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

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 § 54 of the German Copyright Law where copies are made for other than private use, a fee is payable to the publisher, the amount of the fee to be determined by agreement with the publisher. © by Springer-Verlag Berlin - Heidelberg 1971. Library of Congress Catalog Card Number 79-162399. Printed in Germany. Offsetdruck: Julius Beltz, Hemsbach

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 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)
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}. Y

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

1

q


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 _
"

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 (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] O.

15

Consequently,

iSnC×,f)I _
_o

Mf(x)
o

(-~,~)

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

(-~,~)

we have

for every n ~ O,

16

1/p (3.5)

Lemma.

For

1 < p
O, 1 < p
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 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 --

--

-~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

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

"
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
blO]~' I.

(X[~'],~') e

Gk:~ =

m', n > O,

In-X[m]l
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) [

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


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
y}

fE

L (~*) oo

x

]

Hilbert

~*:

x

Let

dt

;

y > O.

Then

I

45 Y -c B Part

re(T) 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
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
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
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
¢ 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)

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)

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


bm_lY.

b£y < C*(p*) < b _ly.

The conclusions

We then define £, 1 < Z


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 >