A Real Variable Method for the Cauchy Transform, and Analytic Capacity
This research monograph studies the Cauchy transform on curves with the object of formulating a precise estimate of analytic capacity. The note is divided into three chapters. The first chapter is a review of the Calderón commutator. In the second chapter
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1307
Takafumi Murai
A Real Variable Method for the Cauchy Transform, and Analytic Capacity
Springer-Verlag Berlin Heidelberg NewYork London Paris Tokyo
Author Takafumi Murai Department of Mathematics, College of General Education Nagoya University Nagoya,464,Japan
Mathematics Subject Classification (1980): Primary 30C85; secondary 42A50
ISBN 3-540-19091-0 Springer-Verlag Berlin Heidelberg New York ISBN 0-387-19091-0 Springer-Verlag New 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 c) r in the complex plane [, Hoo(r
and analytic capacity. For a compact set
denotes the Banach space of bounded analytic functions in supremum norm
i!·liHoo.
The analytic capacity of
y(f) ; s up ] I f ' (00) where
I;
C
with
)
is defined by
c)}, f E Hoo(r
II fllHoo ;; 1,
f'(oo); limz+oo z(f(z)-f(oo».
r
[U{oo} - r (; r
We also define
y+(f) ; sup{O/2rr)! du ; IlcjJllHoo ;; 1,
CjJ EHoo(r
CjJ(z) ; O/2rri)! 1/(1;-z) du G;)
¢ (the
c),
oJ,
u
where
We are concerned with estimating y(.)
(z
and y+(.).
support
0
u) ) •
To do this, compact sets having
finite i-dimension Hausdorff measure are critical. Hence we assume that finite union of mutually disjoint smooth arcs. Let Hausdorff measure (the generalized length). Let space of functions on L;(r)
r
denote the weak L
r
(l;;p;;oo)
wi th respect to the length element 1
space of functions on
r.
is a
denote the I-dimension denote the LP
LP(r)
where the infimums are taken over all compact sets transform on
i· I
r
Idz I,
and let
Put
E
r.
in
The Cauchy(-Hilbert)
is defined by
Then we see that P+(f) ;; p (f) ;; Canst where
1/3
Const P+(f) ;; l/II Hr IlL 1(f) ,L;(f) ;; Canst p+(f),
is the norm of
(Theorem D). Hence the study of
y(r)
Hr as an operator from (n to L;(f) is closely related to the study of Hr'
Here is a history of the study of the Cauchy transform on Lipschitz graphs. 2 According to Professor Igari, the L boundedness of the Cauchy transform on Lipschitz graphs waS first conjectured by Professor Zygmund in his lecture at Orsay in 1960's. Let line. Let
era]
r
{(x,A(x); xEEJ,
l/{(x-y)+i(A(x)-A(y»}. Cj a ]
a(x); A'(x),
where
E
is the real
denote the singular integral operator defined by a kernel
is bounded (from
Then the above conjecture means the following assertion:
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