let q' f s L (-~,~). Let~be the set-function, defined on the measurable subsets E C (-~,~), such that (1.10)

~ (E)

=

(T~)f E

Since T is of restricted type

du

(p,q),~is countably

absolutely continuous with respect to ~ .

additive and

Thus by the Radon-Nikodym

theorem, there exists a unique (almost everywhere) function h

on (-~,~),

such that (I.i13

~(E) =

IE h d~

for each measurable set E C_(-~,~). Define the operator T*, acting on q' L (-~,~), by letting T*f = h. T* is clearly linear; also, it behaves, at least formally, like the adjoint operator to T.

That is, if s is a q' real-valued simple function defined on (-z,z) and f is in L (-z,z), we obtain from (i.i0) and (Ioli) and the linearity of T and T*

(1.12)

(Ts)

f d~ =

In g e n e r a l , however, i t in

s(T*f)d~. p' is not true that T*f is in L

(-~,~)

for all

f

(1.13)

Lemma. Suppose T is linear and of restricted type (p,q),

where 1 < p <

Lq'(-~,~).

~ and I Z q <

~ , then T* is of weak type (q',p'). q' Proof. Let f be in L (-~,~) and h = T*f. If

E

= {x a (-~,u) ] ]h(x)] > y} and k(y) = v(Ey) is the distribution Y function of h, we must show the existence of a positive number B, q' independent of f in L (-~,~), satisfying for each y > 0

Actually, according to our definitions we need only consider T* restricted to the class of real-valued simple functions on (-~,~), but the proof of (1.14) will yield the result for all functions in Lq'(-~,~).

We put EX = E +y U

Ey , where E+ = {x ¢ ( - u , ~ ) 1 h ( x ) > y} Y

E Y : {x e

(-rr,~)

Then, for y > O, + X

h(x)

< -y).

+ + I (y) = la ( E y ) ,

Let

(y)

+

type

X-(y)

= ~ (E).

E+('iy E-y : 0 and X(y) = l.i(Ey) : laCE~) + l.i (Ey) :

-

s = ~

Y

I

-,I-

k (y).

of X (y), (i.12) with

Thus by the definition

Ey+ , Holder's

inequality

and the assumption

of restricted

(p,q) we have

X+ (Y) = Y _<

s

Ey+ du <

I?

y+

h d~ = _

Ey+ (T*f) du=

[

{T~E

;

)f du

lITx ~+yilq "Ufllq, <_ AII~:E; II p " ~filq, • Y P = ~+ (y)] ' / ' ii{+lt

Since

+ ~ (y)

y

But 1 - (l/p) = (i/p');

<

, we have shown,

for y > O,

+ i/p A[x (y)] 8fll q, •

so that this last inequality

can be transformed

into

(l. IS)

X+(y)

fory

By a completely

(1.16)

analogous

X- (y)

> O.

argument we obtain,

<

lif

for y > O,

.

+

Since B=

X(y) = X (y) + X (y) , (i.15) and (i.16) yield

(1.14) with

21/P 'A. (1.17)

types (po,qo) Proof.

Theorem.

and Cf.

(Marcinkiewicz).

(pl,ql), [16]

If T is linear and of weak

then i t is of type ( p t , q t ) .

Vol II chapter XIl.

(1.18)

Theorem.

weak type (po,qo) Proof.

(Stein-Weiss).

and

(pl,ql),

Hence by (1.13)

(q't' P't )

The conditions P o ~

i m p l y p ' o , ~ p~, ,

(pt,qt) for 0 < t < I.

T* is of weak type

0 < t o < s < t I < i.

p~ ~,~ p ' o tl

then it is of type (pt,qt).

Since T is of restricted weak types (po,qo) and (Pl,ql), by

(1.9) it is of restricted type

(k = 0 , 1 )

If T is linear and of restricted

1 < q' < p' tk - tk

1 < q~ <_ p~ (k = 0 , 1 ) .

PI'

Suppose 1 < Pk <-qk

(k = 0 , 1 ) .

Thus we h a v e

Hence we can a p p l y

(1.17)

to T* with (pk,qk)

replaced by (qt 'P~k ) (k = 0.i), k obtaining the result that T* is of type (qs' Ps )" It follows that T* is defined on all of Lq's (-~,~) and is a bounded operator into LPs(-~,~).

Thus the adjoint of T* is also bounded.

But it follows from

(1.12) and the representation theorem of F. Riesz that this adjoint must agree with T on the class of all real-valued simple functions defined on (-~,~).

Thus T is of type (ps,qs).

By making t o tend to O and t I to i,

if necessary, we see that s can be any number satisfying O < s < I. (1.19)

Remark. It is very important for the application of (1.18)

in Chapter III that the reader convince himself that the bounds of the operator in (1.18) Pt L (-w,~)

into

L

(as well as in (1.17)) as a transformation on qt

(-g,~) depend only on t, Po, qo, Pl' ql

constants of the weak type (po,qo) and

(pl,ql) inequalities.

and the

Let Fourier

i L (-~,~).

f ~ series

Let M : functions

for f. P L (-s,s)

on

II.

THE MAIN THEOREM

Let

Snfx;fl.. be the n th partial sum of the

Let

1 < p < ÷

class

of extended

real

valued

(-~,~).

Mf :

(-~,~)

+

[o,~]

Mf (x) = sup { IS ( x ; f )

[n >_o}

n

(2.1)

Remark.

It

In chapter (2.2)

Theorem.

is

easily

3 we g i v e For every

shown t h a t a proof f s

IIMell ic P

(2.5)

The r e a d e r

should

then

there

almost every

exists

CA)

~ >

~f-Pl~

(B)

0 there <

s

(C)

If

on p b u t

<

the

~

independent

following

and

f E

a polynomial 1 _< p _< ~

fnk(x)

LP(-~,~) P c ;

of f.

well

llfn-fl[ p

(n k } s u c h t h a t

For every

exists

1 < p <

recall

If I < P

a subsequence

everywhere.

following

P dependent

spaces:

sublinear.

llf[[ P

a constant

about Lp

of the

LP(-s,s)

where C > 0 is P Remark.

M is

known f a c t s

÷

o,

÷

f(x)

1 <_ p < ~

LP(-~,~) 1 < q < ~

and for

such that , then

p > q

P

implies L q ( - s , s ) ~ (2.4) then

Main Theorem S (x;f) n

+

LP(-~,s). (Carleson-Hunt)

If

f(x)

every

for

almost

f c

P L (-~,~)

x in

(-~,~).

I < p <_ ~

,

9

(2.5)

Lemma. If f ~ LP(-~,~)

0 < ¢ < 1

I < p <

~

, then for every

there exists a sequence {Ek} of positive real numbers and

a sequence {Pk } of polynomials in LP(-~,~) such that O < c~ all k,

=d

Ek

÷

O,

llf-PklIp < ~

Pk(X) 2

+

f (x)

< k

for

for almost every x in (-~,~),

for a11 k

k Proof. This is immediate by (2.3) (A) and (2.3) (B). (2.6)

Lemma. For each k let E k =

where

Ck

1 < p <

and ~

Pk

{x ~

(-~,~) IM(f-Pk) (x) >

are those constructed in (2.5).

we have m E k ~ C ~

E~

where Cp

ck }

Then for

is that of (2.2) and

f ~ LP(-~,~).

Proof.

dx

[M(f-Pk) ( x ) ] P dx = -7

IIM(f-Pk)I[ P

<

(2.2).

P

Jl kII,P < E2kP

But by (2.5)

Proof of Theorem (2.4) Clearly, by (2.3) (C) it is sufficient to show that for almost every x in (-~,~) if 1 < p <

~

Sn(x;f)

÷

f(x)

It is easy to show that

for each k Sn(x;f ) = Sn(X; f-Pk) + Sn(X;Pk) for all n and for all x E (-~,~);

have I S ( x ; f )

so that for each k, for each n and for all

- f(x) I

ISn(X;Pk) - f(x) I.

< tSn(x;f-P k) I

x ¢ (-~,~) we

+

But since Pk(X) is differentiable on (-~,~) for each

k we have for each k limit Sn(X;Pk) = Pk(X) for each x in (-~,~).

for each k and for each x in (-~,~)

Hence

10

limit n

-+

lSn(X;f ) - f(x) t < oo

limit n

.+

IS ( x ; f - P k ) I n

co

+

Pk (x) I

If(x)

But for each k and for each x in (-~,w)

limit

ISn(x;f-Pk) I <_ M(f-Pk) (x)

Hence for each k and for each x in (-~,~) limi--"~ ISn(X;f ) - f(x) I <_ M(f-P k) (x) + n

-+

If(x) - Pk(X)]

co

By (2.6) if x ~ Ek, then M(f-Pk) (x) <__ ek where reEk< Cpp co

0 for all x ~ (J k=l

Consequently, M(f-P k) (x)

Ek = A.

But co

m(A) <

E

co

m(Ek) < Cp

E

oo

ep < Cp

~

)

k = Cp I ~ _ ~

And If(x ) - Pk (x) I where m(B) = O.

+

0

for x in (-~,~) and x ~

B

Consequently,

limit Sn(X;f ) = f(x) for x in (-~,~) but x ~ (AOB) where

since c < I.

III.

For every f ~

A PROOF OF THEOREM (2.2)

Ll(-g,~) let fo denote its 2~-periodic extension

with domain (-4~,4~).

For every

f ~

LI(-~,~)

let

S*(x;f) = P.V. n

(-~,~)

and x ~

- in t e

Ix-tl< ~

fo (t)

x- t

dt

;Inl z

o.

-int

let

Sn(x;f;m i) = P.V.

; 4~,, 4~

In[ > 0 .

For

i < p <

M*:

LP(-~,~)

+

class of extended real valued functions on (-~,~)

(-IT,~)

÷

[0,~]

M*f

:

=

f°(t) d t ; x - t

let

M*f(x). sup ~ISn(X,e,~* l) I [ Inl ~ o} (3.1)

Remark

We introduce the integrals S*(x;f) and S*(x;f;~l) "

n

N

so that in the sequel we can use the machinery associated with the Hilbert transform (cf. Appendix A). complex conjugate of S* (x;f)

for

Note that S~(x;f) is the Inl ~ O and similarly for

-n

S*n (x;f;~l)"

Let

Also, it is easily shown that M* is sublinear.

12

sin (n+~)y if

y ¢

[-~,w]

I 2)

if

y=O

n>"

if

y e

if

y = 0

{0}

2 sin 2 D (y) =, rt

(n

+

sin

Fn (Y) =

I

n

Gn(y) = Dn(y) - Fn(y) (3.2)

{o}

[-~,~]

Y

Remark.

for all y ¢

It is clear that

[-~,~].

Dn(Y )

F (y) '

n

and G (y) are n

continuous on [-~,~].

(3.3)

Lemma.

]Gn(Y)

] < C1 f o r

all

n >0

and f o r a l l

y ¢ [-~,~]

where C 1 > O.

Proof.

Let

•

g(y)

1

i

2 tan y 2

Y

if

y ¢

if

y=O

[-~,~] - {0}

= 0

By L'H6pital's rule it is easily seen that g(y) is continuous and bounded on [-~,~].

13

But if y # 0 we have by direct calculation that

D

n

(y)=

sin(n+l)y 2

s i n ny cos y2 + cos ny s i n i

2 sin •

i

sin ny 2 tan@

+

, m

,

2 sin)'2

2

s i n ny

,

g(y)

s i n ny +

cos ny

1 cos ny = Fn(Y) + g(y) s i n ny + -~ cos ny,

Y

And if y = O, then by direct substitution we have Dn(y) = (n + I) = Fn(Y) + ½

Consequently,

for all

y E

[-~,~]

we have t h a t

Dn(Y ) = Fn (y) + g ( y ) s i n ny + ~ cos ny ; so t h a t IDn(Y) y ¢

Fn(Y) I <__ (]g(Y) I

+ ~) <

C1 f o r a l l

n20

and f o r a l l

[-~,~1.

(3.4)

Lemma

Me(x) <__ Ep( IIfll P + M'f ( x ) ) for 1 < p <

, for

almost

every x in (-~,~) where Ep > O is a constant independent of f but dependent on p where f ¢ Proof.

Sn(X;f)

LP(-~,~).

For each n > O and x E

= }i

i T[ ~

f°(t)

(-~,~)

j <~ fO(t) Dn(X-t)dt = Dn(x-t)dt = ~1 I x-tl

1 f o ( t ) Gn(X_t)d t + 1 I xJ - t l <~f ° ( t ) Fn ( x - t ) d t ftl ~"m- Ix - <w But for every x E

(-~,~)

we have

14

~

in(x-t)

f°(t)Fn(X-t)dt = i_. P.V.

Ix-tl<~

e

(-~,~)

we have

in(x-t)

-in(x-t) o inx -¢ f (t)dt = c 2i (x-t) 2~i

e

~ P.V.

inx 2~i

f°(t)dt-

Ix-tl<~

-inx ¢ P.V. 2~i

e

-int e x-t

Ix tl<~

=

f° ( t ) d t

Ix-tl<~

But for almost every x E 1 P.V. ~Y

-in(x-t)

-~ 2i (x-t)

x~

~ I -tl <~

-i (-n)t fo

(t) dt

X-t

-inx S*(x;f)

-

~

n

S*(x;f)

2~i

;

so that for almost every x E

(-v,=)

-n

and for every n > 0 we have

ISn(x;f) I < -

f°(t) I

~" IX

--

IGn (x-t) I dt + ~1 I S*(x,f) n

1

I + ~ IS*-n (x;f)

<

Consequently, by (3.3) for n > 0 and for almost every x E

n since x e

<

--

ISn(x;f) I

~ ( I s *n ( x ; f ) l

+

+

IS*- n (x;f) l )

(-~,~) we

and

= IS*n(X;f) I we have for n > 0 and for almost every

(-z.~) tbat

•Ign(X;f) I -< C2~If|~I. ,u + C2 'Is*(x;f) • n I where C 2 > (CI+I) But for n > 0 and x e

(-~,~)

S*(x;f) = S* (x; f;~*l) n n -

-

we have _intfo

E

(t)dt; x-t

Ix-~l>~ It]<4~Is~(x;f)I < Is*(x;f;~:I) I + 4 Uf~l --

n

IT

so that

~I > O.

15

Consequently,

iSnC×,f)I _<

for n h 0 and for almost every x ~

llfll÷

C2(I + ~4)

(-w,~)

c21 s* I (x;nf;~:l)

But by Holder's inequality it is easily shown that 1

II

--

Consequently,

p

for n h 0

and for almost every x e

4 (2~) (l-l)ilf C 2 (I + ~) II + C 2 ISn(x;f;~*l)l; n >__O P

ISn(x;f) l < Let Ep

~r)

(2~r)

+ C2

>

O.

Then for every n > O and for almost every x E iSn (x;f) i

_<

Ep

iS* (x;f;w* I)

--

we have for almost every x ~

-n

so that for almost every x -

E

(llfJ + M'f (x)). p

P

]

-

(-~,~)

IS (x;f) I < E (llf~p + sup n -- p inl >_o

Mf(x) <

(-~,~)

we have

(~f~p + ISn(x;f;~*I)_ I)

But since IS~(x;f;~*l) I :

sup n>o

(-~,~)

ls~(x;f;< 1) I)

(-~,~)

we have

for every n ~ O,

16

1/p (3.5)

Lemma.

For

1 < p <

~

~Mf~

<__Ep ([~f|~(2~)÷ ~M*f[~)

for f in LP(-~,n) where E Proof.

> 0 is a constant independent of f. P This is an immediate consequence of (3.4) and Minkowski's

inequality for integrals. In chapter 4 we prove the following (3.6)

Theorem. Let F ~

function of F.

(-~,~) and l e t ~ F

For every y > 0 and 1 < p < ~

kM* ~ F(y) = m{x E

(-~,~) I M * ~ F ( X )

be the characteristic we have

> Y} -<- Bp (mF) p Y-P

where B > 0 is a constant dependent on p but independent of F and y. P (3.7) Lemma. For 1 < p < ~ for each measurable set E C

(-~,~)

we have

ilM~EII p-"

_< Fp~[~E[~p

where Fp > 0 is a constant

independent of E. Proof. This is an immediate consequence of (3.5), (3.6) and (1.9) with

Po = qo = ( ( p + l ) / 2 ) , Pl = ql = (p+l) and t = (I - I/p). Fix integer N > O. Let MNf(x ) = max o
ISn(x;f) l

Let e denote any simple function with domain (-~,~) and range in th {0,i .... , N}. We say ~ is an N order simple function. Let T f(x)

= S

~(x) (x;f)

x ~

(-~,v).

th Clearly, T is linear for every N (~

(3.8)

Lemma.

For

I < p <

~

order simple function.

i , Tl l ~Etlp, <

F p .nU~EYp _

for each

17

measurable set

E

C(-~,~) and for each N th order simple function

where F is that of (3.7). P Proof.

This is immediate by (3.7) and the fact that

liT fuP --< I~Mf~__ for each i < p < each

=

, for each f s

Lp(-~,~) and for

N th order simple function =.

(3.9)

Lemma.

For I < p <

for each measurable set

-

IT~ ~ ~Y) <--Fp Y - P ~ E ~ P

p

E ~ (-~,~) and for each N th order simple function

where Fpis that of (3.8).

Proof. This is an immediate consequence of (3.8) and (i.7). (3.10)

eemla.

For i < p <

~

I~r f ~ < Cp ~[f~p for every simple

function f in Lp (-~,~) and for each N th order simple function ~ where C > 0 depends only on p. P Proof. This is an immediate consequence of (3.9), (1.18) and (1.19) with Po = qo = ((p+l)/2) , Pl = ql = (p+l) and t = (i - l/p). (3.11)

Lemma.

Let f s

N th order simple function. { f n } = Lp (-~,~) T

(f-%) Proof.

(X)

Lp (-~,~) 1 < p <

0 for x e

[]f~p

and

This i s an immediate c o n s e q u e n c e o f t h e Lebesgue d o m i n a t e d there exists

ll%l[p

functions { % } ~

Lp ( - ~ , ~ )

such t h a t

f (x) n

for

(-~,w) and

x ~

be any

(-~,~).

c o n v e r g e n c e t h e o r e m and t h e f a c t t h a t

+

Let ~

There exists a sequence of simple functions

such that l~fn~p +

÷

-

f(x)

(-~,~) and n

>

x e O.

a sequence of simple

+ ilf{ip , Ifn(X)l < If(x)l for

i8

(3.12 3

i n Lp ( - ~ , ~ )

Lemma.

For

1 < p <

and for

e a c h Nt h o r d e r

liT~fl~ ~ Cp lifllp

~

simple

function

~

for

every

where

Cp

is

f

that

o f (3.10). Proof. (3.11).

Fix

Then

by Fatou's

f e

]Tf(x)

theorem and

lIT f~p

< limit n+~

Lp ( - ~ , ~ ) I

.

IlTjnll

{f}

IT f(x)l

÷

(3.10)

Let

be the for

sequence

x e

(-~,~)

of ;

so that

we h a v e

limit

< c

P-

P

n+~

IIfnll

P

-- c Ilfll P

P

Proof of Theorem (2.2)

Fix f N th

O

in Lp (-~,~)

order simple function

for all x e

~fo(X)

It is easily shown that there exists an ~o such that ~T~ofo(X) l = MN f°(x) But

(-~,~)

increases monotonically to

Mf (x) for each O

(-~,~).

Hence

IIMfoll p _ c p 11%11 p-

x in

IV.

(4.1)

A PROOF OF THEOREM (3.6)

Remark. The notation in this chapter is the same as that given at

the beginning of chapter III. In chapter 5 we will prove the following

(4.2)

Theorem.

Let ~ F E ~

Fix:

N > O, 1 < p <

be t h e c h a r a c t e r i s t i c (-4~,4~)

such t h a t

I n i _< N and f o r a l l

x e

~

, y > O and F C

f u n c t i o n o f F.

(-~,~).

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

a set

re(E) _< cPy-P(mF) and f o r a l l n such t h a t (-~,~)

- E we h a v e

[Sn(X; ~ F ; m * l ) l < (C2L)Y where C1 > O, C2 > O, L > O a r e independent

o f N , y , and F, b u t E = E(F, y , p , N) and L = L ( p ) .

P r o o f o f Theorem ( 3 : 6 ) . Let EN = {x e Let

E =

(-~,~)

(x e

It is easily

Isup {ISn(X; ~( F;W*l)! Inl <_N} > y}; N > O

(-~,~)

IM*YF(X ) > y}.

shown t h a t

since E is bounded,

EN +1SEN and E =

re(E) =

lim

m(EN).

0 N=I

EN ;

so t h a t

But by (4.2) we have for

each N > O

EN ~ (C1C2L)PY-P (mF) Consequently,

since

Limft N÷~

(C1C2L) i s i n d e p e n d e n t

m E

o f N > O we h a v e t h a t

~ (CIC2L)PY-P (mF). N

V.

(5.1) A c

Notation.

(-4~,4~).

A PROOF OF THEOREM

I AI

will denote the Lebesgue measure of

For each integer v

equal intervals

(4.2)

~ O we subdivide

(-2~,2~) into 2.2 ~

(called dyadic intervals) of length 2~.2 -~.

The

resulting intervals are from left to right denoted ~jv ' j = I, ..., 2.2 v Let

~I

= (-4~,4~) and for j = I, ...

(2.2u)-I '

will always denote a dyadic interval w* will always

j~

j+l,~.

(except for ~ i ) denote the union of two adjacent Consequently,

m'C

I~'1 =

such

J~

--

~. contained in (-2~,2~). J~

dyadic intervals of common length• ~*

u > O let ~? = ~ ue. '

that

for

some ~

5 0

~

for every ~* ~here exists and

]~*1

= 2

or in other words for every ~* (including W:l ) there exists ~ ' ~

such that 41 'I= l *t

Note that fo=

we have

•

-i

4[[0,2~][

= [mll l.

Also, for some ~ ~ 0

mtj~ #~£U

for all £ -> 0 and for all u _>

;

~*

and -1

and for some j ~ 1 we have O.

For each nonnegative integer n let n[~jv] be the greatest nonnegative integer less than or equal to n2 -v. n[~*l] = n. -

Let b k =

For u > O let -

1 2k

-

;

Let

n[~? ] = n[~l,u+l].

k = O,1,2,

...

21

For ~

real and ~ = ~. let jv

ca(u) : ca(re;f) = ~ For each p a i r

p

:

1 /m

fo

-i2Vax (x) e dx

we a s s o c i a t e

(n,~)

the number

oo

C(p) = Cn(m ) = Cn(cO;f ) : T ~ "

Z

IC(n+~)

(1

(c°)l

+

2)-i

~/_-- - o o

Note that dx) , and --

--

-~
Cn(m;f) = 0 Let

(n+~)

iff

f = O almost everywhere in ~.

C;(m_*l) = Cn(ml0 ) = Cn(m20 )

For each p a i r p* = (n, ~*) we a s s o c i a t e C*(p*) : C * ( m * ) = max{C (m') ] ~ ' ~ n n Note t h a t

C*n [m*] (~*)

By the statement:

:

S_n(X;f;c0* )

S* ( x ; f ; ~ * ) n for

m* 4 ] ~ ' ] '

: Ira*l}

I

"x in the middle half of m*"

S*(x;f;~*) = P.V. n

Note t h a t

the number

max {Cn[m, ] (~')

in one of the two middle fourths of Let

co

--

fw *

we mean x is

~*.

e -intf ° ~t) x-t

dt,

In[ >_ o.

is the complex c o n j u g a t e o f

Tn] > o.

In chapter 6 we will prove the following (5.2) F ~(-~,~). m(E)

~ CP1

Theorem.

Fix N > O, 1 < p <

Then there exists a set E C y-P(mF)

such that 0 < n < N w

and for each

=

,

y > 0

(-4~,4~)

x e ((-~,~)

and

such that

-E) and for each n

there exists four finite sequences:

22

~0"i, 0J*) oJ~) -

,,.)

~0~)

o

...)

CO

3

J+l

n : n_l ) no, nl, ..., nj, ..., nj+ 1 = 0

k_I , ko, k l , . . . , m_l,

kj,

...,

kj+ 1

mo ' m i , ..., m., J ..., mj + 1

(where the last three sequences consist of non negative integers.)

such t h a t x

is in the middle h a l f of m.,* ~* ~ J j+l

kj+ 1 < mj _< kj

and

; n.=3 4-2~-nj[mj]

for each j .

(strictly)

Im J*l-I ," nj+l --< (I + b kj)nj

= iSnj+l(X;

IS*nj(X; £ F; °a~)l

~*. J

' ~ F ; m*j+l )]

+

0 (Lmj bmj_lY )

In a d d i t i o n for each j

IS~(X;~F;m )I

=

0

(Ly)

where C1 > O, L > 0 are

independent of N, y, and F, but E : E(F,y,p,N) and L : L(p)

r(x) and where the equation r(x) = O (s(x)) means ]s~x) l < C where C > 0

is independent of F, y,p, N and x e

(-~,~).

Proof of Theorem (4.2) Case I.

By Case I I .

(n = O)

cs.2) [SoCk; %' F; ~-1 )1

=

0

(Ly)

(0 < n < N)

By (5.2) we have

ISn(X; /' F;

-1

)1

: ISoCX; KF;

*a+l ))

i=l

i-

23 co

But I So* ( X ; X F ; Case I I I .

oa*j+l)]

= 0

(Ly) and i=l~ i b i_l

<

oo

(-N < n < O)

This is immediate by Case II since

ISn(X; ~I~F; (5.3)

3

~*)l Remark.

*.nE

sum

=

IS

-n

(x;~"

F

;

In general in (5.2) for each j we have

In the proof of (4.2) ~ i i=l

bi_ 1

Case II we take the entire infinite

to insure the existence of an independent constant.

Note that the condition k~+ IJ infinite sum.

~*)l

< m.] -< k~j permits the use of the entire

The conditions nj = 4°2~'nj[w~]l

w~l -1J

and n~+ I a ~ (I + bkj)n "J play an important role in the algorithm needed to

construct the four finite sequences in (5.2).

VI.

A PROOF OF THEOREM

(5.2)

The first six sections of this chapter are concerned with the construction of the exceptional set measure.

sense of the word

(T*uU* UV*) (V*

is the "exceptional"

Roughly speaking, the set

(S*UX*UY*)

and

set in the usual

is used only to establish the last statement in

set in the sense that if ~*~

in (5.2) and the estimate of its

We will construct this set as E = ( S * o T * o U * o V * o W * O X * O Y * ) .

Actually, the set

(5.2)).

E

x ~ ~* x

~

(S*UW*U X*uY*) but

W*;

is an "operational"

x ~ E, then

and this fact will

(among other things)

allow certain "pair changing" operations to be performed. the most important function of this "operational"

Consequently,

set is its use in the

proof of (6.40) and (6.41), the pair changing theorems. In the rest of this monograph a lower case letter with a Greek subscript of

(for example: ca)

will denote a positive constant independent

F, y, P, N and x ~ (-~,~).

Context will usually indicate when two

lower case letters with the same subscript denote the same constant. We fix 1 < p < mF>O.

~

, y > O, F C

(-~,~)

and

N > O.

We assume

25

1.

C o n s t r u c t i o n o f the Pk(X;m) Fix i n t e g e r

k > 1.

Consider m= Let Gk(~ro) =

to; r=l,2 ] Cn (mro;l: oF)] _> bkyp/2 }

{ (n, (~ro) I

Note that since

polynomials and t h e s e t s Gk,X ~

limit

Icn(mro ; ~ oF) I

Inl + ~

a finite set. Also, i t is clear that implies

(-n,~ro)

c

Gk(~ro)

Let

Pk(X;~ro) =

For

~

Let

Gk(msl ) = {(n, Wsl) ]

Let

RkCX ; _ msl)_ =

sl

~

= o,

Gk (~ ro ) is

(n, mro) c Gk(mro)

f o r Inl > o.

Z Cn(~ro '• ~ o F) e inx = Rk(X;~ro) (n,~ro) c Gk(ero)

; lxl

ro i C n C m s l ; l ~ _ Pk (" ; mro))l ~

bk yp/2}.

Z c (m . ~ o (. ; m )) e (n,~sl) c Gk(~sl) n Sl" F-Pk ro

i2nx; j Ixl<2~

Let Pk(X;~sl) = Pk(X;~ro) + Rk(X;~sl) = ~(X;~ro) +Rk(X;~sl). For

~t2 C ~sl

Let Gk(~t2) = {(n,~t2) I ICn(~t2;~-Pk('; ~sl))I > bk yp/2}. c (~ ; ~ - P k C - ; e s l ) ) Gk(~t2 ) n t2

Let

Z Rk(X;~t2) = (n,c0t2) e

Let

Pk(X;~t2)=Pk(X;~sl)+Rk(X;mt2)

Suppose ~jv = ~ £ v - 1

and

c

i4nx., I xl

_< 2~

= Rk(X;~ro) + Rk(X;~sl)+Rk(X;~t2 )

Pk(X;~£v-1 )has been c o n s t r u c t e d .

2~

26

Let Let

Let

Gk(~jv ) = { ( n , ~ j v ) ]

~%'(x;Wjv)

lc n (wjv ; / [ o _F p k ( ' ; ~

~y-I

33[ i b k yp/2} i2Vnx

=

z

c

°-p

(n,mjv)¢Gk(mjv) n(~jv ' X F

Pk(X;~jv) = P k ( X ; ~ , v _ l )

•

))

¢

k(''mZv_l

+ Rk(X;ejv ) •

"Ixl

<

•

_

Then

Pk(X;~J v) = R k ( X ; ~ r o ) + R k ( X ; ~ s l ) + " ' + R k ( X ; ~ z , v = l ) + R k(x;o~jv) Continuing in this way we define for each k > 1

and for each

dyadic interval m a polynomial Rk(X;w), a polynomial Pk(X;~) and a set

Gk(~). (6.1)

Remark.

an (~) t h e c o e f f i c i e n t

To simplify the notation we will often denote by

of the ¢

term in Rk(x;~)

c o r r e s p o n d i n g t o t h e element (n,~) E Gk(~); times write a ¢ write a ¢

i~x

ikx

for simplicity

It is to be noted that in the sequel whenever we

as a term in Pk(X;~) we always assume that the terms in

this polynomial have not been "collected"; term of Rk(X; ~') for some ~ ' D have that if a e

]aJ ~

we may a t

~.

that is to say a ¢

With this convention in mind we

i~x . is a term of Pk(X;m) for some ~ C [-2~,2~], then

b k yP/2 .Also, it is easily shown that

Icm(~;X~-Pk(';~))l

Let

< bk Yp/2

Gk = [J{Gk(~ ) I ~ C

(6.2)

Theorem.

for lml S o

and mc[-2~,2~]

[-2n,2~]}.

y: (n ,~o) ~: G

[an (~)

[2

1~1 <

2mF.

k

Proof.

Suppose

iXx . is a

~. = m )v £,v-I

where v > o is arbitrary.

Then

•

27

J

Lojv

I% ~(x)-Pk(X;°Jjv)l2dx

=JmjvlX~(x)-Pk(X;me v_l))-Rk(X;C°jv)] 2

dx.

"

But i t is e a s i l y shown t h a t { ~ o F" pk ( ' ; ~ £ , v - 1 and Rk(.',~jv )

are o r t h o g o n a l over ~jv"

).Rk(.;~jv))

so t h a t by a s t r a i g h t f o r w a r d

expansion o f the r i g h t s i d e o f the above equation we get

i~jvl~{x)-ekfX;~jv)l2 dx

= f~ojv

IXF0 (X)-Pk (x;cO£,v_l) 12dx ]2

( n , ~ Jv .)

e

l~n(~jv) Gk(~v)j

-

l~jvl

Hence

]X~(x)-ek(X;~)12dx=la~l=2~2-z(v-~) I~l~°F(x)-pk(x;~)]2dx E

C %(m) [~1=2~2-v

(n,c~)

lan{~,)t21~l

;

for

v_>

1.

We can now repeat the same argument for the first term on the right in the above equation.

Finally, after a finite number of steps

have

o<

"

E

10JI=2~2-v

If ~°<x)-Pk{X;~)I2dx--12~I~{x>I2dx0~

p

-2~

]an(~O) 12 ]~] • (n,~) E Gk

I~I->2~2-v But since

v 2 o is arbitrary, the result follows.

we

28

Corollary.

(6.3)

Proof.

Since

Z (n,w)¢ lan(~)l

I~oI <_ 2 bk2y-PmF. Gk

>

bk y

p/2

[ a n ( ~ ) l 2 _> bk2 yp ., SO t h a t

b k-2 y -p

z ( n , ~ ) ¢ Gk

! bk2y -p

completes

[-2~,2~]

of

¢ G (w) k

0

if

Note that for each

v > O --

A k(x) =

Let

X

we h a v e

]an(~)]r~2 I.

]a n ( ~ ) 12

,"

Consequently,

Lemma.

2 bk2yp

mF.

Gk

(6.3). v > O if

we define x ¢ ~

and

I~l

= 2~2 - v

°

x is an end point of ~ and I~I = A v (x)

2~2 -v

is a simple function.

k

v Z A (x). v=o k

= {x I k

(6.4)

6k

1 _< bk2y-p

.~o t h a t

lan(~) 12

(n,w)

I

e

and for

z

Let

¢

lan(~) 121~l !

Z (n,w)

the proof

For x ~

Av k (x) =

(n,~)

(6.2)

by

This

I~[

if

-1 A (x) > b yP} . k k -p m Xk < 2 b k y mF.

Proof. (x)dx = 2~

Z v=o

-2~

(~)l ~n(~) 121~I )

A (x)dx = Z k v=o

C

(n,~o) ¢ Gk

1~l=2=2:v E (n,~)

] a (~)121~l n

¢ G k

~1,

0

Q

O

t~

O

N

IA

..~ c.~

(1} ~

t...~.

4

(1}

'~

i

w°

N

v

cr

IA

t,d

'

Ix}

I^

:~

~

,~ ~

Iv

{I} .-~

i

-t~

C] 0

0

t-I-

I:::

O

t,~

~

i-% O

IA ".-~+ II M

~o E

~

t..~,

0"t

tl

~TJ

>.~

I

I^

0 ,t

"<

u"

~o E

~

~

E

ox

Ix}

u

0

i

o"

II

(..,,.t

i

mU

i

q,

,.<

,<

I^

~-~

~

,<

cr

1^ --~

o

,~

:

o

I^

t...a.

~ co

<

S

~

[A

~

O

,

"<

'.~,

~

o

i

v ~.

t..~o

=

o

~°

t..~o

n^

I^

Ix.}

IX}

0

c~

O

,,<

,_,>

N

II M

~

°~

cr

o

~

F'

D~

~

o-,

X

N

4

I^

Iz~ X

I^

X

,.<

IA

X

~

Z~

v

"<

c¢

::r

:g

;}<

B.}

~I

~o

IA

,..-,

Ix}

(I)

ix.}

o

(I}

{I}

Ch 0

30

Ak(x o) ~ b k l y p

-i -p/2 Ak(Xo) ~ (bklY -p/2) (bkly p) = bk Y

IPk(X;~) I <

-2 p/2 bk Y (6.7)

Remark.

It is immediate by the definition of Ak(X) that if

x c Xk, then there exists a dyadic interval ~ each ~ ~

Xk

with x ~ 9.

X k we consider its three left dyadic neighbors ~

and its three right dyadic neighbors

Let X

=

w

1 2 3 ~r,~r,~r

12

,~

For 3

all of lengthlm I

~Iuw2u~3u~U~ U~ U~ r

r

r

~

~

If ~ is located too close to either

2~ or

-2~ , then some or all

of the three left or right dyadic neighbors may not exist.

If this

situation occurs, simply delete the missing terms from the expression for X~ . It is clear that X k* =

Let (6.8)

1X I ~ 71~ I.

U { K w I~ ~

x k}

Lemma. m X* k ~ 14 bky-P inF.

Proof.

(6.9)

m

X* < 7 m X ~ 7 (2 -PmF) k -k -bkY

Remark, Note that if 0J*~ X;,

four subintervals 2.

~'

of

=*

then 0J'~ Xk for each of the

4I='I :

with

Construction of the sets

by (6.4)

G* k

and

I~*I Y*. k

(6.10) Remark. We first note that if Pk(X;~) contains a term ikx -- i(-k) x a e , then it also contains the term a

31

Also, if k >

O, then

for by construction

(X[~'], ~') E

some integer n > 0 and

~. :~ w and jv

v (2 n) [~jv ] = n.

But

for some w = m'; k 2Vn for some integer v > 0 and for

X =

Hence

the other hand for each :n'

l~'l-1

27

(n,~jv) ~

G (w ). k jv

(k[~jv ], mjv ) e

Gk(~jv).

(n',~') e

Then

G

On

G k where n' > O let

n' = X[~'].

For each k > i consider the following two conditions on a pair p = (n,~0): -t0

For some (Ak )

a n d I~t

For some

(~[~'],~') ~

Gk: w o w ' ,

n > O,

In-X[~]l<

bk

> blO]~' I.

(X[~'],~') e

Gk:~ =

m', n > O,

In-X[m]l<

-I0 bk

(Bk)~and there is some term a' ¢ iX'x of Pk(X;w') such that

(q0<_1

!

- Io< u:

%

We let

G k = {(n,~) I m ~ X

and

(n,~)

satisfies

"Ak"

or

"Bk"}

k (6.11)

Lemma.

-19 -P ,x, [wl < C O b k Y mF.

Z (n,w) £

For each

k > 1 m

Gk

let %

G* = { (n,~o*) k

(6.12)

(n,~')

]

Remark.

If ~'

~

Gk,~O*.~ w'

[~*[

Also,

4[~'1

= I~o*l}

is not located too close to either

then there exists two intervals ~* 4l~'I =

and

if w* ~

such that Xk

and

m*=~'

(n,~*)~

G k,

2~ or -2~ ,

and then ~ ' ~ X k

%

and

I~*I

(n,w') ~ Note

Gk

for each of the four intervals ~'c= ~*,

41~'I =

32 %

that

(n,~*_l) c G*k iff

(6.13)

left

G k and

(n,~20) c

Proof.

la*i < c6 bk 19 G* k This is immediate by (6.11).

Let ~

be any dyadic interval contained in

Let

F

Lemma.

(n,~lO) c

%

y-P (mF)

Z (n,~*) e

be the i n t e r v a l

Gk.

[-2~,2~].

of length

2

b

w ! symmetric about the

of length

2

3 b k l ~ t symmetric about t h e

end p o i n t o f ~. Let

F2

be t h e i n t e r v a l

right end point of ~. 1 2 Let Fw = F~ ~ F

I

Let

(6.14)

Lemma.

m

(n,~) ~

for some integer k 8 b y-P mF.

Yk < --

Proof. we have

2.

Clearly,

Z (n,~) ¢

m

Y* < k-

l~I ~ 2

G

n}

k

3 4b k

~ (n,~) E

bk2y-PmF.

I~[. G

Hence

But by

(6.3)

k mY~ ~ (4b~) (2bk2y -p mF).

Gk

Construction of the sets

S*

and

fl(k).

f

Let

S : U{o~ I y-P J~,~ I,~ °F (x) I

(6.15)

Lemma.

Proof.

m S <__y

(6.16)

m

S < 2

dx >__ I~[

y-PmF

~ F (x) I dx =

2

y

mF

Remark. For each ~ C S we consider its three left 1 2 3 dyadic neighbors w ,~ ,~ and its three right dyadic

33 1 2 3 neighbors mr,mr,m r Let

S

m

]w I

all of length

I = mrV~

1 2 ~U~U~£

3

If m is located too close to either

2~ or

-2~,

then some or all of the three left or right dyadic neighbors may not exist.

If this situation occurs, simply delete the missing terms from

the expression for

S

It is clear that Isml ~ 71m I. Let

S* = U { S

Im ~:S}

•

(6.17)

Lemma.

mS* < 14y-PmF.

Proof.

mS* <

7mS < 7(2y-PmF)

(6.18)

Lemma.

Proof.

By definition of

If m 4 2 S ,

then S

by

(6.15).

C(m ,• ~

oF ) < Y

we have that


inequality it is easily shown that

and by Holder's

But s i n c e

C(m • ~ ( o ) '

F

< sup -~.~<~ <~

we have 0

C(m ; ~ F ) <_

Hence

17

J c

(m) l n+~_ 3

<

1

llx

34

c(~; ]( O)F -< (6.19)

4-(#)

Lemma. I f

~*~

k > 1

we have

t h e n f o r some

bkY < C* --

But since

S*

(m*; Z ~) < b

~* ~ S*

we have

[[~Fllp = I ~ ) I / P I I ~ I I

a n d ~ oF ~: 0

n

Since .z/[ 0 ~ 0 F

Proof. ......

(i/p-I)

on ~*,

w' ¢ S

four

a~' ~

0 < cn. ( ~ . ; X °

a~*,

F ) < y"

we know

for all four

41w' I = lw*I. Hence by (6.18) we have all

on ~*,

y

k-i

a.e.

a.e.

p
41c0' t = Im*t.

0 < C , ( ~ , ; ~ .I o~ n) n ,J~' ~

~*,

0 < Cn (w''' ~ ° F ) < y

for

Consequently,

The lemma now follows immediately from

the definition of b k. (6.20) function

p

Lemma.

There exists a positive integer L which is a

only such that if

bkY _< Cn ( m,; ~ o F) We now define for each

e*4

S* , then yp/2 <_ bkL .-1/4

implies k > 1

(k) = {p* = (n[~*], ~*) e

y

the set

G*kL I C*(p*) < bk_lY

and

-I n = 4"2~. 4.

Ico*l

n[w*]}

Construction of the partition Fix

partition

k >__ i.

Let

~(p*, k)

p* = (n[~*], of

m*

our partition satisfy (6.21)

Cn[ ,](m' ) < bk_ I y.

~(p*,k) m*) e

of ~* ~(k).

for each

p*

in

To construct the

we require that each interval

~'

of

~(k).

35

Clearly, each of the four intervals ~' ~ 41~' I =

Io~*I satisfy

(6.21).

For each of these intervals

we consider t h e tWO i n t e r v a l s

~"C

~',

each of these two intervals satisfy otherwise

~'

~*,

21~"l =

te ' l "

~'

If

(6.21), we split ~';

is an interval of our partition.

We continue

splitting according t o the above rule as long as possible or until we reach an interval of length (6.21)

each interval

~'

2~ 2 -N.

In addition to

of our partition will satisfy

I~'I ~ 2~ 2-N

(6.22)

I

-N+I

I if ~ l~*I z I~'l z2~ 2 (6.23)

, then (6.21) does not

hold for at least one of the two intervals

m" •

~',

21 ~"I = I~I, ~d %

(6.24)

If

~*

~

%

~ ::) ~'

and

41~ I !

l~*I, then

(6.21)

holds

%

for ~ . We now define fi(p*,k)

and

m*(x), the center interval c0rresponding to

x

Consider the collection of intervals formed by taking each

~,. s 3v

~. 3+1, v

in the middle half of m*

For each

x

~(p*,k)

~*

and adjoining

w.l , v j

or

there are intervals

%

(a% l e a s t

one)

~*

as a b o v e , which c o n t a i n

x in t h e i r

%

We define

m*(x)

maximal.

We h a v e

c6.2s)

21~*cx)l

(6.26)

which are

as such an interval

~* with

<_ I~*1

x belongs to the middle half

of

~*(x).

I~*I

middle half.

36

(6.27)

I

w*(x)

is a union of intervals of

~(p*,k),

since

I~* (x) I is maximal. I

If

(6.28)

~*(x) = WjvUw j-+l,v, where ~.jv ¢

a(p*,k) '

it follows from (6.21), (6.24), and (6.27) that max ICn[~jv](~. ), Jv q~ If Im*(x)[ >

C

(Wj+l,v)I< bk_ 1 y. n[wj+l ' v]

2.2~.2 -N

it follows from (6.23) that

(6.29)

l

C* (m*(x)) > n[w*(x)]

b

k-1 y •

m* -m*(x) is by (6.27) the union of certain intervals of ~ (p*,k).

(6.30)

For each such interval ~' the distance from exceeds half the length of ~', since

5.

Let

~(p*,k)

En (t)=

(6.31) For each [mm] = x ¢

1 "I im m

~(k)

be the partition of w*.

JW

" e-znYdy

)~F(y )

k > I.

For

t

E

t ~ e

m

m

Lemma [En(t) [ <_ C@ bk_lY ~m ¢

for some

~(p*,k)

let

~

m

for

t ¢

e

~*

we let

a(p*,k)

w*.

have midpoint

t

m

and let

6m" ~*

let

~(x) =

& (x;

~) =

Z m

Let

[~*(x)[ is maximal.

Construction of the sets T* and U*.

Suppose p* = (n[w*], ~*) E

For

x to w'

62 m 2 (X-tm)2+ 6m

C be a fixed positive constant (to be specified in the sequel).

37

Let

T*(p*) = {x ~ ~* [ H*(x) > CL k b n

y}

H* (x) is

where

n

k-i

the maximal Hilbert transform of

E (t)

over

m*.

n

Let

U*(p*) = {x E

~* [

A (x) > C L k}

-csCLk (6.32)

Lemma. mT* (p*) <_ c -CBCLk

mU*(p*) < c a a (6.33)

]~o*]

Theorem.

If

x ¢

(-~,~)

and

x g

(T*(p*) U

U*(p*))

I tSn(X;X F;%)I-INn(X; ~ F; ~*(x) I I~ %(L k bk_lY) where

~*

is any interval which satisfies

O

x is the middle half of ~o' w* ~

co*, c0*(x)~ ~* (strictly)

O

and

~* -

c0*(x)

O

is a union of intervals of

O

We l e t

T* =

0 k=l

(6.34)

Lemma.

Proof.

Since

zl~*l ( n [ ~ * ] , m*) e

~(p*,k).

oo

( U T*(p*)) fl(k) m(T* U

fl(k) C !

and

U* = I J k=l

(U u*(p*)) a(k)

U*) < CA y-PmF. G~L

for

z

(n,~*) ~ G* kL fl(k)

each

I~*l

k >__ 1

- 19 <_ c ~ bkL

we have by (6.13)

y-PmF

Hence by (6.32) ,

m( U a(k)

(T* (p*) O U* (p*) ) <_ (c

-cBCLk

¢

)

Z

I~*t

(n[~*] ,~*)~a(k) -CBCLk - 19 --< ( c ' ~ e ) ( C ~ b k L y-PmF ).

then

38

We now choose

Hence

m(T*U

C

such that

e

~

. bkL

U*) < c' --

6.

cBC > 20

bkL )

y-PmF < c A y-P inF.

k=l

Construction of the excepti0nal

set

E.

We let

X* =

U k=l

Y* kL

(6.35)

Lemma.

Proof.

By (6.8)

0 k=l

X~L

Let

and

m(X*UY*)

m(X* U Y*) 2 c z'

so thai

<__ bkL

( ~ ~

log 2 ;

Y* =

< c Z y-ProF.

--

and (6.14) we have

( k=l Z bkL )

y-PmF < czy-PmF.

] H*

V* = {x E

> Ly} where H * X °(x)F

is the

N

maximal Hilbert transform of ~ 1 6 t % F~ (6.36)

Let

Lemma.

W* =

{x I

mY* -< Kpp

~* -i

e

y-PmF

x is an endpoint of some dyadic interval ~}

(6.37)

Lemma. m(W*) =

Proof.

This is immediate since

Let

over

O. W* is countable.

E = S*uT*~U*uV*uW*uX*uY*.

(6.38)

Theorem.

m(E) < C

y-P inF. P

Proof. 7.

This is immediate by (6.17), (6.34), (6.35), (6.36), (6.37).

The pair changing theorems. Suppose

no,mS,

k

and

x

satisfy

39

x ~

I

(6.39)

(6.40)

X

is in the middle half

2"2~-2

,

exist

%

In[%]

Moreover,

%

-

0

[w:], u

%

m*) ~ 0

%%

'~ = 4"2"rr" Ito*l In[w'l, - no[to;]l if

(n

p: = u

< A bk - 1 , where A i s

~ ; = ( ? [ ~ o ] , too),

for all

n such that

(6.41)

Theorem. exist

and

lno[m;]

-

%

a fixed

+ bk_iY}

n[m;]l £

~,w* and m s u c h t h a t

integer.

then

2 A bk

-2

n o ' ~*o ' k and x s a t i s f y

Suppose

of

p* = (n[a~*] ,to*) e G]~L

ilS oCX; g)t-ls cx;mg T %

there

Gy. KL

S u p p o s e no,to~, k and x s a t i s f y (6.39). % % n and m* s u c h t h a t x is in the middle half

%

and

no[U;] ,

C*(pg) < bk_lY.

bkY i

to*, to*2)too'

-1

~;,no=4-2~'t~;1

of

-N

Ito;l >

Theorem.

Then t h e r e %

E,

x is

(6 3 9 ) .

Then

in the middle half

of

~-* ,to* D to*

0

=

and

4-2~ •

1 ,1-1 [7.2

1 < m < k.

If

,

In[%]-no[%] I

Po i s

<

2A bkl-

given by (6.40)

,

~*:(~[~*],;*)

then

C*(p;)

~

< b

"

Moreover, defined. and

7.

C*(p*)

< bm_lY ,

For this

w* - ~ * ( x ) o

partition

so t h e p a r t i t i o n we h a v e

y.

~ ( p * ; m) i s

~ * ( x ) ~ w* o

is a union of intervals

m-t

mL' G*

(strictly)

o f ~ ( p * ; m).

A Proof of Theorem (5.2).

(6.42)

Lemma.

Cn (~) -< c~ Cn+l(to ) (6.43)

Lemma.

If ~ = mjv,

then for each

n ~ 0

we have

where c~ is also independent of ~. Suppose

m* = tojvLkOj+l, V

and

no [to*] =

no v+l 2

40 v+l Then f o r

all

n such that

In-nol

< 2

and f o r

x ~

(-~,~)

we h a v e

ftSn(X;~ F;~*)I-1S

(X-•F;to*)lt '

n

<

0

(

cB

(6.44)

max

no [to

]

,

•

jv Suppose

Remark.

for

c,.,+

n

some n > 0

0

,V]

we h a v e

n

= n[~*]

0

0

2v~+1 where W*o = (mjvUtoj+l,v)" 0 < (n-no) < 2 v+l,

n[tojv]

= no[tojv]

Then it is easily shown that

no[too] = n[too], and

+ 1,

and

n[0~jv] = no[tojv]

n[to j + l , v ] = no[to j + l , v ]

n[ to. ] = no[ toj+v,v] + i. J+l,v

or

or

For example, if we let

-i n

o

= 4.2~.I~oI

n[too] where

it is easily seen that

(6.45)

~* = (~jvUW ), o j+l,v

then

no = n[to~] 2v+l

Remark. Note that the condition in (5.2) that -i

n .3 = 4 " 2 ~ ' n j [to~]Ito J ~I

some

v -> O

where

implies

to* = j

that

(tojvUtoj+l,v )

nj[to~]

=

and that

n ) .... 2v+l

for

n. = O if and J

only if nj[~*]j = O. (6.46)

Remark. Using the fact that

the condition in (5.2) that (since n i = n)

that

--~--(i + (2)) i=l

nj+ 1 _< (I + bk.)n 3 j

n. < ~ J--i=l

< 2

implies

(I + bi)n < 2N < 2N. ---

Consequently,

_N if

Ito~l ~ 2"2~'2

, then by combining this with the condition in -i (5.2) that nj = 4-2~.nj[to~]Ito~l we have that

41

n j j[m~]2

N+I

so that is

-1

< 4.2~ --

j

nj[w~] = 0 ;

n. ~ 0 J

implies

nj[m?]3 -<

so that

2N '

nj = O.

I~*] > 2.2~.2 j

1 n j [~?] j - .< ~ ;

Hence

-N

An equivalent statement

We now prove

by means of the following algorithm:

Let

n

(5.2)

= n and -I

m* = [-4~,4~]. -I

By (6.19) there exists k such that

bkY _< C*n (re*l) < b k _ l y" (6.47)

that

Lemma.

(n[~_*l],~_*l)

By ( 6 . 4 7 )

the partition

by ( 6 . 3 3 )

since

x }

~

G~L .

~((n[m_*l],~*l); E

k)

is defined;

we have

IS*n_1 (x; /~.F''m*-i )[ = IS*n_ I ( X ; X F ; m * ( x ) ) ] + O(L k bk_lY ) We let

k 1 = m 1 = k,m*(x) = ~* and n -

By (6.28),

-

O

(6.42),

IS*n_lCX; • F:~*I )]-

(6.43) and

= 4 " 2 ~ ' n [ m o ] 1 % 1 -I O

(6.44)we

= IS*n o ( x ; ) ~ F ; ~ o ) [

have

+ O (Lk bk_lY)

Suppose n o ~ O.

Then by (6,46) we have I%1

Consequently,

(6.29) we have

implies that ~

exists

k

by

oF ~ 0 a.e.

such t h a t

Clearly,

Cn[~o ](~o) >_ bk_lY.

on ~* " o '

o

ko < k = m_l "

-

Let

(w~) < b k _1 y •

no[~

-N

;

But this

so that by (6.19) there

b k y < C*

o

> 2.2~.2

]

p*o = ( n [ ~ ] , w ~ )

There are now three possibilities:

o

.

so

42

Po*

Case I.

¢

G*

koL

Then the partition

we have

]S*n ( x ; ~

~(p;, ko) is defined;

F; mo)l =

lSn (x; • F "

0

so that by (6.33)

mo(X))l + 0

(L ko bk -1 y)

0

0

-1 Let mi = rag(x), By

(6.28,

(6.42),

I s* ( X ; ~ F ; mo) I no Case 2.

Po ~

G~oL

mo = ko, and

(6.43), =

n 1 = 4-2W'no[~]

and (6.44)

IS* (x; Z F ; nl and

lmi[

we have

ml) [

no[U;] > 2

+ O(L mo bm _ly) o -2 bko

A

eU

Choose

partition

n

as in (6.40)

~(p*; m)

I S - ( x ; J]~ F'"CO*o )l n

n, w*, m as in (6.41).

The

y i e l d s by (6.33) =

IS--(x; ~ F ;~-*(x))l + n

--

Since

and

0 (L m b

m-1

y)

-1

In[mo]

no[W;] I < 2

AU k

< 2

A

bk 2

O

IIs

IS*(x;_ 0

(6.40) yields

O

÷ bk l Y }

n

o e~

By (6.41) we have we obtain

C*(p;) < bm_lY.

By combining r e s u l t s

IS* (x; X ;mo)] = Is*(x; ~F;~*(x))] no F n

+ O(L m bm_lY)

-1 Let ml = m--*(x), m o = m and n I = 4-2~-n[millmll -2 The f a c t t h a t n o-[m*] > 2 A bk i s c r u c i a l in the p r o o f O

O

of the following (6.48)

By

(6.28),

Lemma. n I _< (i + bko)n °

(6.42),

(6.43) and (6.44)

IS~o(X;XF;~°*°)l = lSnl

(x; XF;m~){

we have +

0

( L m b

o mo- lY)

43

Case 3.

P* ~ o

G* koL

and

no[U;]

< 2 --

A b -2 ko

%

Choose

n

as in

(6.40)

IIs*n (x; X.F;m*) o I o

But by

(6.41)

and

n,m*,

IS*(x; X F , m ; ) I I o

m as in

< C {C*(~;) -- ~

+

Then by

(6.40)

b k _ly} o

we h a v e

Is/cx;

IIS*n ( x ' ~ , F"m; )1 o m

)C

F;mo)[ ]

_< cU {bm-lY + bko - l y }

~''ulear±y, if we

let

IS n (x;mo) I = o

lSg(X;mo) I + O(L mo bm -1 y) o

where

(6.41).

o

it is u n d e r s t o o d

= m = i we h a v e

that

m* * o = ml

We c o n t i n u e u n t i l

Case (3) o c c u r s or u n t i l

y i e l d an i n t e r v a l

~* 3+1

so small t h a t

n~+ 1J

Cases (1) and (2) =

O.

"

APPENDIX A.

THE HILBERT TRANSFORM

For f real valued with domain -~

(Hf)

< a < x < b <

(x) = P.V.

~

(A.I)

f(t) x-t

Theorem.

everywhere

in

(A.2) (a,b).

Let

jb a

(a,b) and

dt =

If

f ¢

limit e + O+

a

1 L (a,b), then

f(t)dt + x-t

I

(Hf)

exists

f(t) x+¢ x-t

(x)

almost

(a,b).

Remark.

H is called the Hilbert

transform of f over

If f = u + iv where u and v are real valued with domain

(a,b), then we define For a proof of

f(t)

will

h(t)

where

(Hf) (x) = (Hu)

(A.I) see

product

L 1 (a,b).

h c

inequality

Since

that

For an explanation

(x) + i (Hv)

[15] page 132.

be the pointwise

by Holder's

(x).

In many of our applications

of g(t)

= c

g c

L~(a,b),

f = gh E

1 L (a,b).

in~t

and

we have

of course

of the notation used in the following

theorem review the first section of chapter 5. (A.3)

Theorem.

Let

~

denote a subinterval

of w*

which

X

contains

x

in its middle half.

transform H* of f over

~H*f) ( x ) - -

sup IP.v. CY

T = {x ~

~*

We define the maximal

J~ U

dtl

;

x~

~*.

(H'f)

(x) > y}

fE

L (~*) oo

x

]

Hilbert

~*:

x

Let

dt

;

y > O.

Then

I

45 Y -c B Part

re(T) <__ c I ~ * l ~

i.

C~

Part

2.

F o r co = co*

m(T) < Cp y - P i i f I J p

-I

P r o o f of Part We

by

may

0 on

i

Ilfll -

p

p

(Taken d i r e c t l y

assume

(1,2~)

from

~* = (0,I)

and

[3]) llf~ ~ =

and denote by E (t)

the

i.

Extend

2~-periodic

f(t)

extension

O

of this Lemma

function.

I.

If in T h e o r e m

(in place

o f condition

[F(x+t) an absolute

+ F(x-t)

(7.15)

([16] Vol

(7.17))

1 page

102) we suppose

that

- 2F(x)]

<_ Alt ,

(0 < t <

~ ,

A 1 > O,

AI

constant),

then

I~~F 3x(r,x)

(_ 1

17(l-r)

F(x+t)+F(x-t)-2F(x) 2 4 (sin t

dt

i <

2)

(0 < r < I, Proof.

As i n d i c a t e d

Q' (r,t)

0 < Q' (r,t)

in Zygmund

precisely

by c o n s i d e r i n g

(

r = l-

~

t_ 4

2

6r

where

the cases

A~ > 0 is an absolute

1 0 < r < -

For

( A

1 0 < r < --2

we o b s e r v e

r)

by ,

0 < t <

constant).

the estimate

- Q' (l,t) < A l

0 < r < 1 and

constant,

an absolute

2 t-4) - Q' (l,t) = 0 (~ r

is able to be stated

for

A2

A 2 > O,

that

(r,t) = 1 - 2r cos t + r2),

A

2'

1 - < r < 1 2

1 (r,t) >_ (l-r) 2 > --4

and for the o t h e r

case we use

A

46

the method of Zygmund. Lemma. 2. I f(x)l

If f is real valued, periodic of period 2~ with

< 1 f o r e a c h x we have

If(r'x)

~_1

- (-

f(x+t)

6r

2 where

- t f(x-t)

< B1 --

tg 2

B I > 0 is an absolute constant,

Proof.

dt)l

gr = l-r, 0 < r < i.

By modifying f by a constant of absolute value ~ I,

we are able to suppose that the indefinite integral periodic.

Under these conditions I

}F(x+t)

+ F(x-t)

an a b s o l u t e

We l e t

F of f is

- 2F(x) I

constant,

=

and

=

f(x+t)-f(x-t) 2

r(t)

=

F(x+t)

~

6r

+

dt = S

(t)

x-t

O < t <

~x(t)

Then I ~

x+t

4(sin t 2

2)

I f ( u ) ldu < B'lt ,

O < B1

~ .

and

F(x-t)-

2F(x)

~ ~p (t)d (

-I

/ =

2 tg t

6r

2

~b(6r)

*

2 tg 6r

2

r

WX ( t ) t tg

hence

f(r,x)

1 - (- ~

(t) ~x at tg t

6r

=

~ 3F(r,x) x

_ (_1 ~

Sx r

~F(r,x) 3x

-

(-

-)2 ~r

~r

4(sin

2-

dt

_ 2 t g -6rT

t tg2

at )--

dt

47

The a b s o l u t e

value of the square bracket

O < r < 1, b y an a b s o l u t e

constant

according

is majorized, t o Lemma 1,

for

and a l s o

the

absolute value of the remaining term. Lemma 3. that

If f is

valued,

periodic

of period

2~ a n d s u c h

%

If(x)l <_ 1 sup

real

f o r each x where f ( x )

I

I

f(x+t) t

A '

where t h e sup i s

tg

2

exists,

then the expression

dt I

5

taken relative

to the

intervals

A ~

(-~,~)

0 e A for which the ratio of the distance of 0 to the extreme right and left of A fixed with

0 <

lies between X

X < i)

I/x (where X

Ill -

A C ( - ~ , ~) A centered

is

is majorized by

sup

where

and

f (x+t) t A 2 tg 2

+ cO

dt

about O

c B d o e s n o t d e p e n d on X .

Let A

Proof.

=

(Xl,X2)

0<~<

(xI < O, x 2 > 0

with we have

Suppose for example IXll < x 2,

i

I~

f (x+t) t

dt

2tg}

<

- _

-

~

+

-

xI

_

~

-x

1

with x <

_Xl

B" 1

ix2 t:1 _Xl t

1

-

B,, iogi l 1

48

where

B" > 0 is an absolute constant, where the result, for 1

the case IXll > x 2 is treated in a similar manner. Lemma 4.

If f is real valued, periodic of period 2~ with %

If(x) l ~ 1 for each x where f(x) exists, the expression

sup

-

f (x+t)

A

dt

2 tg t

where the sup is taken relative to the intervals 0 e A

for which the ratio of the distances of

A~

(-~,~)

0 to the extreme

1/~ right and left of

A lies between k

with

is majorized by

0 <

k < i)

and

(where ~ is

fixed

%

c' sup ~O
c~, c' B Proof.

If(r,x) I +

' cB

do not depend on k . Free to choose the second constant an absolute constant

majorizing the expressions x I 1

-I 3

f(x+t)

-

- ~

dt

t

xI

2 tg

I

f(x+t) 1/3 2 tg t

1

dt

1

(-~

< --

If

A'

x

2

we are able to suppose that

(Xl,X2) = A ~ (- ~, ~)

the interval contained in A

having an extremity in common with

A , the proof of Lemma 3 gives

+

A

--

A'

CH C~

< -

is

1 1

-,-< 3 3

x2<--

~)

49

where

(0 < r < i)

f(x+t) 2 , t g ~t

dt

<_ I f ( x ) l +

-

(r',x) I +

c" ~

+

_ n

dt

I

dt

dt

of

f(x).

]~(r,x)-

<

f (x+t)-f (x-t)

1

-

But

~

l-r

2

B1

l~(r'

sup

+

O
,x) l

< i

by Lemma 2. We now prove

(A.3) in the following way:

1 t

1 2tg t 2

=

O(i)

for

0 < It I <

We infer that

I H*(x) <

E (t) o dt

s~

-

°x

x

Lemma 4 (and the condition

+ B2 ;

x e

(0,1)

2tg-(X~ t)

where 0 < B 2 is an absolute

constant.

Next, with

on o )we see that we have an absolute X

constant

B' > 0 such that 2

H*(x)

<

B,

"

2

sup O < r < l

IE (r,x)]

+

o

B' 2 -ff

We set,

!

1

T f (x+t)-f (x-t) 1-r 2 t g t_

<

f(x+t)-f(x-t) t 2 tg ~-

l-r

a c c o r d i n g t o Lemma 2 and t h e d e f i n i t i o n

_

t

f(x+t)-f(x-t) 2tg~

< sup O < r'<

A'

we have

I ) 1

If 2r is the length of

does not depend on ~.

c"

for the reason of simplification

in writing,

k =~

4

+lf(r,x) l

50

We have (see[16] vol 1 page 254 Theorem (2.11))

e

O

(O < B 3

dx < B 3

an absolute constant)

But then according to [16]vol 1 p. 155 Theorem 7.5 and that and the Poisson kernel, we have,

which is said after about the A r

for

O
I

c

dx

=

. ~

( .

_v: _

Z V=

I E (r,x) IVdx =

0

0

Z V=O

V!

IEo ( r , x )

tVdx

.

e~

z A

v

~

v! lEo(X)[vdx ~ A

V=O

-~

with the

A

> 0

~v

f

(Z

~

~. IEo(X) ]V)dx = A

V=O

f

klE (x) I o dx

which do n o t d e p e n d on v m a j o r i z e d by t h e

V

absolute

constant

A.

We h a v e t h e r e f o r e

%

I ~

klE (r,x) l O e dE < A B

.~

-

(0 < r < I 3

"

A and B '

absolute constants 3

> o)

Under these conditions, we consider the function h, analytic in the open unit disk, defined by

h(r,x)

= e

-+ 21 k i (Eo (r,x)+i

P being the Poisson kernel.

+I ki (Eo (t)+i E o (t)) E o (r,x)) i |. ~~ ~ e-2 =P (r,x-t)dt Since

%;

I c

+I k (Eo(t)+i Eo(t)) 2 -2 i I dt ~- B 3

and since the Poisson kernel of Lemma (7.1) in [16] constant

B' > O 3

P

satisfies the conditions

vol 1 page 154, we have an absolute

such that

51 q~

O<

sup r<

¢

o

dx < B' 3

i

Hence

I

T

sup O
.~

dx<

2 B'.

-

I f we now s e t k'

kl~o(r,x)]

e

k kt = ~

then

B½ '

H*(x) & k

3

s~ O
IE ( r , x ) [ o

+ k

We deduce i

~k'H*(x)

dx ~ f

k e

]E (r,x)] dx =

i e k ¢k .

~

(o,1))

%

o

I

(x ¢

sup O <

r <

0

i

kI~oCr,x) l k sup ~ dx < 2 e 0 < r < 1

-~

B" being an absolute constant > O. 3

B t

3

_-

B ~! ,

3

Moreover,

[

B" > 3 "

~ J{x ¢

Proof of P a r t 2. Let

xC ~*1

(~f)

(0,1)

k' H* (x) ]H*(x) >¢ y} dx

(Taken d i r e c t l y

denote an i n t e r v a l

(x)

=

sup

from [8]) with c e n t e r x, and d e f i n e

x-t

~x

~x

and (Hf)(x)

=

sup x

J ~*'~x

f(t) dt I x-t

52

The Hardy-Littlewood maximal function of

:

~F~[

sup

is

I If(t)ldt

1

~(x)

f

ax

~x

P

P

P

'

But it is easy to see that

(H'f) (x) E (Hf)(x)

+

ca~(x ) ~ 2(Hf)(x) +

c f(x)

The result now follows from the corresponding result for the operator H.

APPENDIX B

P r o p e r t i e s of

c (~), n

C (~) n

and S*(~) -n

This section is taken directly from [8]. The word const, denotes an absolute constant. is that C (~; f) = 0 n

An important property of the numbers Cn(~; f) for some n if and only if

is not shared with the numbers

~.

This property

Cn(~; f).

Cn(m ) and Cn(~) are clearly related. S*(~) as n

x e

f(x) = O for a.e.

Also,

Cn(m ) is related to

Cn(~ ) is related to Sn(~), the n th partial sum of the Fourier

series of f over

~.

Sn Lemma (B.I)

That is, for m = (O, 2~)

Sn-1

=

cn

and ISnl -

]Sn_ll =

O(C n ).

is the technical basis for the above relations.

(B.I.) Lemma

Let

~(t) ¢

C2(~),

I~I = 2~'2 "v

Then we can

r e p r e s e n t ~ (t). (*) where

(i + p2)l~pI Proof.

=

exp{-i2~.3-1.~t}

~ (t) = Z ~

(o,

(m~x I~I + 2 -2v maxm l~''l)-

By a change of variables, We choose polynomials

2~).

I (t)

<. const.

t ~ ~,

=

pl,P2

Pl(t)

[-2~,O)

(t)

(O,2~)

P2(t)

[2~,4~]

¢ c 2 ([-2~,4~]) , ~(k)

(-2~) = ~(k)

t = 2"~,

we may assume

such that

satisfies

(4~) = O, k = O, I, 2.

Then

54

max

I ~ I

+

[-2~.4~]

max

I ~"I ~ const[

[-2~.4~]

The Fourier expansion of ~

over

(t) : Zy~ exp{-i3 -I ~t},

For

,:

0

For

U ~ 0

we have i~I

[-2~,4~]

I~"I]

yields

t e co.

~max

i ~I

"

max i "I

U For any integer n and any co =co. we have J~

Lemma

ICn.2-v(co) l < c o n s t . Proof. = co. ~

max

(0.2~)

we integrate by parts two times to obtain

I;I (B.2)

max I~l +

(o.z~)

Cn[co](co)

We apply Lemma (B.I) to the function Since

0 <_ ~ <

2~ we have

I~ I <

e

const.

-ist

.2 v ,6--n-n[co] ,

(I+u2) -I

Then

%.2

-v (co) =

--

=

Icol

co

......1.... l~i

co

e-Zntf (t) e_iB t

e

dt

-in[~]2

tf(t)

at

z y Cn[col+~/3 (co) la la

Hence) 1Cn. 5" (co) I ~ c o n s t . That i s ,

i f each

A more p r e c i s e

Cn [co] (co)

c (co) i s s m a l l , t h e n each C (co) (~ n r e l a t i o n i s g i v e n in Lemma ( B . 3 ) .

is small.

55

(B.3) and

Lemma Given n and ]Cm(~)] ~

p

g ¢

In-m[ ~ M g M1/2"} .

whenever

Cn (~; g) "< c o n s t . { ~ log M ÷ 1

Proof.

Cn(~;g) = ~" Z

Cn+p/3(~) = iii ~ 1

I

= I~I

Let

e

+iax P

g(x)

~ Z a e k p,k

e

I

a

p,k

=

I~1

I % ~

= I~11

(I + p2)-1

dx

=

kz CkC~).E p , k

ia x o~ e

Then

Then

Cn+ul 3(~) Also,

2).

~I Z

{-2~(n + ~)ix} dx

-iapx

i2Uxk

Ig(t)12dt ~ G2[ml

Suppose I

[Cn+u/3(~)[

exp

~ gCx)

L2(~).

.i2Vkx

P

e

dx

~ exp{-i2~[k - (n + ~) ]x} dx

If k - (n +--~ ) is a non-zero integer, then ap, k k - (a + ~) = 0 then law,k[

= O.

If

ap, k = i. Otherwise,

< 2_ ik . (n +

In an)" c a s e ,

I%,kl-~c°nst. [1+ ] k - n - o~1]-1 ICn+.~/3(~) I <. kz I%C~)11%,k I -1 <

const. k

56

1"~t <. y

Suppose

".

Then !J -i

E

"~1:]

ICk(~O) I[l+ Ik-n-

Ik-nt~iM

<_ ~t

~

9, "1

<_ V(2 + l o g

I

[I + I k - n

- "~I

]-1

I k-n I.<M M)

I~I.<M Also,

z

Ik-nl >M

ICkOO) I [1 + tk-n -

v -i ~I]

g ( ~ [Ck(W) ,[231/2 (

Z

[1 + [k-n - ; [ ] - 2 3

112

Ik-nl>"

k

z M £-2)1/2

~ G(

:

const,

G

I t follows t h a t G

l Cn+v/3(~)l <. const.

for

Igli 2IM.

For

I-~I> l

c

n

MI/%~ ) = q,

(v log M +

M we use the estimate

Icn+vl3(~)l< o.

~z < M I Cn+v/31 ( 1+v2)'1+ (~;g) < m'6" 17 I IlI1' 2 1

~

.<

q(~

1

z (i + 2)-i

+

G

~

M

Hence,

1

1~ i~ M1%+~/31(1 +

(i + U2) -I

IJ

.< q

+

const,

const,

-1 G.M

q.

We w i l l need the f o l l o w i n g p r o p e r t y o f t h e numbers (B.4)

Lamina Suppose

g(x) -- e iXx

and ~=~

~v

C (~). n

Then

v2)-I

57 .%)

I 2

k - nl'C (~;g) _< c o n s t . n

Since Ig(x)l

Proof.

f < 1.

It follows

.M

that Cn (~;g) < i. For

= 1,tCn+p/3(~;g)

H e n c e , we may" assume t h a t

< w-1 ~I2 -2

X

- nJ

12

X - n I _> 1.

we h a v e

12"%)~, - n - ~I ~ ~1 12" v X - nl Icn+~/3c~;g31

I

= l..J-~

I~I

e

ilx-i2%)(n+p/3)x e dxl

-%)

If

2

X - n +--3

(~ O) i s an i n t e g e r ,

then

J C n + P / 3 ( ~ ; g ) l = O.

In any c a s e .%)

12 x-nl'k

+~/3C~;g)i < I2"%)~-nl"

For I~I >~ 12"'x- nl we use 12"%)x-nJCnOo;g)

<- 12%)x-nl"

2

lCn+~/3(~'g)l(l+~2)-I

iF.,,

Ic %)

z

2

2

<-~ Tf

Icn+p/SC~;g31 <_ X. Wen

I-~1 <-. "~12 x-nl

+ 12"x-nl"

+ ~j-1

12%)X-n

•

Oo;g) l (1

+

p23 -1

n+~13

E 1

.%)

(1 + 2 ) - 1

const. I S*(x,~) n

sum o f t h e F o u r i e r

e

-int

f(t~

=

dt

corresponds

to a partial

x-t ~*

series

of

f

over

~*

of order

n[~*].

If

58

n[~*] l = 1 then S*(x,m)

[no[U* ]

and

S* (x,m)

n

differ by the

no

modified Fourier coefficient

C*

n [m*]

(m*).

This statement is made

0

precise in Lemmas (B.5) and (B.6) (B.5) Lemma

Let

S*(x,~) n

=

(j e-intf(t) dt, m, x-t

where

~* =m*O~ = mO Umlv" Suppose 2W+in 0 [m*] = n o . Then

In-no I .< 2v+l

implies in

l einXs *(x,m)

x

o

- e

S* (x,m)l < const, max no

n

{C

no[mo I

(~O~),Cn

o

[mtv

] (~0,) )

Proof. A o

einX I =

-int ~°Ov

[

=

¢

f ( t ) dt - e

e in°(x-t)

in°x I

ino (x-t) }

(x-t)

{ 6

}

e -in°t f (t)' f~

~O~

-

-e

f(t) X-t

f(t)

dt

dt

dr,

e i (n-no)t - i t

where ¢ (t) = We can write (i

x-t

{ein (x-t)

mOv

I mOv

=

e

-'

¢(x-t) = E ~ ua x {m

const.

+

e i2v'3-1~(x-t) , t e mOV ' where

1

(x-t)l

+

COO~

2_2v maxlc~"(x-t)l} C°O~

<

const.

Hence A° =

Z~"

:

zy~

I mOv exp{i (n o + 2~. 3- I "~) (x-t)}f(t) e

i(no+2~'3-1~)x J m 0

i(n :

e

.

~

e - i2v(2-~ no + ~/3)tf (t) dt

+ 2"°-3-11a)x o

dt

1%1

o _v

2 n +~/3 0

2

59

Note t h a t

2 -v n o = 2no[m* ] ,

an i n t e g e r .

Hence

2 n o = no[mOv ]

Then

I Aol <.

ll l'lmo llcno[ O const.

The integral over (B.6) Then

Lemma

] (mo )l

Cno[mOV]

(mov)

mlv is treated in the same way to complete the proof.

Suppose

In-nol < 2 v+l

Im*I = 4-2n.2 -(v+l)

and

implies

inx in x le S*(x,m*) - e o S* (x,~*) I ~ const. n no Proof,

Let m * =

j = i,...,4.

2~+in [m*] = n o o

mlUm2um3.m4

with

41rail =

C* (m*) no[u* ]

Ira*l,

We integrate over each ~. separately and use the fact that ] C* (m*) no [~*]

=

max l& j ! 4

C

(~j) no[~j]

Let us consider the modified partial sums of the Fourier series of the function

f(x) = e

(B 7)

Let

•

Lemma

inx

over m*. eimX) ; S*(x,~* = n

'

m*

e -int e i m t x-t

dt,

where m and n are integers and x is in the middle half of

IS~(x,~* ,eimX) I ~ const.

~*

Then

60

Proof.

center

Let ~*

~*

e

-int imt f~, -int imt • e dt I < ] e • e , dti+ 2 log 3. x-t x-t -

We may assume

x = O.

17.

-i

with x the

Then

of ~*. I

be the largest interval contained in ~*

Then

e i (m-n)t t

dt =

P.V.

J _, ~

sin(m-n)t dt t

But

l~*1/2

lim s->,-O+ I

~

dt = t

e

I

(B.8)

t

Lemma.

Cmo~)I~'1/2

lim .(m-n) ]$* ]12 c+O+ ~ e (m-n)

sin t

-V-

<

dr-

tk

A (x) =

= {~ } be a partition of ~* and let ~k k and length 6k. We define 2 ~k 2+-'2 (X-tk) ~k

A (x; ~) = Z k

Under these conditions if we set, for M ! 0 U = U : M

(x c ~*[

A (x)

> M}

we have -const. M

mCu)

const

Let ~

have midpoint

:

sin t --t

lul ~

const.

~

i~*l

(x e ~*)

dt

61

Proof. may assume We s e t

Taken d i r e c t l y

from [3] .

Without loss of generality we

~* = ( 0 , 1 1 . ~k = ( 1 - r k )

For

(0 < r k < 1).

0 < x < 1

we have

(where P d e n o t e s t h e P o i s s o n k e r n e l )

2 6k

2 6k

<

2+62 (X-tk)

2

k

2 6k

<

2

(x-t~)2

(X-tk) rk+6k

(2sin

2

2

rk+6k

( l - r 2) -<

2 ~k

k 2 (x- '........ ~k +rk (2 sin tk)) 2

=

26k P(rk,x-t k) •

2 It is sufficient to establish the lemma by replacing A (x)

i (x) = kZ 2~k

P (rk' x -

by

tk)

We will utilize for this The inequalit Z of Harnack. With

T

This follows immediately:

designating the Torus of dimension one, let 1

be a function of L (T).

Let

g(x) ~ 0

g(z) = g(r,x) be the corresponding

harmonic function in the open unit disk

(z = r e

ix

,

0 ~ r < i),

defined by g(r,x)

1 2

1 =--

(l-r) (l+r)

g(O)

<

= -2v

( ~ J T P(r,t)

g(x+t)

< 1 2

P(r,t)

(l+r) (l-r)

g(t)dt T

We have t h e f o l l o w i n g i n e q u a l i t y

o f Harnack

dt

(O
62

(l-r ) g(O) < g(r,x) Ifl÷r1 _i=_~ig(O) -

Corollary.

Let

(O
<

U(z) be harmonic and ~

1)

0 in a domain D.

!

Let

Zo,Z o

¢ D and suppose t h a t f o r

closed disk centered at z

a X > 1, t h e

and o f r a d i u s XlZo-Z~I

be i n c l u d e d in D.

O

Then we have

"&A-I which is essentially no different than the first inequality of Harnack. We prove (B.8) by c o n s i d e r i n g t h a t We form, f o r g(x) ¢ LI(T) for

(0,1) i s on the t o r u s T.

w i t h g(x) .> O

x ~ (0,1) t h e f o l l o w i n g e x p r e s s i o n

and g(x) = O ( A I(X) h a v i n g a sense on T).

IT A l ( x ) g ( x ) d x = Z 2 6 k I g(x) P ( r k, X-tk)dX = k

kZ if

(2Tr~k)

g(r,x)

~1

l T

T

g(x) P(rk, X-tk)dX = 2~ kZ ~k g(rk'tk)

(0 ~ r < 1) i s t h e harmonic f u n c t i o n c o r r e s p o n d i n g to g ( x ) .

Since, for and o f r a d i u s

t E Wk, t h e c l o s e d d i s k o f c e n t e r

rk

itk

2 1 r k e i t - r k ¢ i r k I i s c o n t a i n e d in the open u n i t

d i s k , we h a v e , a c c o r d i n g t o t h e C o r o l l a r y o f t h e i n e q u a l i t y o f Harnack,

I 6k g ( r k ' t k )

= ] ~k

g(rk'tk)dt

"< 3

l~k g(rk'tk)dt

"<

3 I~k sup O_
< "

3

( J ~k

sup

O
g(r,t)dt

g (r,t)

dt

63

hence

I T A1 (x)g(x)dx_< c o n s t .

~ T

sup O
g(r,t)dt

o

The last integral is majorized by an absolute constant if Io I g(x)

log+g(x)dx

remains itself majorized by an absolute constant.

(See theorem on maximal f u n c t i o n s in [16] We have a l s o , A1 (x) Al(X)dx < "

=

(2~)

Z k

6

1 ~

k

I

b e i n g d e f i n a b l e on

Al(X)dx =

T

T

P

vol.

(rk '

r. 2~ k

k

T,

P ( r k , x-tk)dX

T

X-tk)dX = 27 Z 6k k

We consider, for M > O, the s e t

1 page 155.)

27

EM defined by

EM = (x c (o,I) IAl(X) > M} We have

IE.[

We put suppose

M < 2~r

We choose

1 for M.> 4~ (implies O < u < 2)

IEMI = u

u +

O.

(If ~

=

O,

=

gM defined by

~)

if

x¢

E

M

otherwise Then we have

o

gM(x)

1 (log ~ )

[1

log P

IEMI <

1

and we

the inequality in vue is evident.)

We consider the function

gM(x)

M > 4~T which i m p l i e s

log ; ÷

log

k

(ill ------" log

log

64 1

S i n c e we have

i Therefore ~o

for

1 0 < V < ~.

~ >

1 (which i s t h e c o n s e q u e n c e o f ~ > log ~)

log -~ log log gM l°g+gMdX = I - ~ log Thus for

~ 0(I)

M > 4~, we have

IT Al(X) gM (x)dx < const. hence

f M IE

g (x)dx = M

J M

and t h e r e s u l t

A (x) < const. gM(X)dx 1

M i < IT log ~ -

follows.

PROOF OF (6.11). (Taken directly from [8].) If (n,~) satisfies a pair

"Ak"

then

(k[~'],a') e Gk, where aca',

pair (k[~'],~') ¢ Gk

(n,~) can be associated with

I~'1

=

bklOl~l.

there are < const, log bk I

For each fixed

different dyadic

lengths lal which satisfy ]~I-> laI-> De-10) ,la [ For each different length there are < const, bk I0 which satisfy

In - k[~]I < bk I0

integers

n

It follows that

:I~I ! const, b -11 I~'T, k where the sum is taken over all pairs (n,~) which are associated with the fixed pair (k[a'], a') E Gk

and which satisfy

"~"

Then

zl~l Ak

<

~onst.

b~ 11

z Gk

I~'1

~ const,

b -13 k

-P mCF) Y "

65

If (n,m) satisfies "Bk" then wc~' for some m' such that (n',~') ~ For each fixed (n',~')

-3 ¢ Gk, Pk(X; m') contains < b k

exponents X,

the number of pairs (X,X,) corresponding to Pk(X;m') is < f i x e d pair ( X , ~ ' ) , X ~ ~ '

b I0 < '

< Const. log bk I m'

I n - X[m]I

k

] k - x ' l - ] ~ ] < b k 20

holds for < const,

It follows that for fixed (n',m') s )aI ! c o n s t ,

bk 17

holds for

bk I0

For each fixed different integers n.

Gk,

(n,a) which s a t i s f y

"Bk" and o~a'

Hence I~l ~ const, bk 17 "B " k

~ ]~ 'I < const, bk19y -p m (F) %

-

combining the contribution from "Ak" and "Bk" the lemma follows.

If ~*¢ 4 Ire'I= Iml

(Taken directly from [8].)

S*, then ~' ~r S

and ~' c

n for each such ~'.

f o r each ~' such t h a t

0J*. Hence by the definition of S we have

c (~') < . -i

f

For each

I~'1

where the sum i s taken o v e r a l l p a i r s

PROOF OF (6.20).

so

~

choices of dyadic lengths I~l.

< bk I0

bk6

Gk-

~' oF(X)dx < yp

i 'm Consequently,

C*(w*)< n

yP.

66

or

y > b,/(p_l)l K

Then

( p / 2 - 1 ) < O;

So that bkY < yP Suppose

1 < p < 2.

so t h a t

(p-2)

(p~-l)

y

< bk

-

/2 (p-l)

=

(2-p)/

bkL 1

/2L I (p-l)

Suppose we can choose t h e ( l a r g e ) p o s i t i v e

(2-p)

such t h a t

1

£

~.

Then we have

integer y

P/2

2Ll{P_l}

2 ~ p

Suppose

L I = LI(p)

-1/4

< b - kL I

y

•

We know

<

C* (to*) < 1 n

so t h a t

bkY <. 1

hence

Y $ bk 1 y

(p/23 -1

Since
-(p-2)/2

p/2

-(p-2)/2L2

b

(large) positive

integer

L2 = L2(p)

-1/4

~ b k L2

PROOF OF 6.31.

y .

Let

L = L(p) = LI(p) + L2(P)

(Taken directly from [9].)

By (B.2)we have

[En(t)I

=

cn.R-~(to.]u )[ < Const. Cn[to -

jP

But (6.21) i m p l i e s

I E n ( t ) l ~ Const.

bk_lY;

PROOF OF 6.32. Immediate by (A. 3) Part I and (B.8). PROOF OF (6.33). We write

such that

~ 1/4

(p-2) 2 L2 y

=

we have

kL 2

Suppose we choose t h e

Then

((P/2) - 1) > 0

(Taken directly from [9]).

t ¢ ~* .

] (c°jlj)

67 -int

I~*-~o* (xl o

. o (t)

X F

=

at

H (X) + R (x) n n

x-t

where En ( t )

Hn (x) = [ ; w -~

dt

x-t

(xl ~-in~ F(t)

Rn (x) = f~o*-o~* (x) o

En(t)

H (xI is majorized by 2 H*(x), where n

H*(x) is the maximal

n

Hilbert transform of

dt

x-t

E (t)

n

over m*.

n

Denote by 6 J t h e l e n g t h o f ~.j f o r each certain

m.ea(p*;k).

j

subset

~

wj and by

t j the m i d p o i n t o f

o - ~*(x) i s t h e union o f a

(x) c f t h e i n t e r v a l s

the n u m e r a t o r in the i n t e g r a n d

~.~(p*;k). Using the f a c t t h a t J o f R (x) has v a n i s h i n g i n t e g r a l o v e r each n

m., we write J (t-tj)

Rn (xl = (x)Z [~j

-int e

,2

.~ F (t)

at

(x-t) (x-t.l J

l -

Z

(x)

(t-t) ~J

(x-t)

j ........

E (t)

(x-tj)

n

at

(6.301 and (6.311 imply t h a t t h e second term on t h e r i g h t dominated by const,

bk_lY •

~ (x)

above i s

8j -t )2+~ J

To

obtain this same estimate for the first term on the right above we introduce the function

68 (t) =

(t-t~) . . . . (x-t) (x-tj)

Note that

E-i(n-2

v )t n[o~j] 1~jl

(i + 2 )

I~p] -< Const.

t e ~. 3,

(6j/{x_tj)2 + ~2.)).3 (The last

inequality follows from the fact that It-xl ~ In-2v n[~j][ <.

2v '

I

t-t,j

1 (x)Z

= 2~.2

x ~ ~.] so ~ e C 2 (~j). According to (B.I) we write

#(t) = E~p exp { -i2v.3-1ut}, where

-v

, t ~ ~j,

~j/2

for

t e mj

and

) With appropriate substitution this yields

E-int yo (t)dtl < Const /'F wj (x-t) (x-t.)3

Z

6 2j (~j) c (x) (x-tj)2+~ 23. n[~j]

Const. bk_lY

E (x)

62 . ] (x-tj)2+~

By adding (positive) terms of A (x) corresponding to the remaining intervals of

~(p*;k)

we obtain

IRn(X) l ~ Const.

b

k-I

A (X).

PROOF OF 6.36 Immediate by (A. 3) part 2. PROOF OF 6.40.

(Taken directly from [9].)

We first investigate the polynomial Pk(X;~) under the condition (n,~) ~ Gk n > 0 (and 4Xk). We write Pk(X;CO) = where

Qo(X;CO) + Ql(X;00)

Ql(X,~) contains the terms a ci~x

of

Pk(X;m) for which

69 -I0

In - x[~]I >

bk

If ~ and

~' are exponents of terms of Qo(X;m), then "B k It implies

< b ki0

This and (6.6) can be used to show

Qo(X;~ ) = pEikx + Ofo8 yp/2), where

o is constant and k If

(n

~*) $

G*

'

x

near

is an exponent of Qo(X;~).

and

~*~

k

X*

we can write

k

Pk(X; ~') = p ' e ik'x + O(b~ yp/2) + Ql(X;m,) for each of the four intervals a' ~ ~*, 41a' I: I~*I. For our ,

purposes, the essential part of Pk(X;m') is the term p G~ and

If (n,~*) ~

~*¢

X~UY~, then the polynomials

i~'x

Qo(X;~')

corresponding to the four subintervals of L0*, 41m' I= I~*I, are To see this suppose Ix[~ ' ] o

Since

- nl

< bklO

(n'~o) ~ ~k'

Pk(X;~o) contains a term a eikx Then ( ~ [ ~ ] , ~ )

e Gk

with

f o r some ~ ~ ' .

"Ak" implies I~I > bkl01% I

identical.

o

zf

does not contain each ~' ' then ~' would be contained in Y* k" Hence, ~ ' Pk (x;m').

for each of the four ~', so a e Hence the terms p

ikx

is a term of each

ikx may be c h o s e n t h e same f o r each o f t h e

four intervals of ~*. Now let

~' o

be the subinterval of ~* o

for which

Cno [mo] (~) : C*(P*)o and ~' be any interval ~ ' ¢ mO'* 41 'I : Let

P

and P be t h e c o r r s p o n d i n g ( k L ) - p o l y n o m i a l s . o We have (I/I~'I I ~ ' l ~ -Pl2dx)I/2 ( 1 --2- p/2 < + DkL)Y

and l C m ( ~ ' F' X, '

- P) I < bke yp/2

and the estimate

yp/2 -< b-i/4 kL y

for all m.

then yield

-I0 (B.3) (with M = bkL )

70

(B.9)

C n (~'"' ~ F - P)~

In p a r t i c u l a r , (B.10)

with

Cno[w~](~o;

bl/2 kL Y

for all

n = n o [ ~ ] , ~' =~ ~, 1/2 Vo ) ~ ~ k - bkL ) y

If every exponent k

of

Po

satisfied

n. t h e above e q u a t i o n y i e l d s

[l[~]

- no[~]]

-> b-Ske' we

would have ,. Cno [~o](~o,Po) a contradiction

< Const ~

to (B.10).

contains an exponent k

°

~ 3 yp/2 < b 2 b L Elan] < Const. bkL kL y' ~

It follows that

with

Po

(and hence each

-5 Ik[~'] - n o [ ~ ] I < bke

Set

P)

n =

O

for such a ~ . Take

~'

~ ~ (n[~'],~') e GkL

Then

for some

with three of its neighbors to form m* ~ ~o*,

middle half.

4"2~ • n[~*] :

]

~' ~

~*

with

] and

~*o x

in it's

e G~L.

r~

Since where

Po ~ G* kL'

we can write

p = p i n x + Q,o(X) + Ql(X)

QI (x) contains only exponents

-i0 I~'[~o] - no[~']lo -> bkL

and

k'

of

Q,o(X) = O(b k? e

Cn[~ , ] (to,,. p - 0ei n x) = O(bke 7 y) for

P

with

y) .

Hence

In[~']-no[m']]

-9 < bee

(B.9) and (B.10) can then be replaced by

(B. ii)

cn[~'](~'; ~F - 0elnx) < Const

and (B.12)

Cn

, (w~; 0c

.% inN)

> Const.

• 1/2

DkL

-9

Y, ln[~']-no[~']l

< bkL ,

bkY

o[~o] (B. II)

with

n = n

(B.13)

[Pl ~ c o n s t

yields C ~ [ ~ , ] ( ~ ' p c znx) ~ c o n s t

1/2 (c* (p*) + bkc y )

71

In particular,

IPl

const, y

<

Ipl

const bkY <

From

(B.12)

we have

Ino[%] - ~[%]1-1

const

,

or

I~[~] - no[~] I -< C o n s t Suppose n s a t i s f i e s

b-lk

Y-I I~I _< A

b-lk

In[~ o] - no[%] I < 2 A bk2

i n x S * ( x ; ~*; I F ) n o

Write

- e i n o x S* (x;. *" • F) I no eo ' %

•~ I ¢ i n x S*(x" ~*; X - pc l n x ) n ' o F

<

inx

Is*(x; n

+

o

)1

- einox~

o i~x)

(X; o;p

+

(x;COo;~f F- pE

inx

)l

I

o

According to (B.7) and (B.13), each of the last two terms are majorized by const IPl <" const {C*(P*) o

+

-hkI/2 L

Y}

The

first term is estimated by using (B.II) and applying (B.6) -

2 A bk2

times.

The resulting bound is S Const 2

-2 1/2 A b k bkL y < bk_lY'

Combining results we obtain the desired estimate. PROOF OF 6.41. Let Z

(i)

(Taken directly from [9].)

denote the collection of all triples * ~* ~ ~o'

x belongs

4-2~.n[~o*] =

to the middle half

(n,~*, ~ ) , where o f ~*

and

nl~*l; rb

(ii)

1 .< ~ ~

(iii)

(iv)

k

and

In[~o] - no[~o] I < A

k Z j=l

-1 b

;

and

J

(n[~*],~*) E G* ~L

We must show that Z (A)

C* (Po) < ~ -1 y ;

If

is nonempty.

C*(P*) < bk_lY ,

then

(n, ~*, k) ¢ Z . (n and ~* are as in (6.40)).

O

(A')

If

C*(P*)o "> bk_lY'

%

we definel ,

1 _< Jt < k, by

bL y _< C*(P*)o <~b-ly"

72 ~

(a)

If

P* e G* , o Z L

(a') If

P* ~ G* o £L

n'

Z.

and

then

~

( n ' ~ * £) e E, o'

~

we apply (6.40) with n and ~ ~ o

We obtain a new pair

nl,m [

--I

n' = 4.2z n[w*]im* I o o k replaced by ~

such that (nl,m[,£) ~ Z.

Note that

Inl[~o]

n^[~*]l~ o

-< Inl[mo]ru

<

ru

In'[~o]

no[mo]l

'I,

no[~o]l

+ In'[~*]o

k .~ b -I j=g j

A

-

Choose

~b

+

Inl[
<_

This proves Z

n'[mo]l

is nonempty. (n,~*, m) e Z such that

m is minimal.

of our lemma are clear, except for the inequality and the statements concerning Let us assume by

If

-

C* (~*) < bin_1 y

~({*; m).

C*(~*) > bm_lY.

b£y < C*(p*) < b _ly.

The conclusions

We then define £, 1 < Z <

p* e G*

zL,

then

(n,~*,£) e

a contradiction to the fact that m is minimal.

If

m,

,

~* ~ G*ZL,

m

we apply

(6.40) with ~o' no' k replaced by ~*, n, Z.

obtain a new pair

nl, w I

contradiction since Z <

such that

m.

(nl,~[, Z) e Z, a

It follows that

We can now assert that the partition Let ~*(x)

and ~* (x).

Since

2-2~2 -N, SO

b Z y <. C({*(x)) < b£_ly (~,~*(x),£) e Z

'

Note that

x ~ W*

or ~*(x) C m* (strictly). o l~*(x) l >

C*(~*) < bin_1

y.

~(p*; m) is defined.

correspond to this partition and then set

~*(x) = (~[~*(x)], ~*(x)). ~* o

We

"

x

is in the middle half of

this implies ~* (x) ~

Suppose

~*(x) ~

C* (~* (x) ).> bm_lY, for some

a contradiction.

Then

(see 6.29).

1 <- Z < m. If

~*. o

If

~*(x) ~ G*

~* o

This implies

~*(x) e G* £L' then

£L'

we can use (6.40)

73

as before t o o b t a i n Since

a contradiction.

Hence

~*(x) is a union of intervals

"~*(x) C ~* o

o f ~(~*;m)

(strictly). it

remains

only to prove that

w * is a union of intervals of ~({*;m). This o f o l l o w s from t h e f a c t t h a t "~*(x) C ~* ( s t r i c t l y ) and by t h e o construction

of "~* ( x ) .

PROOF OF 6 . 4 2 . T h i s f o l l o w s i m m e d i a t e l y from t h e f a c t 1 + (v-3) 2 2

that

c o n s t which i s i n d e p e n d e n t

of

v.

(1.v) PROOF OF 6 . 4 3 .

This follows immediately from (B.5). PROOF OF ( 6 . 4 7 ) .

(Taken d i r e c t l y

from [ 9 ] . )

Suppose n o t .

Then we have ICm( 10) l < b k ~ -i0 for all m such that Im-n[ < bkL Since [ %O.x. F ( )]2dx

-i0 (with M = bkL ) bl/2 Cn(Wl0) < - k L

) 1/2 <

and t h e e s t i m a t e

Y < bkY'

y

P/2 ,

P/2

we can u s e

(B.3)

yp/2 ~ b~[/4 Y to obtain

a contradiction.

PROOF OF ( 6 . i S ) ,. If ~ < n - o i s an i n t e g e r 2

-2 A bk < o

2

-1 A bk o

Hence

we h a v e

no[~]

<

Suppose n o < ~.

then trivial.

n

o 2v+1

2v+l

we have

bk o

-

2

no I 2v+l -2 A bk < o

<2 n

Then since

-i A bko

-i 2 A bko

But since

o ; so that 2v+l

0

O ~

..I-

A

0

I~°

I~ .

~° OJ

0"

0

~I °

~I °

i

4~

APPENDIX

(C.O)

C. THE RESULTS OF KAHANE AND KATZNELSON

We denote by R the additive group of real numbers and by Z

the s u b g r o ~ consisting of the integers. the ~ o t i e n t

The group T is defined as

R/2~Z where, as indicated by the notation, 2~Z is the

g r o ~ of the integral multiples of 2~.

There is an obvious

identification between functions on T and 2~- periodic functions on R, which allows an i~licit introduction of notions such as continuity, differenti~ility, etc. for functions on T. T also c ~

Re

Lebesgue measure on

be defined by means of the preceding identification:

a

function f is integr~le on T if the corresponding 2~-periodic ~nction, which we denote again by f, is integr~le on [0,2~] and we set

f(t)dt =

I

f(x)dx.

O

In other words, we consider the inte~al

[O,2~] as a model for T and

the Lebesgue measure dt on T is the restriction of the Lebesgue measure of

R to [O,2~].

of our ~ u l a s

~e

total mass of dt on T is equal to 2~ ~ d

many

would be s i ~ l e r if we normalized dt to have total mass

i, that is, if we replace it by dx/2~.

T ~ i n g intervals on R as

"models" for T is very convenient, however, and we choose to put dt - dx in order to avoid con~sion. factor

We " p ~ " by having to write the

i/2~ in front of e v e ~ integral. all-i~ortant property of dt on T is its tr~slation

invari~ce, t h ~

is, ~ r

all t ~ T ~ d O

I f(t- to)dt =

If(t)dt

f defined on T,

76 1 We denote by L (T) the space of all complex-valued,

1

integrable functions on T.

llf}}L 1

=

For feL (T)

27! ....... IT

It is well known that

Lebesgue

we put

If(t)Idt'

LI(T), with the norm so defined, is a

Banach space. (C.I)

We consider a homogeneous DEFINITION:

A set

EC

T

Banach space B

on

T.

is a set of divergence for B if there

exists an f e B whose Fourier series diverges at every point of (C.2)

DEFINITION:

I f e L (T)

For

E.

we put

S*(f,t) : n

sup mE n

ISm[f,t)I

S*(f,t) =

sup

ISn[f,t)l

(c. 1) n

THEOREM:

E is a set of divergence for B if, and only if, there exists

an element f e B such that (c.2)

S*(f,t) = ~

for

t ~ E.

The theorem is an easy consequence of the following: LEMMA:

Let

g c B;

then there exist an element f ~ B and a positive

even sequence

{~ } such that ~. ÷ ~ monotonically J J and f(j) = ~.~(j) for all j e Z, where ~(j) J A

with

j

denotes the j

th

complex Fourier coefficient for f,

77

For e a c h n l e t

PROOF OF THE LEMMA:

II°X(n)

IB

X(n) be s u c h t h a t OO

We write

X

f = g +

(g)).

(g -

n=l 7he s e r i e s f(j)

=

defining

2jg(j)

f converges

where

2.--i 3

+

PROOF OF THE THEOREM: for the divergence of Z

by the

1emma.

n=l

h e n c e feB.

Also

[jl/(k(n)

min(1,

+ 1)).

The condition (c.2) is clearly sufficient

~(j)e ijt

other hand, that for some Let f e b and { 2 j } be t h e

i n norm;

~ (n)

for all

tsE.

Assume, on the

gsB, Z~(j)e ijt diverges at every point of E.

function

We c l a i m t h a t

and t h e s e q u e n c e c o r r e s p o n d i n g

(c.2) holds

for

f.

This

follows

to g from:

n > m,

n

S (g,t) n

- S m ( g , t ) = Z (S ( f , t ) m+l J

- S j-1

( f , t ) ) 2 -1 j

= s (f,t)2~ 1 - s_(f,t)2~l I + n m

(c. 3)

n-I -I m+iZ (2 3"

-i 2 j + l ) Sj(f,t),

hence

ISn(g,t ] - Sm(g,t) l.< 2S*(f,t) It

follows

and

that

J

S*(f,t)

<

~ , the Fourier

series

of g converges

t ~ E. Remark:

~.

if

-1

~m+l

:

t&E,

o(n.), J

Let {~ } be a sequence of positive numbers such that n -I -I Z 1 (2. - 2j+l) ~ < ~; then, for all

Sj (f,t) # o(~j).

j

This follows immediately

from (c.S). (C.3)

For the sake of simplicity we assume throughout the rest of this

section that

78

If

(c.4) LEI~4A :

feb

and

neZ

Assume ( c . 4 ) ;

eintfeB and l l e i n t f l l

then

then E is

a set

B =

of divergence

%IflIB

for

B

if, and only if, there exists a sequence of trigonometric polynomials

PjeB

such that

(c.5)

sup j

let v. 3

S * ( P t) J,

~

and

(c.5).

and

on

E.

of a sequence

{P.} s a t i s f y i n g 3 t h e d e g r e e o f Pj and

D e n o t e b y mj

be integers satisfying

vj > v j_ 1 Put

~

= =

Assume t h e e x i s t e n c e

PROOF:

Z ||Pj [[" B <

Z I[Pj[lB <

f(t)

= Z e

iv.t J P.(t). J

+

For

m.j_l

+

n < m. J

Svj+n (f't) - S v

m..3

we h a v e

(f,t) = e

iv .t J % (Pj,t) ;

j -n-i hence Z

~(j)e ijt

Conversely, By r e m a r k function teE.

(c.2)

diverges on assume t h a t

there

exists

feb such that

We now p i c k

E. E

a monotonic

S (f,t) n

a sequence

is a set

>

~n

of divergence sequence

infinitely

of integers

J and t h e n

integers

(c.7)

and w r i t e

uj

such that

~ . > 2 sup S* (ax . ( f ) , t ) ~J t 3 P. = V *(f J Vj+l

where * denotes

- ~k.(f)) j

convolution

and w h e r e

~

often

{Xj} s u c h t h a t

i l f - g x . ( f ) l ~ B < 2-~

(c.6)

~n

for

V

denotes

~

B. and

for every

a

79

de la Vallee Poussin's kernel

(2K2~+l(t) - K~(t)} where

It follows immediately from (c.6) that Z ~ Pj ~ B

Fejer's kernel.

If t~E and n is an integer such that [Sn(f,t) I > j,

~j < n <

k~(t) is

~j+l

~n'

<

then for some

and

Sn(P'J,t) = S n ( f - o k.(f), t) = Sn(f,t) - Sn(ol.(f),t). 3 3 1

Hence, by (c.7),

.]Sn(P.,t)13 > ~-mn' and (c.5) follows.

Theorem:

(c.4). Let

Assume

E. be sets of divergence for 3 E =U E. is a set of divergence for j=l 3

B,

Oo

j = 1,2 .... ;

then

B.

J Let (Pn } be the sequence of polynomials corresponding to Ej.

PROOF:

Omitting a finite number of terms does not change (c.5), but permits us to assume

Z j,n[[PJ[l B <

set of divergence for (C.4)

~

which shows, by the lerrana, that E is a

B.

We turn now to examine the sets of divergence for

B = C(T), the

space of all continuous 2~-periodic functions. Lenuma: Let

E be a union of a finite number of intervals on T;

denote the measure of

E

by 6 .

There exists a trigonometric polynomial

such t h a t

S*(~,t) > ~

(c. 8)

log

36

on

E

80

PROOF: {z;

It is convenient to identify T with the unit circumference

Izl = I}.

the function

Let

I

be a (small) interval on T, I = (e It,

~I = (I + ~- ze

-ito)-I

has a positive real part throughout

the unit disc, its real part is larger than I/3e on

I, and its value at

-i the origin

(z = O)

is

consider the

N

(i + c)

We now write

small intervals of equal length

It - tol ~ e};

2~ such that

EC

Ne < 6

~i

I , the I. being j J

, and

function (z) :

1 + E N

IVI.(X). j

has the following properties:

Re(~(x)) (c.9)

> 0

Izl

fo r

< 1

~ (0) = I

I P (Z)l > Re(~(z))

> i > 3 NE

I 3~

on

E.

The function log ~ which takes the value zero at

in a neighborhood of

z = 0 is holomorphic

{z; Izl ~ 1} and has the properties

llm(log ~(z)) I < ~

on

T

on

E.

(c. 10)

flog V (z)) > log(3~) -1

Since the Taylor series of log P converges uniformly on T, we can take a partial sum valid for (t)

~ =

M n ~(z) = E 1 anZ

in place of log ~ . 1 e-iMtlm(~ ~

and notice that

(eit

)) =

of that series such that

(3.10) is

We can now put

12~i

e_iM t ( M Z 1

a e int_~ a e -int n

1

n

)

81

Theorem:

Every set of measure zero is a set of divergence for C(T). If

PROOF : union

U

E

is a set of measure zero, it can be covered by a

the

In,

I n

being intervals of length ..IInl such

and such that every

IInl < i I 's, n

Grouping finite sets of intervals we can cover

U E such that every E n n n such that IEnl < e -2 Let ~ n E = En

and

S*(Pn,t ) > 2n-i/2~n2 many E ' s n

Pn = n

put

on

En.

that

t e E belongs to infinitely many

often by

for

(eit)I

i = -~I~

ISM( ~ ,t)

2 @n ,

E

infinitely

is a finite union of intervals and be a polynomial satisfying We clearly have " Z

IIPnll

<

(c.8) ~

and

Since every t~E belongs to infinitely

our theorem follows from lemma (c.3).

(C.5) Theorem: the condition

Let (c.4).

B

be a homogeneous Banach space on T satisfying

Assume

B~C(T);

then either T is a set of

divergence for B or the sets of divergence for B are precisely the sets of measure zero. PROOF:

By theorem (c.4) it is clear that every set of measure zero

is a set of divergence for B.

All that we have to show in order to

complete the proof is that, if some set of positive measure is a set of divergence for B, then T is a set of divergence for B.

82

Assume that For

E

is a set of divergence of positive measure.

~ ¢ T denote by E

set of d i v e r g e n c e multiples

the translate of E by

for

B.

o f 2~ a n d p u t

Let

{an } b e t h e

E = I~E

~;

Ee is clearly a

sequence

of all

By t h e o r e m c . 2

rational

E is

a set

of

n divergence, to prove

a n d we c l a i m

that,

we d e n o t e

that

T-~ is

by ~

the

a set

of measure

characteristic

zero.

function

In order

o f E and

notice t h a t ~(t

- an)

=~(t)

for

all

t

and

a . n

T h i s means ^

~(j)e

-ia j . . n eiJt

A ijt Z ~ (j)e

=

J

J or

A = ~ (j)

-i~nJ %

(all

~n )

(j)e A

If

j #

O, this implies

everywhere that

the

almost

and,

since

measure of

all

o f T.

~ (j) = O;

,~ is E

Now

is

hence ~(t) = constant almost

a characteristic either

T -- E

is

function,

z e r o o r 2~ . a set

this

Since

of divergence

implies

E mE, ~eing

E is of measure

%

zero)

and

E

is

a set

(C.6) Thus, for spaces and in particular for

of divergence,

B

hence T is

of divergence.

satisfying the conditions of theorem c.5,

B = LP(T),

either there exists a function

1 ~ p

<

~

,

or

B = C(T),

f~B whose Fourier series diverges

everywhere, or the Fourier series of every everywhere.

a set

f~ B

converges almost

83

Theorem:

There exists

PROOF:

a Fourier series

For a r b i t r a r y

m e a s u r e ~< o f t o t a l

diverging

K > O we s h a l l

everywhere.

describe

mass one h a v i n g t h e p r o p e r t y

a positive

that

for almost all

t eT. (c. ll)

S*(~K, t) =

sup IS (~K,t)) > n

n

Assume f o r t h e moment t h a t

such u

exist;

< it

f o l l o w s from

K

(c.ll)

that

there

exists

an i n t e g e r

Lebesgue) measure greater

than

N< and a s e t

1 - 1/<,

(c.12)

sup I S n ( V K , t ) I > K n
If we write now ¢

=

UK*

VN< (VNK

E<

such t h a t

of

for

(normalized t e EK

being de la Vallee Poussin's

kernel), then ¢< is a trigonometric polynomial,II¢
> sup n
ISn(¢ ,t) 1 =

-

sup I S n ( ~ < , t ) I > n
A p p i y i n g t h e lemma c . 3 w i t h

pj = j - 2 ¢ 2 j

E =~ m U

of

almost

m fijE2J

all

T,

is a set

divergence

we o b t a i n

for

K on

EK

that

L1 (T).

Since

E

is

K o l m o g o r o v ' s t h e o r e m would f o l l o w from t h e o r e m c . 5 .

The description of the measures u however, for the proof that

(c.ll)

<

is very simple;

holds for almost all t e T,

we shall need the following very important theorem of Kronecker. Theorem XI~

...~

(Kronecker):

Let

Xl, ...,

xN

be real numbers such that

XN,~ are linearly independent o~er the field of

84

rational

numbers,

there exists

Let ~ >

an integer

n

and

such

J - e

We construct

}I <

N

be r e a l numbers, then

that

¢ ,

j = 1 ......

now the measures

integer,

let

x I .....

XN, ~

such that

Ix. - (2Zj/N) l < I/N 2 , 3

1/N Z6

1'

i~.

inx. le

O

x., J

j = i, .... ,

are linearly

~

as follows:

K

N

N.

let

be real numbers

independent

N be an

such that

over the rationals

and let ~

and

be the measure

X.

J For

t ¢ T

S (V,t) n

For almost

we have

=

Jf

Dn(t - x)d~(x)

all t ~ T,

1 N

N Z

= 1

N

N

~

the numbers

are linearly

independent

there exist,

for each such t,

e

~

=

1

D n (t

( ,x)l sin

j

(t-x )

1. . . . . . .

...,

t - Xn,~

By Kronecker's such

< } ,

j = i, ...,

1 sin(n+-)(t-x ) .... 2 )

1 sin ~(t-x.) 3 It follows

that

>

1 i ~

sin

t-x" 1 ) -i

2

theorem

that

2 hence

i

sin ~ (t-xj)

over the rationals.

xj) _ i sgn

:

",,,

1

n

xj)

sin(n + ~)

t - Xl,

integers

-

for all

j.

N ;

85

(c.13)

Sn(la,t) > " ~

j=~l

I sin

and since the x.'s are so close to the roots of unity of order N, J the sum in (c.13) is bounded below by

1 g

Isin t / 2 1 - 1 d t

> log N > <

, p r o v i d e d we t a k e

N large

1/N

enough. (C.7)

In [4]

Y.M. Chen modifies the classical construction of

Kolmogorov's function that is given in [i] section 17 in Vol. 1 to obtain a function of the class

L(log + log +

L)I-¢ for any ¢

>

0

whose Fourier series diverges almost everywhere.

Also, a l l the m a t e r i a l

in t h i s appendix has been t a k e n v e r b a t i m

with the author's permission from [ii].

BIBLIOGRAPHY i.

Bary, N.A.

Treatise on Trigonometric Series, Vols. 1 and 2.

Pergamon Press, Inc., New York (1964). 2.

Carleson, n.

O__nnconvergence and growth of partial sums of

Fourier series. Acta Math. 3.

Carleson, L.

ll6 (1966), 135-157.

Sur l aconvergence et l'ordr__~e d_~egrandeur des sommes

partielles des series de Fourier. 4.

Chen, Y.M. class

5.

(unpublished Marseille notes)

An almost everywhere divergent Fourier series of the

L(log+log+L)l" ~.

Edwards, R.E.

J. London Math. Soc., 44 (1969), 643-654.

Fourier Series a Modern Introduction, Vols. 1 and 2.

Holt, Rinehart and Winston, Inc., New York (1967). 6.

Halmos, P.R.

Measure Theory. Van Nostrand, New York, (19SO).

7.

Hewitt, E. and Stromberg, K. Springer-Verlag,

8.

Hunt, R.A.

Real and Abstract Analysis,

Berlin (1965).

On the convergence of Fourier series.

(unpublished

Chicago notes) 9.

Hunt, R.A.

On the convergence of Fourier series.

Expansions and Their Continuous Analogues Edwardsville,

Ill.

Press, Carbondale, 10.

Hunt, R.A.

(1967))

pp. 235-255.

Orthogonal

(Proc. Conf. Southern Illinois Univ.

Ill., (1968).

On L(y,q)

spaces,

Enseignment Math. 12 (1966),

249-276. ll.

Katznelson, Y.

An Introduction to H a r m o n i c ~ .

John Wiley and Sons, New York (1968). 12.

M~tt~, A. functions.

The con yergence of Fourier series of square integrable Matematikai Lapok 18 (1967), 195-242.

(in Hungarian)

87

13.

Rudin, W.

Real and Complex Analysis,

14.

Stein, E.M. and Weiss, G.

McGraw-Hill New York (1966).

An extension of a theorem of

Marcinikiewicz and some of its applications.

J. Math. Mech. 8

(1959), 263-284. 15.

Titchmarsh, E.C.

An Introduction to the Theory of Fourier Integrals.

Oxford University Press, New York (1948). 16.

Zygmund, A. Trigonometric Series,

Vols. 1 and 2.

Cambridge University Press, New York (1959).