Lecture Notes in Mathematics Edited by A. Dold and B. Eckmann
1307 Takafumi Murai
A Real Variable Method for the Cauch...
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Lecture Notes in Mathematics Edited by A. Dold and B. Eckmann
1307 Takafumi Murai
A Real Variable Method for the Cauchy Transform, and Analytic Capacity
Springer-Verlag Berlin Heidelberg NewYork London Paris Tokyo
Author Takafumi Mural Department of Mathematics, College of General Education Nagoya University Nagoya, 464, Japan
Mathematics Subject Classification (t980): Primary 3 0 C 8 5 ; secondary 4 2 A 5 0 ISBN 3-540-19091-0 Springer-Verlag Berlin Heidelberg New York ISBN 0-387-19091-0 Springer-Verlag N e w York Berlin Heidelberg
This work is subject to copyright. All rights are reserved, whether the whole or part of the material is concerned, specifically the rights of translation, reprinting, re-use of illustrations, recitation, broadcasting, reproduction on microfilms or in other ways, and storage in data banks. Duplication of this publication or parts thereof is only permitted under the provisions of the German Copyright Law of September 9, 1965, in its version of June 24, 1985, and a copyright fee must always be paid. Violations fall under the prosecution act of the German Copyright Law. © Springer-Verlag Berlin Heidelberg 1988 Printed in Germany Printing and binding: Druckhaus Beltz, Hemsbach/Bergstr. 2146/3140-543210
PREFACE
The purpose of this lecture note is to study the Cauchy transform on curves and analytic capacity.
For a compact set
F
in the complex plane
denotes the Banach space of bounded analytic supremum norm
il-i!H~.
The analytic capacity of
IIflIH~_;
Cu~H'(rC),
u >= 0},
where C~(z) = ( 1 / 2 ~ i ) 7 l / ( g - z )
d>(¢)
We a r e c o n c e r n e d w i t h e s t i m a t i n g y ( - ) finite
and y + ( . ) .
finite union of mutually
disjoint smooth arcs. Let
(the generalized
space of functions on
F
length). Let
p(r) = i n f ~ ( E ) / i E I,
transform on
F
~)).
compact s e t s h a v i n g
Hence we assume that i'}
F.
P
is a
denote the l-dimension
LP(F)
(lip! ~)
with respect to the length element
denote the weak L 1 space of functions on
where the infimums
support of
To do t h i s ,
l-dimension Hausdorff measure are critical.
Hausdorff measure
LI(F)
(z¢(the
denote the L p
Idzl,
and let
Put
% ( r ) = i n f ~r+(E)/I~i,
are taken over all compact sets
E
in
F.
The Cauchy(-Hilbert)
is defined by
Hrf(z)
= (i/~)
p . v . IF f ( ¢ ) / ( ~ - z )
id~]
(z~r).
Then we see that p+(F) __< p(F) =< Const p+(F) I/3,
Const p+(F) __< I/]IHFIILI(F),LI(F ) 0,101%~) i(r,0)
z
Cr (.)
and radius
(0 e T[a](x,y)f(y)dy.
Calder~n showed Theorem A ([3]).
For any
f ~ L 2,
T[a]f(x)
(1.3)
IIall" --< Const llT[a]ll2,2
(1.4)
llT[a]l12,2 ~ Const llalI~ ,
exists a.e.
and
where
llT[a]l12,2
is the norm of
In §1.2, we show (1.3). §1.2.
T[a]
(as an operator from
Ps (x) =
E c ~, 171+
XE
to itself).
In §1.3-i.ii, we show various proofs of (1.4).
Proof of (1.3) (Coifman-Rochberg-Weiss For a set
L2
[15])
denotes the characteristic function of
{fl s(A(s)-A(t))dt}dsl
E.
We put
(x 6 ~,
s > 0),
lim ~ ~ 0
@ / 3 = Const lal
s where
I+ s = (x, x+s) , I_s = (x-s, x).
We have, for almost all
x,
Then
a.e.
to
P (x) = Ill+s ~l_s T[a](s,t) {(s-x) 2 + 2(s-x)(x-t) + (x-t) 2} dt} ds I 0, we estimate
(TE[a]g,f) = f2Te[a]g(x)f(x)dx, where
Te[a]
is an operator defined by a kernel
assume that e = X[0,~ ).
A(x) = fx a(s)ds.
Then
Ys(x-y) T[a](x,y).
We may
A(x) = f ~ e(x-s)a(s)ds, where
We have
r (x-y) {e(x-s)-e(y-s)}g(y)f(x)dydx]ds.
(T~[a]g,f) = f_~a(s)[ f-~f-i
(x-y) 2
Set f+(z) =
1 2~i
f_~
f(x) x-z
dx
Im z l > 0 )
-
0}.
k
Fixing a sufficiently large
T, We prove
(1.5)
ix; ~(v)(x) > ~x , A(v)(x) ~ ~/~ I (Const/T 2) ix; m(x) > k I
Let
W(k) = {x; re(x) > k},
with a sequence
M k = {Ik}
6(k) = IW(k) l .
Then we can write
W(k) = U]= I Ik
of mutually disjoint open intervals.
sufficient to show that, for each (1.6)
(X > 0).
IE I - yx }
for any
intervals
~ • ~/2.
~{J(x )}
x~I
Yx } "
{ ~n ]v12
0 < Yx < 21II
Iv(~,yx) I ~ Iv(x,Yx) I - Const A(v)(x)
There exist
such that IEI ~
R = QO N U A(x ), where
large enough so that the last
x E E, there exists
J(x) = {(~,yx);
(~,yx) ~ J(x), we have
- Const X/T
•
m(v)(a) ~ k, we have
Hence, for any
J(x) = (x - (Yx/5), x + (Yx/5)),
where
A(v)(~) ~ X/~ , we have,
IV(a,y) - v(x,y)] ~ Const A(v)(~) =< Const k/~ ,
Iv(x,y) l ~ 2X (x E I, y ~ 21If).
-
E = ~.
~ ~I; otherwise
quantity in (1.7) is less than
I~
To do this we may assume that
x ~ I, y ~ 2111 ,
(1.7)
~X
X/~}.
E = {x ~ I; m(v)(x) > ~h , A(v)(x)
~ ]VVl 2 d~
dr],
R
is the inner normal derivative and {ff ,
IVvl 2 d~ d~} i / 2
ds
where
is the length element. ~*(x) = {(~,~);
Let
l~-xl < q/lO}.
(x) N R Then a geometric observation shows that is a point which is nearest to
x in
AR(V)(X) ~ A(v)(x ) ~
{x }.
k/~ ,
where
xv
Hence the right-hand side of (1.8) is
dominated by: Const fl ~ (v)(x)2dx~C°nst(X/~)2]I] We divide
oR
~ Const k 2 ]I I.
into the following three parts:
DR 0 =
8R N U J(x ),
DR I = {(~,D);
~ ~ I, ~ = 2111},
0R 2 = oR - (oR 0 U oRI).
~IV v(~,~) I ~
Const
By the definition of
any
X/~
on
(~,D) ~ oR, Iv(~,~)I ~
IfDR ~
oR. ~X
+ Const
~-~-[~ds] 8n =< Const
fOR
X/~ ~
mlVvllvl
Note that
Yx (x ~ E), we have, for
Const ~X •
Thus
ds
Const (k/~) ~k /oR ds ~ Const ~2 ill. Since
Iv(~,O) l G Const k
on
OR I,
These estimates yield t h a t 4 R 0 U o R 2 O~/On ~ 0
on
OR 2, 0H/0n = i
on
we have If0R I On
0R 0
8~ 8n
Ivl 2 ds i ~ Const ~2 iii.
Ivl 2 ds G Const k 2 I I [ . and
Iv(~,~)l
~ ~k12
on
Since 8R 0, we have
2 k21El =< Const~8R 0 ~nn 8~ Ivl 2ds =< Const/oRoU OR 2 8~ ~nn Ivl 2ds = 1/2} . A geometric observation shows that, for any
Y O ~c ~ ~ , where
Y = [y-s,y+s](C ~).
IY N EI/IY 1 --< MXE(X 0) -_s.
y c Y
and
Then
This shows that
{ ffa(x) Ih(y's) I2 dy 2ds
} dx
s
>
]h(y,s)12 dy ds
ff^ ^
We now prove Theorem 1.7. ~k = {x; MXEk(X ) > 1/2}
Q.E.D.
S
I--~
Given
h E T, we put
(k = O, ±i, ...).
the totality of components of p(k)(y,s)j = (2-k-I/iI(k) i)j
~k
For each
E k = {x; S(h)(x) > 2k} , i~lj T(k) ~7j=l be k, let
and let
h(y,s) xj(k)(y,s)
(j >-_1),
13
where
X! k) is the characteristic function of ~!k) _ ~!k) J J J Then we obtain the following T-atomic decomposition of h: h(y,s) = p(k)j
Each
Z k =-~
Z j =l
2k+l I£4~(k)jI~ P~k)(y,s)-
is a T-atom, since
ff~!k) I P~k)(y,s)I2
supp(p~ k)) c i(k)j
dYsdS
2-2k-2 II!k) I-23
fl (k) E j
S(h)(x)2 dx =< i/ll~k) l
- k+l
Hence Lemma 1.8 shows that
IIR(h)iII --< k =-~E j=IE 2k+l Ijl(k)
Const
0
C6
David-Journe-Semmes
[40]).
[20]
These theorems give
for various constants depending
differs in general from one occurrence to another.
0 < 5 ~ i, we say that a kernel
kernel if there exists
T1
K(x,y) (x # y; x,y E ~)
such that
is a
8-standard
16
tK(x,y)l
N B/Ix-yl , - 0}.
since
d~Ss + 2 f;
Vs(X-t)(Vs*b)(t)Us(t-y)
Us(X) = (i/s)e -]Xi/s ,
L1. = 2 f~ U
el(L) N Const I]BIIBMO.
is given by
L
where
and
be an operator defined by
L(x,y) = Censt +
b ( BMO, there exists an anti-symmetric l-standard kernel
El = b, []elI2,2 ~ Const IIBIIBM0
is a Carteson
measure with
constant
Const
2
ITbtlBM0 ,
HblIBMo .
I t r e m a i n s to p r o v e
el(L) ~ Const IIbIIBM0 •
I n t h e same manner a s i n
17
Lemma 1.2, we have [L(x,y)l ~
IIVs*bll = ~ Const IIblIBMO. Const HDIIBMO
flu
Us(X-t)Us(t-Y)
d
Const IIBIIBMO S_~
Since
t
(Ix-tl + it_yl) 2
IVs(X)l ~ Us(X), dt ds s
So
e-I/s ds
i
s3
=< Const IIbllBMO/I x-yl Since
lU's(X)I -h~+l(V)}
(Const 2k)
{lh~(u)h~(v)I + lh~+l(U)h~+l(V)l}du
du dv I
dv
Const 2-k < C 8 2-k/{l + Ix-yl2-k} I+8 =
Thus (1.17) holds.
If
Ix-yl ~ 4 " 2k,
then
18~ Kk(X,Y) l If _~~ 7 -~"{K(x-s,y-t)-K(x,y)} {h{(S)hk(t) /_~ /_~ C8-Jsl 6 + It15
C 8 2 (8-1)k ~ C 8 2-2k/ {I + Ix-yl2-k} I+8 Ix-y[ < 4 • 2k,
then
' t )} hk+l(S)hk+l(
ds dt I
{lh~(s) lhk(t ) + lh~+l(S) lhk+l(t)} ds dt
Ix_yj1+5
If
-
20
[fi ~(x,Y) [ ~
=
V
[f_~ f_~ {fo f0 K(x-s,y-t)dsdt}{h~(u)h~(v)
- h~+l(U)h~+l(V )} du dv [
Const 2-2k ~ C8 2-2k/{I + ]x-yl2-k} I+6 . Thus (1.18) holds. We now show that, for
(1.20)
k ~ g,
[(KkKe)(x,y)[ -U 0 (See (1.23).)
Lreal,~
The norm
The operator
C[a]
C[a] = (-~)H
+
Tl[a] = T[a]
Meyer commutator (2.2)
= {a E L®;
the singular integral
We put a
is real-valued}.
showed
IIC[a]II2,2
is bounded if
a E Lreal"
is expressed formally in the following form Z (-i) n Tn[a], n=l
(the Calder~n commutator)
(n ~ 2), i.e., Tn[a ]
Tn[a](x,y)
C[a]
This is called the Cauchy transform of
{(x, A(x)); x E ~}.
Coifman-McIntosh-Meyer
Theorem B ([7]).
where
a, we define a kernel by
and
Tn[a]
is the n-th Coifman-
is an operator defined by
= (A(x) - A(y))n/(x-y) n+l.
Prior to this theorem, the following three theorems were shown. that
llTl[a]II2,2 ~ Const Ilall= (a E L~), Coifman-Meyer (2.3)
IITn[a]II2,2 ~ Const n! llalI~
CalderSn showed
[9] showed that
(a E L ~, n ~ 2)
and Calder~n showed that (2.4)
IIC[a]ll2,2
is bounded if
llall~ (a E ereal)
is small enough.
At present, there are three proofs of Theorem B; the original proof, a proof by the Tb
theorem [40] and a proof by perturbation.
contained proof by perturbation. Calder~n
[4] and David [17].
In this chapter, we show a self-
A proof by perturbation was first given by
Improving their methods and repeating a simple
perturbation method, we shall deduce Theorem B only from the boundedness of ([17], [42], [45]).
(See APPENDIX II.)
H
32
§2.2.
Two basic principles
(Zygmund
Here are two basic principles Coverin$ Lemma. IUxEA
~I < = •
Let
{~}X
[54]) in real analysis.
E A be a family of intervals in
Then there exists a sequence
{I~k}k=la
~
such that
of mutually disjoint
intervals such that
lIXkl.
iux~ A ~I-< 5 k=iE
The proof is as follows. larger than the supremum of
~i' "'''
~k-i
llxI
Now we show that
{j;
{IXk }
we have
> 211X. I, j # ~ . Let k
211kk I ,
the same midpoint as
If
{IXk }
be an interval such that
over all
X E Ak_ I,
where
(k ~ 2). (If
is the required sequence.
is
211~kl
A0 = A
is
and
Ak_ 1 = ~, we stop our
We first assume that
Since the intervals are mutually disjoint and
For
IX,
there exists
X ~ A.. Hence J be the smallest integer in the set.
IX. 3
which implies that
which gives that I~ k
211XII
Suppose that
according to the definition of our choice.
1% n Ixk # ~,
IUhE21 Ikl
IXk
lim k ~ ~ llXk I = 0.
[IxI
X ~ Aj}
IIxI ~
, we have
Let
X E A .
(i ~ j ~ k-l)}
is an infinite sequence.
IUk= I IXk I < ~
be an interval such that
over all
II~I
Ak_ 1 = {X E A ; Ik n IX. = ~ J induction at k-l.)
such that
IXI
have been chosen.
larger than the supremum of
{IXk }
Let
IX c IXk , where
and of length
--< I U IXk I --< 5 k=l
51
I"
IX k
Then
Since IXk
X ~
Ak,
is the interval of
Thus
% llXkl • k=l
is a finite sequence, each
IX
intersects with
in the same manner, we have the required inequality.
U
IXk"
Hence,
This completes the proof of
this lemma. Risin$ Sun Lemma. ~ a(x) ~ ~
for any
( ~ ~ T ~ ~ ),
Let
a
be a function in an interval
x E I, where
~ ~ O.
we define a function
the infimum is taken over all functions
B ~
Let in
I
such that
A by
I
such that
be a primitive of
a.
For
B(x) = inf ~(x), where ~ ~ A, ~' ~ Y
a.e. on
I.
33
Let
b = B'
and
components of
~ = {x 6 I; A(x) # B(x)} ~.
(2.5)
=
Uk= I I k ,
where
are the
{Ik}k= I
Then
Y _-< b(x) _-< ~
(2.6)
b(x) = Y
(2.7)
(a)l k
a.e. on
I,
(x 6 ~), I
(2.8)
_-< Y
I~I--< ~ -
((a)l k =
- (b)l T
~
III
fl k a(s)ds, k >-_ i), i "I~[ ~I b(s)ds).
((b)l =
I I
i
Inequalities
(2.5)-(2.7)
are easily seen.
(b) I III = 71 b(s)ds =
which gives (2.8). of Type i
71_62
We have
+ 7~
For the sake of convenience, we call this rising sun lemma RSL
(7-r~y,8-~e£~t);
an open set
z
we shall use later various rising sun lemmas.
~ , we denote by
{I~ ,k}k=l
its components.
For
The following two lemmas
are also the rising sun lemmas for integrable functions. Lemma 2.1 (The Calder~n-Zygmund k > 0.
decomposition
Then there exists an open set
~
[35, p. 12]).
=
in
(0,~)
such that
by
k
and
(k => i),
X
A(x) = f0 If(s) Ids
If(x) l < k
~c.
a.e. on
(x > 0), and define a function
B
B(x) = sup ~(x), where the supremum is taken over all functions
~ & A,
l~ll
f E LI
=
I~I < llflll/k, (Ifl)l~,k TO see this, we put
Let
such that
~
If(x) l ~
~' ~ k 1 ~
a.e. on
f~ If(s) Ids
a.e. on
Let
(Ifl) •
X
(0,~).
= ~ l~l,k
(0,®) - ~I "
~I = {x > 0; A(x) # B(x)}. (k ~ i),
Then
34
Considering Then
f(-x),
we obtain, in the same manner, an open set
~2
in
(0,~).
~i U {-x; x E ~2 } is the required open set. In the same manner, we have
Lemma 2.2. satisfy
Let
f
be an integrable function in an interval
k > (Ifl)l.
Then there exists an open set
i I~I _-< ~ 71 If(x)I ds, If(x) l < k
a.e.
M
is defined by
the supremum is taken over all intervals M
I
k > 0
such that
I
The (non-centered) maximal operator
denotes the norm of
in
and let
(k>= i),
(Ifl) I =< ~,k
on
~
I
I
Mf(x) = sup(Ifl) I,
containing
as an operator from
Lp
x.
For
to itself.
where
p > i,
NMIIp,p
The following lemma
is deduced from Covering Lemma. Lemma 2 . 3 ([35, p.7]). For
f ( L I,
IIMIIp,p~ Cp
X > O, we put
can choose an interval
Ix
(p > i). Ek = {x; Mf(x) > ~}.
containing
x
so that
For each
x ( EX, we
(Ifl) I > ~ . Covering Lemma X
shows that there exists a sequence that
IE~) ~ 5 %k= I Ilxk),
f ( L p and
fk(x) = 0
if
HM f[]
=
=
k~ I /i k if(s) ids ~
k > O, we define If(x) I ~ k/2. = Cp
f0
of mutually disjoint intervals such
which yields that
Ix; Mf(x) > ~I ~ ~5 For
{Ixk}k=l
fx
by
5 ~
[iflll "
fk(x) = f(x)
if
If(x) l > k/2
xP-l]x;
Mf(x) > k]dx
Cp 15
k p-I {Ix; Mfx(x)
Cp 75
xp-I ix ; MfN(x) > X / 2 )
dX
Cp /5
xp-2 llfkllI dX = Cp f~
xp-2 {fk/2 Ix;If(x) I > s I ds} dk
®
> X/21 + Ix; M(f-fk)(x) > k/2 I} dX
2s
Cp f0 Ix;If(x)] > sl { /0
which gives that
IIMIIp,p
and
Then
xp-2 d~} ds = Cp llfll ,
Cp.
At last we note John-Nirenbergts inequality, which was used in Chapter I. This is deduced from RSL. Lemma 2.4 ([32]).
Let
(For the proof of Theorem B, this is not necessary.)
f ( BMO
and
I
be an interval.
Then
35
Ix E I; If(x)-(f)l I >
-function
§2.3.
kI
=< exp(- Const >OIIl
(7~_> l).
([8], [35], [54])
In this section, we show a fundamental
the sake of simplicity, we deal with only kernels
K(x,y)
(See §i.9.)
in (1.22),
We use the notation
~(K), ff(l,K,f)
standard kernel, we define an operator =
K f(x)
I f l 1x '_' y
sup E > 0
K
For
inequality for standard kernels. satisfying
(1.21).
(1.24).
For a
by
> g K(x,y)f(y)dy I .
We show Lemma 2.5 ([35], p. 49). Then
Let
be a
K(x,y)
6-standard kernel (satisfying
(1.21)).
IIK I12,2 ~ Const o(K) + C 6 ~5(K). We begin by showing (2.9)
where
~(K ) ~ Const
~(K )
is the supremum of
and intervals I,
we put
if(K) + C 6 ~6(K),
I.
For
¢ > O,
(l/Ill) /I K (Xlf)(x)dx f E Lreal,l,
over all
an interval
J' = (x - s/2, x + s/2), J = (x - s, x + g), g =
h = XI_ J f. If
0 < ~
s K(x'y)(XIf)(y)dyl
I
f E Lreal,l
and a point %1 fl J f
x
on
and
s E J' ,
= IKh(x) J ~ IKh(s) J + IKh(x)-Kh(s)J
IKh(s) I + C 5 ~5(K) ~ IK(Xlf)(s) I + IKg(s) l + C 6 ~6(K)
= IX ,(s)K(Xlf)(s)l
+ IKg(s) I + CO ~b(K),
I where
I
is the double of
I.
Taking first the square roots of the first
quantity and the last three quantities, respect to
s,
and taking next their means over
J'
with
we obtain
Iflx-Y I > s K(x'y)(XIf)(y)dyII/2 M(I X , K(XIf)II/2)(x) I If
S >= III, then
+ (IKgll/2)j, + C 6 ~6(K) I/2.
/Ix-yl > s K(x'Y)(XIf)(y)dy
= 0.
for all
Hence this inequality holds
s > 0, which shows that this inequality holds with the first quantity replaced by K * (Xif) (x)i/2 . Taking the squares of both sides of the resulting inequality,
and using Shwartz's inequality, we obtain
36
K (Xif)(x) _-< Const
M(IX , K(Xif)I1/2)(x) 2 I
1/2 2 + Const(iKg I )fl, + C6 ~6(K). Since
( M(IX I ,K(Xlf) iI/2) 2) I =< Const
0.
~(K) + C 6 ~6(K),
I
of
*
Mf(x) > ~k
(2.11) (See §1.4). (x0
on
To prove this, it is sufficient to show that,
i
Mf(x) ~ ~Ikl =< - ~
~ E I,
llI.
=
g = Xjf
and
is the left endpoint of
1
-iO
i1 l '
h = Xjcf,
I).
where
IX E I; K g(x) > k I
+ Ix E I; K*h(x) > 2k I
(= L I + L2,
Note that
J = (x 0 - 2II I, x 0 + 2III)
Then we have
Ix E I; K*f(x) > 3X 1
3X}
k I
I, this inequality evidently holds.
for some
Let
k
{x; K f(x) > X} ,
Ix E I; K f(x) > 3X, If
I@f(x) =< ~k I ~_ - ~
is determined later.
for each component
M f( 3k,
~ > 0
which implies (2.9).
We show the following good
*
(2.10) where
~5(K),
say).
K h(x 0) ~ >~. For ~ > 0
and
x E I, we have
37
Iflx-y I > e K(x'y)h(y)dy - 7 1 x 0 _ Y 1 > g K(x0'Y)h(y)dy I ~
IK(x,y) - K(x0,Y)IIN(Y)I
dy + Const
oos(K) M f ~ )
=< C6 oos(K) Mf(~) =< C5 oo6(K) ~ . Since
g > 0
is arbitrary, we have, with a constant
(2.12)
depending only on
5,
K*h(x) s K(x,y)(g(y) - g(y))dy I = If(i IN 12) n (x-s,x+g) c K(x,y)(g(y)-g(y))dY +
=
- 1121 ->- . . . .
We have,
(k > i),
L 1 =< Z ]((K-T)(XkU), XkV)~l k=l ®
+
Z k=l
(= Let
xk
~
l((K-T)(XkU), Xjv)~ I +
Z
j=k+l
Z ]((K-T)(XkU), XkV)~] k=l
be an endpoint of
*¢ y ( (Ik ) N Ij,
k < j,
Ik
k-I
Z
Z
k=2
j=I
+ LII + LI2,
such that
xk ( I -~
l((K-T)~kU), Xjv)~I
say). (k >-- i).
If
x ( Ik,
then
IK(x,y) - T(x,y) I = IK(x,y) - K(Xk,X j) + T(Xk,X j) - T(x,y) l C8 llklS/[x-ylI+6
* c . If xj ( (Ik)
(Here we assume IK(x,y) - T(x,y) l ~ LII
+
C8 llklS/Ix-yll+8.)
Z
Z
k=2
j=k+l
f
Hence
*c I(K-T) (XkU) (x)v(x) ]dx (Ik ) N Ij
Z fl {fie k= 2 ke C8(¢o8(K) + cos(T))
+
then we have evidently
--< Z f , l(K-T)(XkU)(X)V(x)Idx k=l Ik-I k
=< C8(¢o8(K) + ~6(T))
+
* xj ( Ik,
Z k=2
fI k
[ Z k=l IIk]6 ix_yll+8 {
>~ k=l
lu(Y) l Mv(y)dy}
f * Ik-I k
{flk
dy}
Iv(x)l
lu(y) Idy} Iv(x) l dx ] f , IH(X k u)(x)v(x)I dx Ik-I k ~- C6(es(K) + ¢o8(T))
_- 0)
aE(~) = sup {o(E[a]); a ( ereal,6}
and show that NO
(2.36)
~E(~) - 0).
a ( Lreal, in
~(E[a]) = - 0
f ( Lreal,l
Type i (-~/3- A.,B- a.)
and
~E(1) ~ Const.
and
I
Let
be an interval.
shows that there exists
b ( Lreal
RSL of such that, with
= {x ( I; A(x) # B(x)}, - 613 ~ b(x)
6 -
Using Lemma 2.7 with
~
6
a.e.
(b) I
on
I,
b(x)
= - ~13
on
~
,
3
we have
K = E[a], T = E[b], 5 = i,
~(I, E[a], f) _~ a(l, E[b], f) +
Z k=l
O(Ik, E[a], f)
+ Const {$(E[b];e) I/2 + ~01(E[a]) + ~Ol(E[b])} Ill, where
I k = I~, k
~(I, E[b],f) = Since
(k >_- i).
~(I, E[~], f).
b(x) = -~/3
on
Put
~ = b - (6/3). We have
Then
II{II~ =< 2~/3
~, we have
D(E[b]; ~) =
~2~(H;~) =< Const.
Thus
~(I, E[a], f) ~ OE (
) +
~E(~) + Const 6 ,
which yields that ~E(6) _-< OE( that is,
) +~
and
001(E[a]) + O~l(E[b]) =< Const ~ .
aE(~) + Const ~ ,
53
(2.37) Let
n
OE(~) < 4 ~E(23 ~)
+ Const ~ .
be the minimum of integers
n =< (log ~)/(log 3/2) + Const.
k->- 1
such that
Inequality
OE(~) =< 4 n ~ E ( ( 2 ) ~ )
(2/3)k~ ~ I.
Then
(2.37) shows that n-i % k=0
+ Const
4 k (2)k3
-_< 4 n {~E(1) + Const (2)n ~} _-< Const where
4n_-< Const (i + ~) N0 ,
N O = (log 4)/(log 3/2).
Thus (2.36) holds.
This completes the proof of
Lemma 2.12.
Q.E.D.
We now give the proof of Theorem B. (x ~ ~),
Since
i/(I + ix) = f0
e-iXs e-S ds
we have C[a] = f~
E[-sa]e -s ds.
By Lemma 2.12, we have IIC[a]I12,2 =< f0 IIE[-sa]ll2,2 e-s ds NO Const f0 (I + sllall )
NO e -s ds _-< Const (i + IIall~)
This completes the proof of Theorem B. §2.7.
Estimates of norms of
E[.]
and
C[.]
In this section, we show Theorem C ([44]).
For any real-valued
(2.38)
NE[a]II2,2 ~ Const (i + IIalIBMO),
(2.39)
IIC[a]I12,2~ Const (i +~IIalIBMO).
In [7], (2.38) was given with was given with
__~MO
replaced by
[44] via [42], [43], [50]. use RSL repeatedly.
a(x), I, ~, ~
and
we define analogously aretan
y.
Then
IIalIBMO
in
BM0,
9 IIalIBMO,
and (2.39)
This theorem was established in
In this note, we deduce Theorem C from (2.36).
The lower bound of A(x) B(x)
a
replaced by
8 IIaIIBMO.
(The sun also rises!)
Type i (y - r.,$ - a.) Let
function
We
RSL in §2.2 is called of a(x)
is independent of (2.5)-(2.8).
be the same as in RSL in §2.2.
For
0 ~ T ~ ~ ,
by using the sun at the right upper infinity of angle
54
~ b(x) ~ 7 (a)ik ~ T
a.e. on (k g i),
I, I~I ~
b(x) = 7
(x ( I k, k ~ i),
(-~) * (b)l
III.
(-~) + 7 This RSL is called of Type 2 ( y - h a y , ~ - d g 6 e e ~ ) . independent
of the above estimates.
For
0 ~ 7 ~ ~ ,
where the supremum is taken over all functions a.e. on
!.
We define
b, ~
-- 0).
In this step, we show that, for
(2.40) where
Recall
Put
~E(~) ~ ~E(@~) + ~
(0 < e < i) and
~(I, E[a], f).
~E~T ^ .l+e ~) + (Cs/e)
is determined
an interval
Since
I,
~ ~ i,
we
later. study
For
0 < 5 ~ i,
~5{~E(~)
+
a E Lreal,~,
an a-priori
~(l,E[a],f) = ~(l,E[-a],f),
estimate
~5}
f E Lreal, with
respect
(a)ik ~ Using Lemma 2.8 with
b(x)
=< ~
e~
a.e.
on
(k => i),
I,
I~I
K = E[a], T = E[b],
b(x)
Since
~o6(E[a]) _< C5 ~5,
III
(x E I k,
k >
b ~ L ~real
i),
( ~ = (b)l)-
we have
~(I, E[a], f) - 0. RSL of Type i (- e~- ~., ~ - ~.) shows that there exists (a) I = such that, with ~ = {x E I; A(x) # B(x)} = Ok= I Ik (Ik = l~,k). - 6~ ~
,
% k=l
~(Ik, E[a], f)
C 6 AI(E[a] , E[b]; ~) IIl. 0~6(E[b] ) < C5 ~6
$(E[b]; ~)i/2
and
= Const $(H;~) I/2 < Const =
we have AI(E[a ],E[b];~) = {&(E[b];~)I/2 + ~5(E[a]) + ~5(E[b]) } × {~(E[a]) +
~(E[b]) +
~5(E[a]) + ~5(E[b])}
_--eF,
I "
In the same manner as above, i+@ ~(l,E[a],f) ~ ~(l,E[b],f) + ~E ( T ~)
{ £ llkl + E Z llk,~l} k=l k=l ~=i
^
% % k=Ig=l
% ~(Ik,g,m, E[a], f) m=l
+ C5 ~5{aE(~) + ~5} {lll+
% llkl+ k=l
% k=l
Z Ilk,~I} • g=l
57 ^
Since
^ in
-
o(l,E[a],f).
Let
Le~=na 2.2 shows that there exists
such that
(IaI )i k ~ 2~
(k ~ i),
I - ~ .
We put
b(x) =
Then
b E
e =real,2~
~)
~(x)
(x ~ I-
~(a0)ik
(x l )~ ~). Ik' ( xk >
Using Lemma 2.7 with
K = E[a], T = E[b],
we have
~(l,E[a],f) = o(l,E[a],f) i) '
(x
(~),
(e = I/2).
II~I, l+e
Lemma 2.8 shows that g(l,C[a],f) ~ ~(l,C[b],f) + gC(-6~,~)Ill +
Z $(Ik,C[a],f) + C5~8{~C(B) + ~5} ii I k=l
Z g(Ik,C[a],f) + C5~5{ffC(~) + ~8}III . k=l
For each k ~ I, we use RSL of Type 3 (0-&.,-~- d.). Since exist b k 6 Lreal such that
and an open set
- ~ ~ bk(X) ~ 0
a.e.
(a)l ~ - @~,
~
there
k
~k = U~=I Ik,~ (Ik,g = l~k~~) in Ik
on
Ik,
bk(X) = 0
(x ( ~k ),
+ (bk)ik (a)ik, ~ 0 (~ ~ I), B + 0
Lemmas 2.8 and 2.13 show that ~(Ik,C[a],f) ~ ~(Ik,C[bk],f) + -- .1)
e~ .
b(x) = e~
I~I < e~ + g
b 6 Lreal
(Ik = l~,k),
e~ + (b) I (a)l k
(e = ~/2).
{Oc (~) + ~ }
(x ~ ~), e~_
I~I ~ e~+e~
½ I~I =
I~I'
Lemma 2.8 shows that E $(ik,C[a],f) + C8~8{~C(~) + ~8} k=l
$(I,C[a],f) ~ $(l,C[b],f) +
~c(-el3,el3) I~1 + For each
k ~ i,
there exist such that
~(Ik,C[a],f) + C858{~C(~) + ~8}II I. k=l
we use RSL of Type I (O-r.,~-a.).
bk 6 Lreal
0 ~ bk(X) & ~
ii I
and an open s e t
a.e. on
Ik,
Since
~k = Ug=l Ik,~
bk(X) = 0
(ak)ik a e~,
(Ik,g = I~k,g))
(x 6 ~k ), (a) I
~ 0
(9~=>l),
k,i
- (bk) Ik
II
~
~ - e~
llkl
= (l-e) llkl.
Lemmas 2.8 and 2.13 show that ~(ik,C[a],f ) __- i) (a) I
(x ~ R).
Then ~(l,C[a],f) =< ~(l,C[b],f) +
+
Z ~(Ik,C[a],f) k=l
c5(i + 825 ) Ill.
IITn[']ll2,2 -- 0
A locally chord-arc
a locally chord-arc
such that
Fn
curve with constant
D(z,e)
is a chord-arc curve with
curve is not, in general,
compact curve with constant
i00.
M, if, for
connected.
We define
Let
F
be
72
(3.3)
p(F) = inf y(E)/IEI,
p+(r) = inf y+(E)/IEI,
where the infimums are taken over all compact sets N'II (i ~ p < ~) L p (r)
be the same as in §2.10.
r
with supremum norm
on
F
li"IIL~(F)
and let
E
Let
L (r)
on
L~(F)
F .
Let
be the
LP(F), L"
space on
be the space of functions
f
with norm
NflIL(F)
=
If()l>
sup
The Cauchy transform
HF
on
operator from
to
LI(F)
LI(F)
F
>
;
is defined by (2.50).
HHFIIL1(F) ,L~(F) "
is denoted by
w relations among
p(F), o+(F)
HF
The norm of
as an
Here are
and IIHFIILI(F),LI(F)"
Theorem D.
(3.4)
C°nst/IIHFIrLZ(F),L~(F)
~ p+(F) ~ Const/llHrHLl(F), L~(F),
(3.5)
p+(F) ~ p(r) ~ Const p+(F)
1/3
.
We begin by showing the second inequality in (3.4). llfH
= i.
k > 0,
For
Let
f E L2(F),
- ~ < e ~ ~, we put
Ll(r) Ek, e = {z ~ F; HF f ( z ) ~ D(ke i e , k / 4 ) } , There exists a compact set
Fk~ 8
in
exists a non-negative measure
EX, ~
on
FX, e
d~ ~ H Since IIC~II
IFx,el ~ IEx,el/2.
such that
There
such that
y+(Fx,e)/2.
k,e
~ i,
d~ = hldz I
we can write
with
h 6 L~(F),
0 0
is arbitrary,
flIL~(r )~
IIHr
Const/ o+(r)
(f ~
L2(F),llfNLl(r) = i).
A standard argument shows that this inequality holds with g E LI(F),
IIglILI(F)
= i.
f
replaced by any
Hence the second inequality in (3.4) holds.
The proof of the first inequality in (3.4) was essentially given by Davie [21], Marshall
[37] and Davie-¢ksendal
[22].
Here is a tool for the proof.
Lemma 3.1 (the separation theorem [53, p. 108]). convex sets in the Banach space
II'IIL.(F). Then
C(F)
Let
P, Q
be two compact
of continuous functions on
there exists a complex measure
~ on
F
F
with norm
such that, for any
f E P, g E Q, Re fF f d~ >
Re fF g d~.
Since F is a locally chord-arc compact curve with constant i00, there exists F N D(z,s 0) is a chord-arc curve with constant i00 for any ~0 > O such that 8 z E F. Let H F (0 < s < SO/2) be an operator defined by
H r~ f(z) = ~i f
f(~l ~ - z
r n D(z,c) c
Id~I
We show that there exists an absolute constant C O (3.6)
Let
M F2g
s IIH FII ~ ~ LI(F) ,e~(F)
C0
F(z,~) = F N D(z,~). IIMISlILI(F ) i
For
such that (= C0m 0, say).
be a maximal operator defined by
MF2g f(z) = 0 < nsup~ 2s where
llHrllnl(r) ,L~(F)
(f e El(F))
f ~ LI(F)
with
~
,n (F)
~
~
i
/F(z,n)
if(~) i id~l
Then Const.
IIftlLl(F) ~ i,
we put
E = {z ~ F; IHF f(z) I =< k ' M2~F f
Taking the supremum of
Re
Since
Re IF g d~
IF
g d~
Como},
is the constant in (3.6). P, Q
We show that
are compact and convex in
C(F), Lemma 3.1 shows that there exists a measure Re /F HFs f
3
~
on
F
such that
(f E ~ g (Q)
over all
g E Q,
Re IF H F~ f d~ ~ 3 C0m 0 IFid~I
we have
(f E F)
which implies that - Re IF f go
IdzI { 3 Com 0
(f E F),
where
g0(z) =
(7 i r Id~l) -I fr, I< - zp > ~
By (3.6), we have, for any
h E LI(F)
Iz ( F;IH ~ h(z) I ~ Since the kernel of
H Fe
2 Como/IEiI ~
Iz E F; Igo(z)I ~
Then
~
with
IIhIILI(F) ~ i,
IEI/2.
fFld~I ~
2 Como/iEiI ~
F = {z E E; Igo(z) I ~
function.
d~(-- "'''
Thus
P N Q # ~.
there exists Let
limn -~ ~ Sn
{On}n= I =
0
and
f~ E L'(F)
such that
fs E
F,
be a sequence of positive numbers such that {f
gn
IdzI}~=l
converges weakly (as a
77 sequence of measures). Idzl; we write by
IIHr
fOllL~(r )~
Const m O.
0 < 2s e < e k < SO/2.
Sk St 1% f %)I
+
Let
ISr, _ F(Zo,CoI2)
st fk
ek, e e
Id~lt
Z
1
(
1
1
1
~ - z
IdYll
fF_F(Zo,ao/2) f ~
-
ee
)
+ Const
(~)
IdYll
) fse(~)
fee(K)IdYll sO c~ M F f (Zo)
Idol
IdYll + Const {i + (ek/t~)Irl}.
fee(~) - r(zo,Ck)
~ - z
~ - z0
fee( 0
such that, for any
We say that a set
E c ¢
is thick, if there exists
z ( E, r > 0,
I/M ~ IE n D(z,r) I/r ~ M. The 1-dimension Caldergn-Zygmund decomposition is applicable to thick sets. thick sets are also natural objects.
From the point of view of §3.2 and
Hence
84
"thick sets", we define (thick) cranks. An interval
i
in
I 0 = [0,i)
is called a dyadic interval if
I
is
expressed in the form
I = [(j-l)2 -g, j2 -g) with integers g >-_ 0, 1 =< j =< 2g. A m R = {Ik}k= 1 of mutually disjoint dyadic intervals is called a m I0) if I 0 = Uk= 1 I k, For a positive integer q and two coverings
finite sequence covering (of n
~
R' = {lj}j=l,
m
= {Ik}k=l,
we write by
R'
k I
~ = C O {~(TF) + i}
{Ik}k= 1
by the largest dyadic interval in
Using this inequality, we obtain
and
CO
(f E L 2,
is a suitable constant.
by a suitable sequence of dyadic intervals.)
immediately yields
k > O) (Replace
Inequality
(3.18) Q.E.D.
IITFII2,2 ~ Const {~(T F) + i}.
For a non-negative
integer
o(n) = sup {~(TF);
F
we put
crank of degree
~(n) = sup {$(I O, TF,f); F
n,
& n} ,
f E Lreal, 0 ~ f a i,
crank of degree
~ n} .
We show Lemma 3.4.
For two positive integers
(3.19) Proof.
Let
satisfying
Let
(3.10)
F
with
g ~ n-l,
Without loss of generality we may assume that
be a crank of degree
and let
A I, ...
A '
(3.12).
g
~(n) ~ $(g) + ~(n -g - i) + Const ~(n). f E Lreal, 0 ~ f ~ i.
supp(f) c I 0.
n,
be
n, n
R 1 . . . . , Rn be
A F, = A 1 + ...+ An_g_l
,
m
Rn_ g Then
F'
= {Ik}k= I.
is a crank of degree
n - ~ - i.
We have
$(I 0, T F, f) = (TFf , TFf)fd x = ~(I0, TF,, f) + ((T r - Tr,)f, TFf)fd x + (TF,f, and
coverings
functions satisfying
n
Put F' = {(x, AF,(X)); x ~ I0} ,
n
(T r - TF,)f)fd x
(3.11)
87
((T F - TF,)f , TFf)fd x m
=
Z flk(T F - Tr,)(X ik f)(x) TF(Xlkf)(x) f(x)dx k=l m
Z flk(TF - TF')~Ik f)(x) T F ~ c f)(x) f(x)dx k=l Ik m E f c (rr - TF')(Xlkf)(x) rrf(x) f(x)dx k=l Ik ( = L I + L 2 + L3, Since
say).
TF,(x,y) = i/(x-y)
(x,y E I k, i _-< k =< m)
and
~(n) ->- Const, we have, by
Lemma 3.3, m
ILl1 ~
m
Z ~(I k, TF, f) + ~ % flk IH(Xlkf)(x) k=l k=l
Tr(Xlkf)(x)Idx
m
~(I k, TF, f) + Const IITFII2,2
Z k=l m
^
~(Ik, TF, f) + Const o(n).
E k=l
Extending coordinates, we see that, for each 0 ~ fk ~ i
and a crank
Fk
of degree
g
Ik,
there exist
fk ~ L ~real'
such that
$(I k, T F, f) = llkl ~(I 0 , TFk , fk ). Hence m
ILl1 ~
% llkl ~(I0, fk ) + Const o(n) k= I TF k ' ~(~) + Const a(n).
Recall (3.16) and (3.17). have, with
Since
x k = (the midpoint of
TF(X,y ) - TF,(x,y)
m
IL21 =
is anti-symmetric, we
Ik) , I *k = (the double of
I Z 7!k(T F - TF,)(Xlkf)(x) k=l
× {TF(× *c f)(x) - TF(× *c f)(xk)} Ik Ik
f(x) dx
Ik)
88 m
+
Z flk(TF - TF,)(Xlkf)(x) k=l
TF(× ,
Ik-Ik
f)(x) f(x)dx I
m Z Ilk I(TF - rF,)(Xlkf)(x)l Mf(x) dx
== ~ - ~12 -q I Re
is odd).
i ~} dy I x-y
dy [(k-l)2-q,k2 -q) {(x-y) + i(a-~)2-q}(x-y)
I
I
I [(k-l)2-q,k2 -q)
7
dy
k even
[(k_l)2-q,k2-q)
f2-q +I 2- q
dy y2 + (~_6)2' 2-2q
In the same manner, we have, for any IT~ XIo(X) I a
Is - ~I/lO.
~
0
......
(x_y)2 + (~_~)2 2-2q > I S _ BI/10. =
x ( [(ko-l)2-q, k02-q) (k0
is even),
Thus ~°(I) = $(I°' t1'
Lemma 3.8.
(~
= I[o
For two positive integers
(3.21) Proof.
XI0) >
)2 dx > Is - ~12/I00.
lO
=
n, g
with
Q.E.D.
g ~ n - i,
T0(n) e ~0(~) + ~0(n - g) - Const q-l~-~-.
We write Ik = [(k-l)2 -qg, k2 -qg )
(i ~ k ~ 2qg).
We have ~o(n)
=
Izo Ir~ ×~o(X) t 2 dx
+ fl0(T~- T ~ ) (XI x )0 = LI =
= ~o(~)
T n0 XI 0 (x) dx + fl 0 TZ0 Xl0(X) (T~ - T~) XI0(X) dx
T0(g) + L 1 + L 2, 2qg 0 T~(x,y)dy} dx k=iZ /ik {/ik(T~(x,y) - Tg(x,y))dy}{flk
03
2qg Z 0 flk {flk (T~(x,y) - T~(x,y))dy} {flo_lk T~(x,y)dy}dx k=l 2 q&
k=IE fl0-1 k {flk (Tn0(x'y) - TO(x'y))dY}
T0n XI O(x) dx
= LII + LI2 + LI3.
Note that
0 T~(x,y) = 0
(x, y EIk, 1 ~ k =< 2q~)n
T (x,y) = [(x-y) + i
Z = g+l
and
(A (x) - A (y))]-i
1 x-y
(x, y E Ik, i ~ k ----2qg) Hence, extending the coordinate axes, we have 2qg
2q&
LII = k=iZ flk [flk T~(x,y)dy] 2 ^
Let
p be the integer such that
integer. where Let
dx = k=iZ Ilk] ~0(n - L) = ~0(n - £).
~k
q4 < p ~ 2q4 and (log p)/log 2 is an For each i ~ k ~ 2qg, we write Ik = Ik, 1 U... U I , 2 k,p 2 P 1 are mutually disjoint dyadic intervals of length p-2 2q~ {Ik,j}j= denote the closed interval of the same midpoint as
Ik and of length
(i + p-l) Ilkl, and let Ik,j denote the closed interval of the same midpoint as Ik,j and of length (i + p-4) llk,jl . We have, with x k = (the midpoint of Ik) , Xk, j = (the midpoint of Ik,j), 2qg LI2 = k=iZ flk {fiE T~(x,y)dy}
{fl0_lk T~(x,y)dy} dx
2qg
k=iE
fl k {flk T~(x,y)dy}
{f(l0 N ~k)-Ik T~(x,y)dy}dx
2qg z k=l
% .(T~(x,y)-T~(Xk,Y))dy}dx flk {Ilk T~(x,y)dy} {fl0_(10 N ik)
= LI21 + LI22 ,
ILl211
2qg 2 k=iZ fie Iflk Tn0(X,y)dyI (fik_l k
dy ~
) dx
04 2q&
2 q8
Z ~(Ik ' TO )}i/2 { Z flk (f~k_lk ~ k=I n' XIk k=I
{f~
Const Const
)2 dx}i/2
log2 dx
f(lk N ZIk,j)-Ik,j {/Ik,j } {fI0-(l0 Nik)}
2qg p2
Z j=l Z k=l
f Ik 0 ~ck,j {fIk '3}
{fI0-(~0 n ik)}
= L1221 + L1222 + L1223 • Since
n 2 IX-Xk,j I +t ~ (A~(x>-A~(Xk,j)) 1 ~=~+I ...... ]TnO(x,y) - T0n(Xk,j, Y)] _-< [x_yI IXk,j - y] Cons¢ {p-2 2-q8 + 2-q(8+i)} / ]Xk,j _ y12 Const p-2 2-q%/iXk,j _ yi2
(x ( Ik,j, y £ ~),
(3.13) shows that 2qg p2 IL12211 = I Z Z
k=l
j=i
fik,j{fIk,j T~(x,y)dy}
× {fi0_(i0 fl Ik) 2q8 p2
Const
2q~ p2
Const
-2
P 2-qg dy) dx Z Z fI Ifik,jT~(x,y)dy] (f~ k=l j=l k,j ~ IXk,j-Y]2 p-i Z Z fIk,j]fI T~ (x'y)dyl k=l j =i k,j
95 Const p-i NT~H2,2 ~ Const p-i ff . Since ITnO(x,y) - TOn(xk,Y) I ~ Const p
2-q~llxk-y[2
(x E I k, y E ~k ) ,
[TO(x,y)] =< Const 2-q(g+l)/Ix-yl2 s
libn I]. N C 0 n/g . Let F - I 0 - E.
:o C (s>ds) = ),i
E
be the
Since
X
From the definition,
..,
C[b n ](x,y) = TF0(n)(X,y)
(x,y 6 E).
Hence T h e o r e m E s h o w s t h a t
{rE IC[b~*] ×E(x)]2dx}l/2
=
{rE Irr0(n> ×E(x)12 dx}l/2
=> {rE ITFo(n)×IO(x) 12 dx}i/2 - {fIoITpo(n) XF (x) 12 dx}i/2 {Const n - /F ITF0(n) Xl0(X) 12 dx}i/2 - Const V ~ V ~ . By (3.13) and (3.18), we have IITF0 (n )if4,4
~
Const{o( TF0 (n )) + l} ~ Const V~,
and hence,
7F ITFo(n) ×lo(X) l2 dx _-= Constq-f llC[bn
(n => i),
Const VT}.
99
which yields (3.22). §3.5.
Analytic capacities of fat cranks For
With
p > 0, z 6 ¢
0 ~_ ~ - 0, -~ < e ~ ~)
x cos @ + y sin @ = r. = #{E N L(r,@)},
elements of
For
less
than
e.
Cr (E) =
and
we write
is the (cardinal)
0 < a ~ i,
finite
line defined by the
E C ~,
number of
we put
Na{u~=lD(Zk,rk)}(r,@)~
where t h e infimum i s t a k e n o v e r a l l radii
set
#{E A L(r,O)} e > 0
Cr(e)(E)~ = i n f f _ ~ { f ~
denote the straight
For a compact
where
E n g(r,8).
quantities
dr} dS,
coverings
{D(Zk'rk)
k=l
of
E
with
We p u t
lim
Cr(e)(E)
(0 < ~. < i),
~ + 0 Bu(E) = If
lim lim c÷Oa÷O
E c D(0,1),
then
is the probability I01 ! ~)
Bu(E)/2~
intersecting
to compare a compact
is called the Buffon needle probability;
(measured by
curve. Then Crofton's (3.7), we have
Cr ( ¢ ) ( E ) .
with
dr d@/2~)
E.
formula
Suppose
~(.) E
with
Cr (.)
such that
that
E
L(r,O)
is a locally
[49, p.13] shows that
T(E) ~ Const CrI(E ).
set
of needles
(and with
Bu(.)).
and
chord-arc
compact
CrI(E ) = Coust I E I.
From this point of view,
y(E) = 1
this
(0 < r < i,
By
it is interesting
It is known that there exists
Bu(E) = 0
(cf. Jones-Mural
[34]).
We shall show Theorem F. y(E) = 1
For any and
Kakutani,
The author expresses
(3.28),
Galton-Watson
and Professors
process
various
set
E
such that
Coifman,
integralgeometric
time of the sun's rays integralgeometric
= l)
P r o b ( X ~ = 0) = 1 - B H
to use the to Professor since
RSL is the first stopping to try to give an
C,
Cr (.).
For
= -1)
0 < ~ < I,
let
(X~}~= I
on the standard probabillty
such that = erob(X~
According
Hence it is interesting
random variables
(F 0 = [0,1])
Cr (.).
cancers outside bodies.
we study
sequence of independent
eroh(X~
Steger who suggested
formulae are used for surgeries,
(or needles).
proof of Theorem
In this section,
his thanks to Professor Kakutani who
[30] for the estimate of
X-rays react to, for example,
(PO,S,Prob)
there exists a compact
Cr (E) = 0.
Acknowledgement. communicated
0 < ~ < 1/2,
=
(n a 1 ) .
/2,
be a
space
106
We put S~0 = 0'
S~n = X~l + "''+ X~n
This is a model of random walks. (3.27)
Then
y0~(x) = i,
Prob(y~n >_-0) = i
(n > I).
We define a Galton-Watson process
y~n(X)= y~n_l(X)+ S~y~n_l(x)(X)
(n >_-0).
c (n) = 2~+I
{Yn}n= 0
(n ->-i, x E F0).
We put
fO/2 { Z
k s b~n)(t(e))}cos Od@
(n >__l, 0 ~ Pl }
Idl(Qn)- fO01 @i If0
Const Pl
~ .
such that the measure s 2 -n"
is less than
Z" k=l
k ~ b. k(n)(tan
{/O N , (r,@) ~ dr - % F k=l
If
of
gap(~n)
then
~- PI'
@) cos @ d@ I
k s bk(n) (tan @) cos @} d@ I
n
l{f F + /(0,01)_ F} { Const For an integer integer
pj
2 ~n IFI + Const
s
j # I, we can choose,
such that,
if
8_1 = 0.
&
Const
that if
as above,
a positive
then
Z k ~ " (n)(t(@)) k=l ok
This shows
s .
in the same manner
gap(~ n) ~ pj,
81Jl Idj ( ~ ) - T e l j l _ l
where
} de I
gap(~n)
cos
@ del
< =
s /(i + j2),
is sufficiently
large,
then
(3.30)
holds. §3.7.
Q.E.D. Proof
of Theorem
F ([34])
Here are three lemmas necessary 0
= i, w
n
on
there exist a crank F
n
such that
w
n
Fn
of degree
is a constant
n
on each
113
~(r0, r[) ~ ~0' cr(r[) ~ ClI(~ n~-~), (3.31) llWnll i * L (Fn)
I,
llWnll~ ~ L (r[)
where
C1
Proof.
Leamms 3 . 1 1 a n d 3 . 1 2 show t h a t
satisfying
C I, film H , w~lle.(F~ ) ~ Cn~'n~, Fn
is an absolute constant.
the first
there
two i n e q u a l i t i e s
in
exists
(3.31).
II~rn,
~
ConstV~-,
tlHFn,
~
Const V~.
IIL2(Fn),L2(Fn)
a crank
F
of degree n n ( 3 , 1 3 ) shows t h a t
Inequality
which yields
fiLl(m) ,L~(Fn )
Thus, in the same manner as in the proof of Theorem D, we obtain a non-negative f u n c t i o n on satisfying F* n mean o v e r e a c h c o m p o n e n t o f Lemma 3 . 1 4 .
and
w
Let
~0 ~ 1
and
three
m
~j = 0
such that
Fm.
n ~ 1.
in
the required
Let
F
m
(3.31).
function
Taking the w • n
be a c r a n k o f t y p e
rm
such that
Then there exists a crank
(ra+l N j N ~ - n )
Wm+ n
inequalities
we o b t a i n
be a non-negative function on
on each component of with
the last F , n
and a n o n - n e g a t i v e
mw
function
{ 6 j } ~0=_
is a constant
Fm+ n
is a constant on each component of
Q .E.D ,
•
of type wm+n
.m+n
16j~j= 0
on
Fm+n
Fm+ n,
rm [[ rm+n' c(rm' Fm+n) >= gO' Cr (Fm+n) -_ Const,
Y(E([)) ~ Const.
Let .101.(1-(l-a)~o)n0mk ~i--OO)
gk = (C0/~) Then
limk ~ ® gk = 0 Cr (F~)
(k ~ 2).
and i01 n0mk-i
(2c0/~)(i~)
.101.-(l-~)~0n0mk-i
~
2 gk_ 1 . We can inductively choose
~0 = {gk}k= I 0®
Cr(i/k)(E([0))~ ~ 2 gk-l'
which shows that
satisfies
Y(E) > 0
Remark 3.16.
and
Throughout
so that, for any Cr (E([0)) = 0.
the note, we use Theorem D to estimate
below.
Here is a weaker inequality than Theorem D. Then
(cf. [29, p. 19]).
I 7r f
dzI2/{Irfll 2(F)
E = E([ O)
Let
F
Y(')
from
be a locally chord-arc
+ llHr(f dz/Idzl)[l 2(r)
This is also useful to estimate
we can deduce (3.23) and
Thus
Cr (E) = 0.
curve.
Y(F) => Const
k ~ 2,
{IIC[a]I12,2; a ( ereal}
= ~
Y(r)
from below.
In effect,
from this inequality.
APPENDIX
For
I.
A N EXTREMAL PROBLEM
s I . . . . , s n E ~, w e define
Ts I,
..., s n
(x,y) = i/{(x-y)
+ i(Asl '
..., s n
(x) - A
s I'
--., s n
(y)},
where 0
A
x ~ I0 :
[0,i)
(x) s I, ..., s n
k-i k ( - -n =< x < -n'
Sk
1 =< k =< n).
Put
(4.1)
ex (n) = max {~(Tsl ' ..., s ); Sl'
.... s n E ~ }
•
n
(See (1.22).) Theorem G.
We show
Const
~TOg(n+l)
The first inequality a positive E c U~=_
integer
n,
[I 0 + ik/n],
to
I0
are mutually
by
x + iAE(X ) E E
the projection
of
TE(X,y)
~ ex (n) ~ C o n s t ~ g ( n + l ) is shown in §3.4.
F n E
denotes
For
(x E pr(E)) to
I O.
Let
such that
Proof.
and
T E''
and their projections
we define a function
AE(X)
= 0
AE(X)
(x ~ pr(E)), where
on
pr(E)
is
E E Fn on
for the proof.
and let WI
W I, W 2
and
be two disjoint
AE(X) ~ 0
f W 1 IrE(XW2f)(x)I 2 dx ~ Const
We define an operator
g Let
E E Fn,
For
such that
= i/{(x-y) + i(AE(X ) - AE(Y)) } .
AE(X) ~ 0
(4.2)
E c ~
We define a kernel by
Here are three lemmas necessary Lemma 4.1.
W e prove the second inequality.
the totality of sets
has a finite number of components
disjoint,
E
(n ~ i).
T E'
~ f_~ g(y)/{(x-y)
denote the adjoint
on
W 2.
Then,
IIXw2fiI~ .
by
- i A(y)} dy.
operator of
T E'
subsets of
Then we have
for any
pr(E) f E L2 ,
118
I T"g(x)I
< H*g(x) + Const Mg(x), nT~ llp ,P =< C P
which shows that
AE(X) - AE(Y) ~ AE(X) ~ 0
(p > i).
Hence
(x 6 W I, y 6 W2),
lIT~Ilp,p =< C P
we have,
(p > i).
Since
in the same manner as in
the proof of (2.9), I TE(Xw2 f)(x) I =< Const {M(T~f)(x) (x £ Wl),
+ IIrE[14/3,4/3' M(I Xw2fl 4/3)(x)3/4} which gives
(4.2).
Put 1
co
~(n) = sup {T'/UT-6TT'Ip~tLjl ~(E,f);
E E u~-' f 6 Lreal, 0 ! f O° Put
with respect to
'
Idzl
125
r
Sl' .... Sn
= {(x, A
Corollary 4.4.
Sl' .-., sn(X)); x E
II Sl, ..., s n
; s I, ..., s n ~ ~ } L (FSl '
), BM0(Fsl ' °°'~
=< Const ~log(n+l)
BM0(F).
Theorem G immediately yields
.
Const'~og(n+l)
max{IIH r
where
I 0)
IIHpII
SN
) °''~
S n
(n ~ i) ,
, L (F),BM0(F)
is the norm of
HF
as an operator from
L'(F)
to
APPENDIX II.
PROOF OF THEOREM B BY
P. W. JONES-$. SEMMES
Quite recently, P. w. Jones-S.Semmes gave a proof of Theorem B by complex variable methods.
The following note is their lecture in M~y, 1987.
(The author
expresses his thanks to P. W. Jones-S.Semmes who permitted the author to write here their proof (cf. P.W. Jones [33]). Lemma 4.5.
Let
F = {x + iA(x); x ( ~}
= {z 6 ~; Im z > A(Re z)} . extension, say simply
(4.4)
where
da
Here is a fact obtained by C. Kenig. be a Lipschitz graph and
Then, for any
g(z) (z ( ~ ) ,
to
g (L2(F)
having an analytic
~ ,
IIglIL2(F) ~ CM {ifzig'(z)I 2 dis(z,F)d~(z)} I/2,
is the area element and
CM
is a constant depending only on
M = IA'II. For
z ( ~,
z * = z - 2i(Im z - A(Re z)).
we write
Cf(z) = C(f d~IF)(z) (z £ ~) 1 2hi
~f(z) = For at
z ( F, z.
f(~) ~ - z
IF
we write by
~f(z)
z 6 F U ~,
we have
For
~f(z) = - i ~ 1 2~i
Cf(~)
(~ ~ ~ )
(Cf)"(z + i t ) t dt d~} dt
(Cf)'(w) --- - {Im w - A(Re w)} {~ + iA'(Re w)} do(w) (z - w*) 2
be a sequence of mutually disjoint cubes (with sides parallel to
the coordinate axes) such that 6(Qk )
the nontangentlal limit of
I~ {I F (If)'(< + it/2)t (z+(it/2) - ~)2
2
{Qk}i= 1
d~.
(Cf)i(z + i t ) d r = ~
~-y IIi Let
f 6 L2(F) , we put
For
, i.e.,
is the length of a side of
-
SSQk
~ = Ok=l Qk' Qk
and
dk/CM ~ Z(Qk ) ~ CMdk (k { i), where
d k = diS(Qk,F).
(Cf)'(w)
(= - w*) 2
Then
{Im w - A(Re w)}{~ + iA'(Re w)} do(w),
IfQk Ilm w - A(Re w) I do(w) ~ CM diS(Qk,F) 3
(k ! i).
127
Hence ~ " dk 2 ][CfIIL2(F) --