Lecture Notes in Mathematics Edited by A. Dold and B. Eckmann
911 Ole G. Jersboe Leif Mejlbro
The CarlesonHunt Theorem on Fourier Series
SpringerVerlag Berlin Heidelberg New York 1982
Authors
Ole Greth Jersboe Leif Mejlbro Department of Mathematics, Technical University of Denmark DK2800 Lyngby, Denmark
AMS Subject Classifications (1980): 43A 50 ISBN 3540111980 SpringerVerlag Berlin Heidelberg New York ISBN 038?111980 SpringerVerlag New York Heidelberg Berlin This work is subject to copyright. All rights are reserved, whether the whole or part of the material is concerned, specifically those of translation, reprinting, reuse of illustrations, broadcasting, reproduction by photocopying machine or similar means, and storage in data banks. Under w 54 of the German Copyright Law where copies are made for other than private use. a fee is payable to "Verwertungsgesellschaft Wort", Munich. 9 by SpringerVerlag Berrin Heiderberg 1982" Printed in Germany Printing and binding: Beltz Offsetdruck, Hemsbach/Bergstr. t2141/3140543210
CONTENTS.
PREFACE
3
CHAPTER I
4
i.
Interpolation theorems.
2.
The HardyLittlewood maximal operator.
10
3.
The StelnWelss theorem.
15
4.
CarlesonHunt's theorem.
17
23
CHAPTER II
Py
and
Qy .
24
5.
The operators
6.
Existence of the Hilbert transform and estimates for the Hilbert transform and the maximal Hilbert transform.
7.
33
Exponential estimates for the Hilbert transform
40
and the maximal Hilbert transform.
45
CHAPTER III 8.
The dyadic intervals and the modified Hilbert transforms.
46
9.
Generalized Fourier coefficients.
51
lO.
The functions
S~(x;f;m*)
and the operator
M* .
69
CHAPTER IV II.
Construction of the sets
12.
Construction of the sets
13.
60
Gk
and
Estimates of the index set
Y*
S*
and
70
~L "
Pk(X;~)functions and the and
Pk(X;W)
~
and
74
X* .
and introduction of 80
G~ .
14.
Construction of the splitting
15.
Construction of the sets
16.
Estimation for elements
T*
~(p*,r) and
p* ~ G* rL "
V*
of
~* .
87
and
EN .
91 101
IV
17.
Final estimate of
e . o. Sn(X,XF,~_I)
18.
Proof of theorem 4.2.
9
II0 118
REFERENCES
122
INDEX
123
CHAPTER I.
This chapter is composed of four sections. In w 1 we introduce the concept of (weak and strong) type of an operator, and we prove an interpolation theorem, which is a special case of a theorem due to Marcinklewicz for the general formulation). mal operator
0
(ef. [9]
In w 2 we introduce the HardyLittlewood maxi
and prove that
@
is of type
p
for all
p ~ ] I,+~[
.
In w 3 another classical interpolation theorem is proved, namely the SteinWeiss theorem, and finally, in w 4 , we prove .the CarlesonHunt theorem under the assumption that some operator all
p~ ] i,+~[
M
defined below is of type
p
for
.
For technical reasons we shall always consider realvalued functions defined on a finite interval, although their Fourier expansions will be written by means of the complex exponential functions. This assumption will save us for a lot of trouble in the estimates in the following chapters, and we do not loose any generality, since for a complexvalued function we may consider the two realvalued functions shall further assume that
f
and more complicated, full generality.)
and
is integrable , f eLl(1)
finite interval mentioned above. this chapter also hold for
Re f
, where
I
is the
(We may note that most of the results in
f ~LI(R)
f
Im f instead. We
, but their proofs may be different
so we have avoided to prove the theorems in their
w i. InterDola_lion theorems, Let
f
be a realvalued function defined on an interval
pose that
f ~LI([A,A])
.
The Lebesguemeasure on
R
is denoted by
depending on the function (i.i)
Ey = {xE
understand the function
(1.2)
and sup
m . We introduce the sets
E
Y
under consideration by [A,A] I [f(x) l > y} , y e R+
By the distribution function
Definition I.i.
%f : R++ [0,2A]
~f(y) , y c R+ , we shall
defined by
~f(y) = m(Ey) = m({x~ [A,A] I [f(x) l > y})
Clearly, we have and
f
[A,A]
%f(y)
0 =y}) y}) y} < ( ~ ) P ~
IIf[l~
and
llTfll < A llfll
,
then IITfllp < p " ~ =
Proof.
9 A ~
9 PPo
llfll
9
We use the same notation as in the proof of theorem 1.9. We choose
the constant
A
as
~
. Then
IlfYll~ =< ~ y
, IITfYII~ =< y , and conse
.co
quently
;~Tfy(y) = 0 .
This gives
the estimate
~Tf (2y) ~ ~Tf (y) ~ A o Y
y_Pol
fy(X) Ip~ I dx ,
and (exactly as in the proof of theorem 1.9) 2P
=
~
pop
1 PPo
9
\A /
proving the theorem.
j
D
w 2. The HardyL~ttlewood maximal operatoK. In this section we shall consider the HardyLittlewood maximal operator and derive estimates for this operator using theorem i.i0. Let
f ~LI(R)
. We define the maximal operator
G
by
I Ix+t 8f(x) = sup ~~ If(y) Idy , x E R ts + ~xt
(2.1)
Clearly, the function
Of
is measurable for each
f e LI(R) , because
@f
is lower semicontinuous as a supremum of continuous functions, and the operator
0
is a sublinear operator, defined on
Theorem 2.1.
The operator
precisely,
satisfies the following estimates
O
0
is of type
+~
LI(R)
and of weak type
I . More
11
(2.~)
llofll. ~ llfll , 4
(~.s)
xoffy) = max IOf(x) >y}) 2 ~ llflll
for all
y e R+
From theorem 2.1 and theorem i.i0 we infer at once that the operator of type
p
for all
p ~ ]I, +~[
g
is
. In the applications later on we only need
functions of compact support~ so let us assume for the proof that
f
has
compact support. It is obvious that (2.2) is satisfied. In the proof of (2.3) we may as well assume that xs
I Of(t) >y}
f
is nonnegative. Let
there exists an interval
I
y s R+ . To each
(with center at
x
x ) such
that I
f(t)dt > y .m(l x) . I x
As
f
has compact support, the set
pose that all the intervals
I
x
{Of >y}
are contained in an interval
We claim the existence of a sequence intervals
I
x
is bounded, and we may sup
(In )
[B,B]
(extracted from the class of
above) of pairwise disjoint intervals, such that
m
In
U Ix
> [m
n I
x
Assume for the moment that this has been proved. Then the proof is finished as follows:
m({x I Of(x) >y}) __<m
4 =< Y
U Ix x
< 4m
4 I f(y)dy = ~ n
;
U In n=i
= 4 I m(l n) n= I
4
U I n n
f(y)dy < ~llfll I =
9
Let us prove the assertion above (this is a theorem of Besieovitchtype; results in that direction can be found in e.g. [3]) Let
SI
be the class of all intervals
a I = sup{m(l x) II x r S I} m(ll ) > 3 a I sect
. Let
I I , let
S2
I
considered above. Let x and choose an interval I I from S I with
be the class of intervals
a 2 = sup{m(l x) II x ~ S 2}
Ix
and choose
that do not inter12
from
S2
with
m(l 2) > ~ a 2 . In this way we continue. If the process stops after a finite number of steps, the result is obvious, so we may assume that we obtain a
12
sequence
(a k)
of real numbers
disjoint intervals
(Ik)
Let us consider an interval denote
the first
ak + 0
with I
and a corresponding
m(l k) + 0
for
and let
x
index for which
sequence of
k + +
k
I x ~ Sk ; Ik. I
in that case
Ix n l k _ I ~ @
i
and
m(Ik_ I) ~ ~ m ( l x)
. Then we have
where
Ik_ I
Jki
and
ter and where
i
i
i
i
Ix c J k _ l
have the same cen
m(J k i ) = 4m(l k i ) . Using , and the theorem is proved. It
s h o u l d be n o t e d t h a t
possible one.
Note that
the constant
4
appearing in (2.3) is not the best
D 9
is not of type i (if we e.g. let
e
f(x) = X]0,1[(x)
, then
e
is not even integrable')
Corollary 2.2.
The operator
e
is of type
p
for all
p ~ ]i, +
. More
precisely we have (2.4)
llefllPp __< 4p. 2p . ~
Iifiip
for all
p E ]1, +~[
o
As mentioned above this follows from theorem 2.1 using theorem i.I0 with Po =I
If
, A ~ =4
g(x)
and
A
is a realvalued
g+(x) = max{g(x),0}
Theorem 2.3.
(2.S)
=i
.
~ .
function, we define another function
. A wellknown example is
The operator
e
log +
satisfies
[ (of(x) 2)+~ ~= 8 [ if(x)llog+if(x)j~ J
J
g+(x)
by
13 Proof. We may of course assume that f ~ 0 and that f log+f grable. We define functions fY and r by Y fYtx~/ = ~ f(x)
L
0
0
if f(x) ~y fy(X) =
otherwise
is inte
if f(x)~y
f ( x ) otherwise
Then clearly Ofy ~ y and so m({x
[email protected](x) >2y}) ~ m({x JOfy(X) >y}) Using (2.3) on the function f we get Y (2.6) m({x 1Of(x) >2y}) ~ 4 ] fy(x)dx . An integration of (2.6) from I
to
+~
J 0f(x) >2y})dy _
i}
I +~ m({x jef(x) >t})dt = ~If (0f(x) 2)+dx . The left hand side equals 2S2 and thus the result follows.
Later on we shall also need the following two simple lemmata:
Lemma 2.4. Let F
x
on
R
by
f~LI(R)
. For any fixed
we define a function
Fx(t) = S~f(x+y)dy 9 Then
JFx(t)[ ~
[email protected](x)
Proof.
xER
for all
x~R
, a~l
t ~R
The len~na follows immediately from
irx+t (x+t [x+,t, JFx(t) J = Jx f(y)dy < Ix f(y) dy < JxJtJ Jf(y) Jdy ~ 21tJ "Of(x) .
Le~ma 2.5. If
feLl(R)
, then
if(x) I ~ t
If(y) Idy
for all
t9
Jxt and
[x+t
I 2t ~  t
If(y) Idy > If(x)]
where we have used Lebesgue's (cf. e.g. [3]).
for
t~0 +
differentiation theorem for
x 9R ,
Llfunctions
D
From lenm~a 2.5 and theorem 2.1 follows that if get
for almost every
f 9
nL=(R)
we even
[[fll= =
[email protected]~
Finally we shall prove an exponential estimate for f(x)
Of , when
f ~L ~
and
is zero outside a compact set.
Theorem 2.6.
Let
c 9 R+
be any positive constant and let
f
be any es
sentially bounded function, the support of which is contained in an interval
I
of length
A . Then for
y 9
(2.7)
XOf(y) = m({x Igf(x) >y}) y)
and
dist(x,l) = d > 0 , then an
application of (2.1) gives
y < Of(x) < ~
If(T) IdT __y}) j A + 2
Let
t
=
9
A llfIl~ (llfll~) Y A. i +  y 
~F77Fl Y e ]0,I[ . Then it is enough to prove that
15
1 1 i + ~ =< 2exp c. ~ 9 exp(ct)
,
tE ]0,I[
,
which is equivalent to the trivial estimate (l+t) exp(ct) ~ 2. expc and the theorem is proved.
for
t E ]0,I[
,
D
w 3, The SteinWeiss theorem. We shall now introduce an operator
T~
associated with the operator
T .
In the following we assume that all functions considered are defined on a fixed interval p , where
[A,A]
p e ]I, +=[
. Let
Let
q
E ~[A,A]
be the number conjugate to
A p ~ R+
such that
,
p , i.e.
. We define a set function
contained in
[A,A]
y
! + ! = 1 , and let P q on the class of Borel sets
by
(3.2)
y(E) = I (T•
H~Ider's inequality implies that that
operator of restricted type
IITXEIIp =< AplIXEII p = Ap[m(E)] I/p
f eLq([A,A]) E
linear
be a
, i.e. there exists a constant
for every measurable set (3.1)
T
X
y
is welldefined,
and it is easy to see
is a countably additive set function which is absolutely continu
ous with respect to Lebesgue measure. Using RadonNikodym's theorem we then get an (up to nullsets) uniquely determined function
(3.3)
~(E) = ~ h ( x ) d x
for all Borel sets
E S[A,A]
Because of this relation we
of
such that
= IXE. hdx = ITXE" fdx ,
.
define
(3.4) Clearly,
h
an operator
T*
on
Lq([A,A]
by
T*f = h . T*
is linear, and
T . We have e.g. that if
(3.5)
formally g
Tg 9 f dx = A
T~
behaves as the adjoint operator
is a simple function, then
Iig9 T * f d x
9
16
Lemma 3.1.
Assume
p ~ ]I, +~[
where
Proof.
that the linear operator
. Then
T~
is of restricted q , where
By assumption there exists a constant
(3.6)
type
P ,
~ + ~ = I . P q
A oR+ , such that P
][TXEH p =< ApI[XEI[p = Ap[m(E)] I/p
We shall prove the existence of a constant
(3.7)
XT, f(Y) ~ and let
[Ifll
Let
f EL q
E
(x I lh(x) l >y} 9 We put
Y
T
is of weak type
h=T*f
for all
and E
Bq~ R+ , such that
f
and all
y~R+
.
%(y) = %h(y) = m(Ey) , where as usual,
= E + uE , where Y Y Y
E + = (x lh(x) >y} , Y
E = {x [h(x) <  y } Y
Finally, we define X+(y) = m(E~) , Clearly,
A(y) = m(Ey) .
X(y) = X+(y) +X(y) , and for
%+(y)
we get the following esti
mate
Y
Y
ffi I TEE+y" f dx
Y
< IIT• p y
= Ap[m(E~)] I/p
tlfllq
where we have used the definition of
IIfllq ~ ApIIXE~IIp [IfI[q
= Aq[X+(y)] l(I/q)
B
= 2 I/q .A q
X+(y) = 0
X (y) , and hence
X(y) = X+(y) + X(y) ~ 2 which is (3.7) with
0 P
Hft[q
,
T* , H~Ider's inequality and (3.6).
From the inequality above we deduce that either [X+(y)] I/q ~ (Ap/y) 9 Ilf]lq We have a similar estimate for
Y
IlfI[
or
17
If the operator
Theorem 3.2. (SteinWeiss).
ed weak type
Po
p
p c ]po,Pl [ .
for every
Proof.
Let
Pl
p c ]po,Pl [
stricted type T*
and
p'
is of weak type
q'
for all
Tf = T**f T
for
T .
" then
T
T*
T** = (T*)*
" Then
T*
is of weak type
ql'
is a bounded linear operator on is bounded on
T**
T
is of re
are linear and
T**
and and
T*
Pl' qo' '
is of type
L q , which
L p . From (3.5) we conclude that
for all simple functions, i.e. on a dense set in and
is of type
Po'
q' r ]ql,qo [ . We choose
theorem (theorem 1.9) we infer that
q , and especially that implies that
i8 linear and of restrict
p' e ]po,Pl [ , and then lemma 3.1 tells us that
Po < p o < p Ek/P 1 9 Then
m(E , k) < ~p Ek
be the poly
20 Proof.
From lenm~a 4.1 we get
IIfPe,k[l~F < e
k
I) k
X}) ~ ~ IlfllI ,
(6.5) for all
X ~ R+
and all
f e LI(R)
with compact support (cf. a remark in
w 2 and definition 1.4). In general, (6.6)
f=f+f
, where
f+, f~ 0
Ifl =f+ +f 9 Furthermore,
m((x IH*f(x) >X}) ~ m({x I H*(f+)(x) >~}) + m({x ] H*(f)(x) >~}) ,
so we may in the following assume that Let
and
X ~ R+
be given. For each
e 9 R+
E + = {x I supNyf(X) >X}
c
Y~
[Note that even if
f
'
is nonnegative,
f>= 0 . we define the two sets E = {x I sup (Hy f(x)) > X}
s
Y~
H f need not be nonnegative; Y E + and prove that
cf. (5.1).] We shall only consider the set
:
m(E ) ~ ~~
f(t)dt , as the proof for
E~
is analogous.
35
For any finite interval of
I , and
Ic
Let us consider
I
of the real axis let
the complement
of
c(1)
denote the center
I .
the family of open intervals
I , for which
I f i ~ J I c f(t) 9 c(1)_~dt > X .
(6.7)
By the definition
E + these intervals I cover E + , and as E + is c c E (here we use that f has compact support) already a finite number
bounded
of
of these intervals cover
E + . Using the same reasoning of Besicovitch type E as in the proof of theorem 2.1 we can find disjoint intervals I I, . . . , I n such that (6.8)
m(E~)
n !im(lj)
~ 4 J
and
(6.9)
1 I&l~ f(t) 9 )~dt c(lj1 > X ,
j = 1,2 . . . . . n
.
J The function tends to
0
gx(t) = ~ as
I
Itl + + ~
is uniformly . Thus,
to any
continuous
for
~s ]0,i[
Ixtl ~ E , and it
we can find a decompo
sition of the real axis into a finite number of small intervals
J
infinite
and
each
intervals
such that for each of the small intervals
j =1,2, ... , n
~i I c. f(t).
(6.10)
J
and two for
,
d t  ~ ji
n .=@ J f(t)dt, c(lj)c(J)
J
< 6 9I .
J
We can even choose
the intervals J in such a way that for each I. either J j c I. or J A I. = @ . In the following we shall suppose that this has = j J been done. Due to the facts that f has compact support and gx(t) tends to
0
as
Itl >+~
vals mentioned
we shall never need to consider
the two infinite inter
above.
We define
g(x) = ~I
J f(t)dt 9 xc(J) I
and
gj (x) = ~I J
. = j
The function
J f(t)dt 9 xc(J)
36 g(x)  gj(x) ffi i
f(t)dt 9 xc(J) Jn
J
is clearly decreasing in the interval get for
x=c(lj)
I. , and from (6.9) and (6.10) we J
,
g(c(lj))  gj(c(lj)) > (I6)X , for
x
in the left half of
I.. J
so
g(x)  gj(x) > (I~)X
Thus we deduce that
n i .I ~m(lj) < m({x I g(x) > 89 ~=i =
+
n I m({x ] gj(x) < j=l

~(I~)}) 2
.
An application of len~na 6.1 then gives !llm(lj)< __ 2 11 j =~((T~6)J ~
2 ~i fJ f(t)dt+ j=l~~ X(I~) J I. = J
f(t)dt~}) < 2 c A 9 exp  I06~ = II proving theorem 7.1.
D
)
42 Theorem 7.2. f
There exist positive constants
cI
and
c2
such that, if
is any essentially bounded function, the support of which i8 contained
in an interval of length
A , then
(?.$)
m({x I H~f(x) >X}) ~ c I A~exp~ C2]T~T~~ ] .
Proof.
We remark that it suffices to prove (7.3) for large values of
We may assume that
f ~0
(write
f= f+f )
E + = {x I sup Hy f(x) > X}
and that
E~ = {x I sup (Hy f(x)) > X}
y~ where
mate
X >0 for
and
m(E )
X .
IlfIlm = I . Let
y~
g >0 . We shall give an estimate for
m(E~) , the e s t i 
being similar.
~s in the proof of theorem 6.2 we can find disjoint intervals
II, 12 , ..., In ,
such that n m(m~) X ,
j=l, 2 .... ,n .
J Let
fj(x) = f(x) Xl.(X) , j= i .... ,n , and consider the function
I[
f(t) dt xt
gj(x) = H f(x)  H fj(x) = ~'I~ J The function
gj
by (7.5), hence
is decreasing in gj(x) >X
lj
(because
in the left half of
f ~0 ) and
gj(c(lj)) >
lj . Thus we infer that
n
I
X ~m(lj) < m({x I Hf(x) > m({x I H fj(x) <  { } ) j=l = ~ }) + j=l From theorem 7.1 we deduce that n
2
12 J~I. = m(lj) 6} tha t
(8.8) As
IIH
llf~
H f(x) ~ [Nf~
f~
~ Cp Hf~
and
IIH* f~
l + H * f~ ~ c*I1f~ P
= 41/Pllfllp , we get from (8.8) for IIH flip
lIH f~
+ llH* f~
, say, and
p e ]I,+~[
,
~ 4 I/p (Cp + C;) llfllp
Finally, we shall introduce the modified maximal Hilbert transform
A H
sub
ordinated the dyadic intervals introduced above. Consider a given
m~j v
and let
x
be an interior point of
~ jv , x E intw~j v . r (Ox)r=~,~+l,.. " of inter
Then there exists a uniquely determined sequence vals, such that each
o r is a smoothing interval from level r , o~ c~* x belongs to the middle half of each o r . Note that x depends on x and that ~ ~v . A The modified maximal Hilbert transform H. with respect to ~J~ is dejv and such that
x
fined by (8.9)
where
Hj~ f(x) = sup [ I (pv) f~ r_>~ or xt x f ~LI(] 7, 7])
and where
fo
dt I
'
x ~ int m
,
as usual denotes the periodic exten
49 sion of
f
to
]4~, 4~] . We shall sometimes for short write (8.9) in the
form f(~)
=
I I sup l;(pv) o x
A H
We shall prove that
f~xt dtl q
also i s of t y p e
x
p
for every
normally is a skew interval with respect to
p E ] 1 , + ~ [ . As r x x , we shall first prove the
f o l l o w i n g lemma.
Lemma 8.2. f
to
Let
f eL1(]~, ~])
1 4~, 4~] . By
A=
]a, b[ ,  ~ <  a
by
~*
~
and let
~
be the periodic extension of
we shall denote the family of intervals I a , satisfying the condition ~ < ~ X}
l >~} U {x~
With trivial modifications
l + H* f~
+ 3@ f~
is contained in
[H*f~
> ~}
this is also true for
U {x~*
I ef~
> ~}.
w* W[l
9 Thus, if
f ~ L~
f o l l o w s i m m e d i a t e l y from theorem 2 . 6 ,
we also have an exponential
Theorem 8.4.
t h e o r e m 7 . 1 and t h e o r e m 7 . 2 t h a t ^ estimate for Hw, . In fact,
There exist positive constants
is any essentially bounded function and transform with respect to (s.13)
~
cI
and
c2
such that, if f
is the modified maximal Hilbert
~* , then
m ( { x e ~ * [~f(x) > x}) ! ci" m(~*) 9 T
" exp c a
A We note that m ( { x ~ * I H f(x) > X}) < m(m*) , and as the function i ~(t) = ~ e x p (  c 2 t ) , t E R + , is decreasing and ~(t) ~exp(e2t) for we can omit the factor
1
llfll.
t
1
in front of the exponential
the right hand side of (8.13), provided have the following corollary.
that
c I ~ exp c 2
t~l
function on
. Thus we also
,
51
There exist positive constants
Corollary 8.5.
f
is any essentially bounded function and
bert transform with respect to
The constonts
cI
and
c2
and
c2
such that, if
is the modified maximal Hil
~* , then
I~f(x)> X} ~
m({xe~*
cI
H
clm(~*)exp
c 2 9 ~
do not depend on the choice of
~* .
w 9 . Gene_ra_alized Fourier coefficients. In the following
~
will always denote a dyadic interval as introduced in
w 8, and
m.jv will denote one of the dyadic intervals of length 2~. 2 v contained in ]2~, 2~] . Similarly, ~* will denote a smoothing interval (including
m~i = ] 4~, 4~])
intervals of length For each
n eN
o
m~. jv
4~. 2 v .
[n~ 2 "~r
equal to ~jv
denotes the greatest nonnegative
is any smoothing interval from level
(9.2)
~*(n; ~ m* =m~l
Remark 9.1. ~(n; ~jv)
jv
integer less than or
v c No ' let
mk,v+l
n*[~v]
for
~*(n; ~;v)
n[~jv]
for
. This has been done in order to
For the same reason we have avoided the use of
the Lebesgue measure,
.
~*(n; m*l ) = n , which is consistent with (9.2).
Here we have changed the usual notation, which is and
be any
(8.4) . We define
n . ) = ~(n; ~k,v+l ) = [n * 2 vl] = [~~m(mjv)]
we define
avoid confusion.
nlml
'
n 9 2 v
of the four dyadic intervals satisfying
and
m. we define j.v
n m(~jv)] = in " 2v] ~(n; ~'~)3 = [ ,2~_
where
For
will denote one of the smoothing
and each dyadic interval
(9.1)
If
and
since it may be difficult
. In our notation they are written
I'I
for
to distinguish between
~(n; m)
and
n .m(m)
n[~]
. In
chapter IV they will both occur in formu]m, which are very much alike.
52
For an arbitrary function
f ELI(] ~, ~])
the periodic extension of
f
We define, for
e E R , e = e . 3~
Fourier coefficient
e =n E Z
we let (as before)
]4~, 4~]
and
ca(u;f) f~
We note that if
to
with period
f E LI(]  ~, ~]) , the
~'th generalized
exp ( i 2V~x)dx = ~
f~
exp
 im~/ax 9
we have the ordinary Fourier coefficients
of
f
~ ) , but in the following it will be neces
sary also to deal with the generalized coefficient, ~ = ~ , p EZ
denote
by
(with respect to the interval
case where
fo
2~ .
especially with the
e
We first remark that we of course have the estimate
(9.4)
lca(co;f) l 0
Iml + max
[0,2~]
Im"l} ,
[0,2w]
in the len~na is due to the
change of variable and the fact that we differentiate
Lemma 9.5.
I~"I
twice.
such that we ~or
D
n ~ NO and
have IOn.2_v(~;f) I < c 3 9 C~(n;~) (~;f) 9
Proof.
Let
8 =n~(n;m)
9 2 v = n_2VLn.2Vjrl
Using lemma 9.4 on the function t E ~ . jv
k0(t) = e iBt
. Then we have and
~=m.
jv
0 &~8 < 2
~
we get for
'
~ Z Y~ exp
~
~!Z ~p exp
with (l+~2) Iyp[ ~ c2~ {maxl~] + 2 2v maxl~"]} ~ c 2. {i+2 2v 82 } ~ c2{i+22v'2 2v} = 2e 2 9
57 Then we have _~(m;f) = ~ I
e
Ir176
I I eiBt exp[i~(n;m)2vt] 9 f~ (t)dt = m(m)
~EZ U ~(n;~)+~ and hence ICn.2_~(m;f)[ < ~ Iy~['Ic
:~EZ
~(n;~)+~
(w;f) l < 20c2.~0
=
~ Ic
p~Z ~(n;~)+~
(m;f) l. ~ I
I+p2
= 20c 2. C~(n;w) (~;f) proving the lemma.
D
Lemma 9.6. Let n E Z and feL2(~) and M ~ 2 be constants such that
be given where
~=~j~ . Let A, B c R +
I If(t)[2dt ~ A 2m(w) and [Cm(~;f) I < B
for
Inml < M .
Then we have
Cn(oJ;f)89
iCn+21 3
[
9
1 ~
I
~
I~[> y 12~;~nl
I+~2
. In w 4
S~(x;f;~*)
2 2~.2 ~

I
2m(P)
.
m(e)=2~.2 As this is true for all
~ eN
we finally get for
~§
that
O
I Ian(m) I2m(m) ~ 2m(F) . (n,m) E Gk
~ m(~) ~ 22k+lypm(F) (n,o) ~ Gk
Corollary 12.4.
Proof.
If
(n,~) ~G k , then
Jan(m) I ~ 2kyp/2
D
.
and so i ~ 22kyPlan(~)I2 .
Prom lenma 12.3 follows that m(m) ~ 22k 9 yP (n,~) E G k
~ lan(e) 12m(~) ~ 22k+l yPm(F) . (n,m) e Gk
We shall now introduce the exceptional sets Gk
we define for each
(12.9)
ken
a function
X*
Ak(X)
and on
lan(~) 12XJX)
Ak(X ) =
Y* . Using the sets ] 2~, 2~]
by
,
(n,m) E Gk and we define the set (12.10) If
Xk
by
X k = {xe ]2~, 2~] IAk(X) > 2kyp}
x e X k , one can find a finite subset lan(m) 12X~(X) > 2k y p
(n,~)
I of the index set Gk, such that
, and as the left hand side is a step func
~ I
tion, there exists a dyadic interval for all
~' such that x~ m' and Ak(Z)> 2ky p
z ~ m' . Hence we have proved the existence of a dyadic interval
79
m' , such that
xs
=c Xk,
so
is a union of dyadic intervals. Using
Xk
the notation from (ii.i) we define
(12.11)
x~ =
u g
and
X~ =
~X k
m(~)
Theorem 12.5.
Proof.
U ~. ~L k=l
o
210km(w')
(~s.?)
m(~)
It should be noted that we write Furthermore, if
~=~3~"
and
~/(~';c0) (= ['~(~'; m ' ) " m ~
~' =~s
at most
10k possible choices of the
m' . On the other hand, for given
i.e. for given
~'
and
2 9 210k i
w'
]) in (13.6)
it follows from (13.7) that
220k
(n,m) ~ G~~ , condition (13.10) can
I~'~"I" m(m) i~ < 210k
for any two exponents
~'
or
and
~"
occurring in
Q~(x;~) . But I~'~''I" re(m) 2~
I~ , m(m) _ ~,,. m(m) I
=
"
2~
f#
~ I~(~';~) ~(~";~)I
In~(%';m) l + In~(%'';m) I +I < 2 .210k+l
+ 1
< 220k ,
so the latter possibility is ruled out, and we get I~' X"I" m(m) i~< 210k
(13.14)
We may of course write I j=l where
il.x J
'
j=l
I ~i6 j = ~ 9 2v we get 6,
~
I00 .
belongs to the middle half
= 2 .2~ .2 ~ " ~ =< N  ]
and assume that there exists an
rEN
. Let
, such that
~ G*rL and , o 2  r y =mo>ml
> ...>mj
0,
0 < n j = ? . I f
nj ~0
, then
m(~) J
> 8 9 2~ 9 2  N
[ log(l+2 i) < [ 2 i = i , we have i=l i=l
N >= 7
, then the conditions
kj+ I < kj
and
so if
.
J)nj
imply for
8~
9
~*(nj,mj)" ~ ~ ~I , and as
=
120 9 2
o , then choose
~
according to
O
theorem 16.1 and
n ,~* ,r
according to theorem 16.2. E s p e c i a l l y ,
r ~k ~
Using theorem 15.7 we get (17.12)
S~(x;• I n ~
o
= i00 , we get , S~o (x;XF;mo)o
(17.14)
* o 1~0 .2
2k o
. Then
k n1 ~ ~ ~ (1+~
If
Proof.
n