Mathematics and Its Applications (Chinese Series)
Managing Editor:
M. HAZEWINKEL Centre/or Mathematics and Computer Science. Amsterdnm. The Netherlands
Volume 3
Vector-Valued Functions and their Applications by
Chuang-Gan Hu Department of Mathematics, Nankai University, Tianjin, People's Repuhlic of China
and
Chung-Chun Yang Department of Mathematics, The Hong Kong University of Science and Technology, Kowloon, Hong Kong
••
KLUWER ACADEMIC PUBLISHERS DORDRECHT I BOSTON I LONDON
Library of Congress Cataloging-in-Publication Data
5a Chuang-g n. 193 . Vector-valued functlons and their applications I by Chuang-Gan Hu and Chung-Chun Yang. em. - - (Mathemat ics and its applicatlGns. Chinese series ; p.
Hu,
3)
Includes bibliographical referen c es (p. ) and Index. acid free paper) ISBN 0-7923-1605-3 (HB I. Yang. Chung-Chun, 19421. Vector valued functions. II. Title. III. Series Mathematics and Its ap�lications. series : 3. QA331.7.H82 1992 5 15' .9--dc20
Chinese
91-46130
ISBN 0-7923-1605-3
Published by Kluwer Academic Publishers, 00 AA Dordrecht, The Netherlands. P.O. Box 17,33 . •. • \! ..
,
'
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© 1992 Kluwer Academic Publishers No pan of the material protected by this copyright notice may be reproduced or utilized in any form or by any means, electronic or mechanical, including photocopying, recording or by any information storage and retrieval system, without written permission from the copyright owner.
SERIES EDITOR'S PREFACE
'Et moi•...• si j'avait su comment en revcnir, je n'y semis point aUtI.' Jules Verne Tbe series is divergent; therefore we may be able to do something with il O. Hcaviside
One service mathematics bas rendered the human 1lICe. It bas put common sense back where it belongs. on the topmost shelf next to the dusty eanister labelled 'discarded nonsense' . ErieT.Bell
Mathematics is a tool for thought. A highly necessary tool in a world where both feedback and nonlinearities abound. Similarly. all kinds of parts of mathematics serve as tools for other parts and for other sci-
ences. Applying a simple rewriting rule to the quote on the right above one finds such statements as: 'One service topology has rendered mathematical physics .. .'; 'One service logic has rendered computer science ...• ; 'One service category theory has rendered mathematics ...•. All arguably true. And all statements obtainable this way form part of the raison d' etre of this series. This series. Mathematics and Its Applications. started in 1977. Now that over one hundred volumes have appeared it seems opportune to reexamine its scope. At the time I wrote "Growing specialization and diversification have brought a host of monographs and textbooks on increasingly specialized topics. However. the 'tree' of knowledge of mathematics and related fields does not grow only by putting forth new branches. It also happens. quite often in fact. that branches which were thought to be completely disparate are suddenly seen to be related. Further. the kind and level of sophistication of mathematics applied in various sciences has changed drastically in recent years: measure theory is used (non-trivially) in regional and theoretical economics; algebraic geometry interacts with physics; the Minkowsky lemma. coding theory and the structure of water meet one another in packing and covering theory; quantum fields, crystal defects and mathematical programming profit from homotopy theory; Lie algebras are relevant to filtering; and prediction and electrical engineering can use Stein spaces. And in addition to this there are such new emerging subdisciplines as 'experimental mathematics'. 'cpo'. 'completely integrable systems'. 'chaos. synergetics and largescale order'. which are almost impossible to fit into the existing classification schemes. They draw upon widely different sections of mathematics.•• By and large. all this still applies today. It is still true that at first sight mathematics seems rather fragmented and that to find, see. and exploit the deeper underlying interrelations more effort is needed and so are books that can help mathematicians and scientists do so. Accordingly MIA will continue to try to make auch books available. If anything. the description I gave in 1977 is now an understatement. To the examples of interaction areas one should add string theory where Riemann surfaces. algebraic geometry. modular functions. knots. quantum field theory. Kac-Moody algebras. monstrous moonshine (and more) all come together. And to the examples of things which can be usefully applied let me add the topic 'finite geometry'; a combination of words which sounds like it might not even exist. let alone be applicable. And yet it is being applied: to lltatistics via designs. to radar/sonar detection arrays (via finite projective planes). and to bus connections of VLSI chips (via difference sets). There seems to be no part of (so-called pure) mathematics that is not .in immediate danger of being applied. And, accordingly. the applied mathematician needs to be aware of much more. Besides analysis and numerics, the ttaditional workhorses. he may need all kinds of combinatories. algebra, probability. and so on. In addition. the applied scientist needs to cope increasingly with the nonlinear world and the extra
VI
mathematical sophistication that this requires. For that is where the rewards are. Linear models are honest and a bit sad and depressing: proportional efforts and results. It is in the nonlinear world that infinitesimal inputs may result in macroscopic outputs (or vice versa). To appreciate what I am hinting at: if electronics were linear we would have no fun with transistors and computers; we would have no TV; in fact you would not be reading these lines. There is also no safety in ignoring such outlandish things as nonstandard analysis, superspace and anticommuting integration, p-adic and ultrametric space. All three have applications in both electrical engineering and physics. Once, complex numbers were equally outlandish, but they frequently proved the shortest path between 'real' results. Similarly, the first two topics named have already provided a number of '·wormhole' paths. There is no telling where all this is leading - fortunately. Thus the original scope of the series, which for various (sound) reasons now comprises five subseries: white (Japan), yellow (China), red (USSR), blue (Eastern Europe), and green (everything else), still applies. It has been enlarged a bit to include books treating of the tools from one subdiscipline which are used in others. Thus the series still aims at books dealing with: a central concept which plays an important role in several different mathematical and/or scientific specialization areas; new applications of the results and ideas from one area of scientific endeavour into another; influences which the results, problems and concepts of one field of enquiry have, and have had, on the development of another. As the authors write in their preface, vector-valued functions tum up everywhere and they are a fundamental tool in physics, spectral theory, approximation and many other fields. Yet there was, so far, no systematic treatise on the topic. Here is one by two authors who have done quite a bit of wode in the field. At first sight one could think that a vector-valued function (instead of a scalar-valued one) would not bring so much new in the way of problems and theory. Just look at the components. It is far otherwise. There is a great deal more to vectorspaces, finite and infinite dimensional, then to scalars, and this takes an added emphasis in the case of vector-valued functions. To try to explain how much would take me far beyond the confines of an editorial preface; even if I could. It seems better to leave that to the authors, and the reader who wants to find out is therefore encouraged to pernse this unique volume. 1be shortest path between two truths in the real
domain passes through the complex domain.
J.Hadamard
Never lend books, for no one ever returns them; the only books I have in my bbrary are books that other folk have lent me. Anatole France
La physique ne nous donne pas seulement
I'occasion de ~udre des problMles ... eIIe
1be function of an expert is not to be more right
nons fait presseuIir Ia solution.
!ban other people, but to be wrong for more
H.Poincar6
sopbisticated reasons. David Butler
Bussum, 9 February 1992
TABLE OF CONTENTS
SERIES EDITOR'S PREFACE
v ri
PREFACE Chapter 1. Theory of Normal Families
1
1. PRELIMINARIES 1.1. The definitions of n(r. a). N(r. a). b(r,f). q(r,f). m(r. fl. mir,f) A(r. f). S(r. f) and T(r. f)
1
1.2.
Convex function
2
1.3.
1.4.
The properties of A(r,f) The properties of S(r. f)
3 5
1.5.
The properties of m(r,f' If)
10
1.6. 1.7.
The relationships among T(r,f). T(r. f') and N(r. fl. etc. The infinite product representations of a meromorphic function
16
22
2. THE NORMAL FAMILY OF MEROMORPHIC RJNCTIONS 2.1. The concept of normal family 2.2. The properties of the normal families of merom orphic functions
26 26 27
3. THE DISTANCE OF A FAMILY OF RJNCTIONS AT A POINT
38
4. ON MEROMORPHIC RJNCTIONS WITH DEFICIENT VALUES
42
5. THE APPLICATIONS OF THE THEORY OF NORMAL FAMILIES
53
6. APPLICATION TO UNIVALENT RJNCTIONS
60
viii Chapter 2. HP Space
68
1. HARMONIC 1.1. 1.2. 1.3. 1.4.
68 68
1.5.
AND SUBHARMONIC FUNCTIONS Harmonic Functions Boundary behaviors of Poisson-Stieltjes integrals Subharmonic functions The convexity theorem of Hardy Subordination
2. THE BASIC STRUCTURE OF lP'
71 73 76
78 79
2.1.
Boundary values
79
2.2. 2.3. 2.4.
Zeros The meaning of converging to the boundary values Canonical factorization
82 84 88
3.lP' IS A BANACH SPACE
3.1. 3.2.
Poisson integral and HI Banach space
Chapter 3. Vector-Valued Analysis
90 90 92 94
1. VECTOR-VALUED FUNCTIONS 94 1.1. Vector-valued bounded variation 94 1.2. Vector-valued integration 95 1.3. Vector-valued HOlder condition 100 1.4. Vector-valued regular functions 102 1.5. Compactness and convergence in the space of vector-valued regular functions 107 1.6. Boundary properties of a class of the vector-valued regular functions in S(O, 1) 110 1.7. Vector-valued elliptic functions 113
IX
2. VECTOR-VALUED BOUNDARY VALUE PROBLEMS 2.1. 2.2.
Vector-valued Cauchy type integrals Vector-valued singular integral equations
2.3. 2.4.
Vector-valued doubly-periodic Riemann boundary value problems Vector-valued boundary value problems with the boundary being a straight line
2.5. 2.6.
The solutions of the vector-valued disturbance problem The vector-valued boundary value problems in lp
3. THE ANALYSIS OF LOCALLY CONVEX SPACES 3.1. 3.2.
G-differentiability Vector-valued regular functions in locally convex spaces
117 117 119 123 127 133 137
145 145 148
mBLIOGRAPHY
151
INDEX OF SYMBOLS
154
INDEX
157
PREFACE The theory of vector-valued functions with one variable is one of the fundamental tools in studying modem physics, the spectral theory of operators, approximation of analytic operators, analytic mappings between vectors, and vector-valued functions of several variables. It seems that, thus far, no book specialized in dealing with vector-valued functions of one variable has been published in the West, except for some brief introduction to such functions in some books dealing with functional analysis or function theory. The present book will be a remedy for satisfying such a need. It consists of three chapters: theory of normal functions, If' space, and vector-valued functions with their applications, and it incorporates a lot of original research work obtained by the authors and others. The reader of this book is assumed to have a basic knowledge in real and complex analysis. The material of this book has been used as lecture notes for an optional course for senior mathematics and physics students at Nankai University. The approach of combining together function theory, functional analysis, and the theory of boundary value problems and integral equations has not only made this book unique, but has also benefited students in developing their ability in abstracting and generalizing mathematical theories. We would like to express our sincere thanks to Dr. DJ. Larner, the publisher of Kluwer Academic Publishers, for his interest and support in publishing this book. We'd like also to acknowledge the excellent typing job done by the Apparatus Office of Nankai University and Tianli Company, and to thank our students Mr. ZJ. Wang, Ms. D.P. Zhang and Mr. G.T. Ding for checking the final proof of this book. C.G. Hu C.C. Yang
CHAPTER
1
Theory of Normal Families
1. Preliminaries
1. 1. THE DEFINITIONS OF n(r,a), N(r,a), b(r,f), q(r,f), m(r,f) , mz(r,f),
A(r,f), S(r,f), AND T(r,f)
Let fez) be a function meromorphic in the disc IzlO, the following inequality
1 ~7f2 (k+
i )~1 +
(log k)2+ 7f 2
holds. Hence
1 ~7f2 ( If(z) 1+ Iffz)
I )~1 +
(log If(z)
I )2+n2~1 +
llogf(z) 12,
consequently
f' (z) 1+n2 2 I~I ( - 2 - ) 0+ llogf(z)
If' (z)1 2 If(z) 12)2 •
12)2~0+
The assertion of the theorem follows from this and the definitions of A(T,log fez)), and A(T,f(z) ).
Theorem 1. 1. 9. Let f (z) be a mel'omarphi.c fUnction in the finite complex plane with f ( 0) #1 0,00 and If(O) I>e, If(O) l>e(e>O). Then there exists a constant K(e,j(O)) depending
on f ( 0) and e such that A(r ,log
f)~ (e+-!.Y A(r ,f) +~ {S(p,f) + K(e,f(O)) } e p-r
(r 1 / e. Thus
It follows that to any rand p satisfying rr
A(p,wl) +p+r 2(p-r) 2 :::>-: 1 -::- 27r(p-r) Thus by the corollary of Theorem 1. 1. 2 and the above, we have
Wg{ A(p,wl) +p+r} 2(p-r) 2
~Wg
{27r(;-r)
f:
1+IL12
fo- (I + I~d 2) dt dO}
1
THEORY OF NORMAL FAMILIES
11
(1. 1. 3) Also
(1. 1. 4) Since
it follows that
).,+e. Thus
ttm T(1',f)
O.
o
r_oo
Consider the case that f(O)#O or
00.
r ,+1
Then for f(z)#O,oo, by Poisson-Jensen formu-
la,we have
loglf(~) I~+z
log f(z)=J... f2n 2:rr 0
~-z
dO-
L;
log
1• .I';;p
a~z-p2
p(z-a~)
+ Ib.I';;p L; wg b"Z- p +ic (~=pe~). p(z-b.) 2
0.1.43)
Taking derivative with respect to z, we obtain from above d wgf(z)
dz
+ L;
_1__
IG.I';;;p z-a~
L;
_1_.
Ib.I';;;p
z-b.
0.1.44)
By taking the p-th derivative of the above, we have d,+1 dz'+ 1 log (z) -(-1)',
=
(p+ 1)! J2n 2~ 2:rr 0 log If(~) I (~_z)'+2dO
p!
(1. 1. 45) In order to derive the limit of the right side of (1. 1. 45) when p-oo, we denote
the right side of the above equation as =/ 1 +/ 2 +/3 ,
CHAPTER
24
First of all, for Iz I 0,
L; of radius 1/2,
the values w of w=f(z) have the
there exists positive no (e, z) such that whenever m>
n>no, the corresponding spherical distance of two points fm(z) , f.(z) satisfying d (f.,(z) ,f.(z))
<e
{f. (z)} is said converges spherically uniformly meaning in general sense that if G is
an arbitrary closed subdomain in D, then the previous mentioned quantity no(e,z) will ofl-
27
THEORY OF NORMAL FAMILIES ly depend on E and e, but independent of the
Zl
s that belongs to G.
Definition. Let F be a family of functions that are meromorphic in D. F is said to be normal at a point
zE D if it is normal in some neighborhood of
z
0. e. ,in an open disk cen-
tered at z with sufficiently small radius). Theorem 1. 2. 1. Let F be a famuy of functwns that are me1'Omarphic m a dOllUJin D. A TlRCes-
8Ul'Y and suJJICIRnt conditwn far F to be naT'mal m D is that F Proof.
t,8
narllUJl at each point z m D.
The necessity is obvious.
We now prove the sufficiency. According to the hypothesis, to each point z in D corresponding a open disk C, such that F is normal in C,. We can construct a closed disk W, that lies inside C,. Then any sequence of functions {f. (z)} from F , a subsequence {f•• (z)} can be so chosen that it converges spherically uniformly in the general sense in W, I
noting that D can be covered by a countable subset {W.,> of the family {W.}(zED). Let {f.(z)} be an arbitrary sequence chosen from the family {f.(z)} , then there exists a subsequence {f1 .• (Z)} of {f.(z)} which converges spherically uniformly on W'I' Next from {f1 .• (Z)} a subsequence {f2 .• (Z)} can be chosen so that it converges spherically uniformly on W'2' And so on , in general, if a subsequence {f.-l .• (Z)} has been selected, then from it a subsequence of {f•.• (z)} can be selected so that the subsequence of (f •.• (z)} converges spherically uniformly on W'.' Obviously, {f•.• (z)} converges spheri-
cally uniformly on W'I
U W'2 U ... U W•• ' Now
choose, from the family of sequences {f•.•
(z) } (k= I ,2 ... · ,n= 1.2",,) constructed above, the sequence {f •.• (z)} (n=
1,2,,,,),
We are going to show that {f •.• (z)} converges spherically uniformly in the general sense on D. Let W be an arbitrary closed domain in D. Then a finite number of the W, s will COver D; let ko denote the largest subindex appeared on the W.s. Then clearly for
n~ko,
the corresponding subsequence {f.o" (z)} is a subsequence of the sequence {f•.• (z)}, therefore, it converges spherically uniformly on W.
1
UW, U·" UW, 2
10
and hence. on D.
2·2. THE PROPERTIES OF THE NORMAL FAMILIES OF MEROMORPHIC FUNCTIONS
Let d (WI' W2) denote the arc (interior one on the great circle) length of two points
28
CHAPTER
W' I ,W' 2 on point
WI
1
the Riemann sphere, which are stereographic projections of the two planar
,W2 respectively. Now let d(wl ,W2) denote the length of the chord WI' w'
2,
then it
is easy to verify that
Let
then d(WI ,W2) =-J(~1-~2)2+ (711-712)2+ «(;1-(;2)2 IWI- W21
-J 1 + 1WI 12 -J 1 + 1W2 12 1
I wl -
1 W2
1
O. 2.
1)
Define
and note
d(z, 0) =dO/z,oo). It follows that when (1. 2. 2)
or
(1.2.3) then
(1.2.4) or
0.2.5) We note that when
-
d (WI' W2)
< 1-
2
-Jl +A
, A> 1
(1. 2. 6)
2
holds then either (J. 2. 2) or (1.2.3) must be true. The reason is as follows: suppose
THEORY OF NORMAL FAMILIES
29
that neither 0.2.2) nor (1. 2. 3) is true, we may assume without loss of generality that
1
1WI 1~A, -1-1 ~A. Then W2
from
we have
1
= 1-
~ ---;:=:====
1+-1WI-1 12
1
1 A
1
2
This contradicts with (1. 2. 6). From the above observation we see that the statement: sequence {f. (z)} converges spherically uniformly in a domain D is equivalent to say that either If.,(z) - j.(z) 1 <e or
If)z) - j)z)
I<e must
be held for m>n>n(e,w), w is an arbitrary closed subset of
D, which also is equivalent to say that to any point
Zo
ED, there corresponds a neighbor-
hood _V(zo) so that either {f(z)} or {j)z)} converges uniformly in N(zo). It follows that if {f.(z)} converges spherically uniformly in a domain D, then its limit function is always meromorphic in D.
Theorem'.
2. 2.
If the sequence oj meromorphic junctions {f.(z)} converl}f!S spherically unir-
farmly to fo(z) on the closed aomain W,then the sequence
+
. afo(z) /l verges spherically umfarmly to vjo(z) +0'
PrOOf.
{~j:~:;t;}
(ao-/lv=J:.O) afro con-
CHAPTER
30
Now if
1ao-
1
py 1 #0, then to any w,
1+ Iwl 2
20. Now choose 1! sufficiently large, then according to the hypothesis of the uniformly convergent, it follows that 1f. (z) 10, and by the hypothesis of the theorem, f.(z) fo(z)
has no zero on
1 z-
Zo 1 ~ 2To; {f. (z)} converges uniformly to f 0 (z) on
1z
- Zo 1 ~ 2To.
Therefore, {fl .(z)} converges uniformly to fl o(z) on Iz-zo 1 =To. Now since on IzZo 1 =To, 1 f. (z)
1
>0,
and I' • (z) / f. (z) converges uniformly to
I'
0
(z) /10 (z), therefore
for sufficiently large nl, we have, for n>1Il .(z) _f' o(z) In> N.;. we have
THEORY OF NORMAL FAMILIES
33
0.2.9) Choose
~V=1TI£lxN •• l~i~,
Now to any
zEE, it belongs to some
C •• Hence
I
,
It follows, from the hypothesis of spherical equicontinuity,
d(f •.• (z) ,
1
f •.• (z.,»n>N, d(f •.• (z)
,f..... (z) )<e,
where .:Y depends on e and W, but is independent of z. This shows that {f •.• (z)} is spherically uniformly convergent on W.
Theorem ,. be normal
~n
2. 6. A necessary and sufficiRnt cmulitUlTl far a family of mermnarphic functinns F
to
a domam D is that F is sphl!rically equicontin1U.JUS on arbitrary cwsed suMmnam of D.
Proof. The sufficiency follows from Theorem 1. 2. 5. We only need to prove the necessity. Suppose that F is not spherically equicontinuous on a certain closed subdomain D. This assumption implies that there exist a positive number e
> 0,
we
two sequences of
points {z,} and {z'.} in E, and a sequence of functions {f.(z)} in F such that
Iz.-z' .I 0, and sequence of functions {f. (z)} and sequence of points {z.} can be chosen from F and W respectively such that
d(f.(z.) ,f.(z.' ) )~e
(Iz.-z.'
1 1 0, to any
fEF whenever
Iz'
-zol0.
Proof. Since O.({f},zo)=w>O, a sequence of functions (f.(z)} and sequences of
points {z.} and
{z~}
can be chosen such that
Iz. -
Zo
I< -.L, Iz~ n
Zo
I< -.L, n
and
1
d(f.(z.) , f.(z'.))>Zw.
The fact that meromorphic functions are continuous spherically and the condition Z.-Zo, z' '-=0 indicates that none of the functions in (f.(z)} can occur infinitely many times. It follows that any subsequence of (f(z)} will not be convergent spherically uniformly in
z- Zo I ~o; 6 an arbitrary positive number. We now know that when (f.(z)} is not normal at point zo, then (f.(z)} will not be spherically uniformly convergent in a neighborhood of zoo However, we note that there may be a subsequence of (f.(z)} convergent spherically uniformly in a neighborhood of zoo For instance,
ty on
Iz I =
a function f (z) that is regular in Iz I< 1 having at least one singulari-
1 can be represented by a Taylor series:
CHAPTER
40
1
fCz)= ao+alz+···+a.z·+···,
which converges uniformly in the general sense 0. e. , in closed subsets of Iz 1< 1). It fol-
.
lows the sequence {S.Cz)}
= { L>.;!} .=0
of the partial sums of f converges uniformly in the general sense in Iz I
< 1.
But it is pos-
sible that there is some subsequence {S., Cz) } of {S. Cz) } satisfying suitable condition will be convergent at a neighborhood of a point on the circumference
I z I = 1,
say z = 1.
When this phenomenon occurs it is called overconvergent.
Ostrowski's theorem. Let
~a'z'
fCz)=
• -
be regular in
no 0 be
gwen
and
< ... O+e)nU-1
Suppose now that whenever n salisfiRs
n2t-1
Ck=I,2, .. ·).
< n< n2' Ck = 1 , 2 , ••• ) then a. = 0 in the above expan-
sinn. Then '21+1
{S'Cz) } = { ~ a.z' } • = 0
converges unifarmly in a sufficienay small neighborhood of zoo
Pr()()f. Choose integer
p> l/e,
and let 1
z=2zoC~'+~'+I) ,
fCz)
then determine bo , bl
, ... ,
=gC~),
b.. , ... , so that the following equations hold:
Ck=O,I,2· .. ). Thus 00
"21+2- 1
gC~)= ~ ~ .t --
0 --"21
b.. ~
THEORY OF NORMAL FAMILIES
Noting that
~0 Ia.1 z·
Iz I < 1,
converges in
• =
Iz I< 1. It follows
is convergent in
41
we can conclude that
that (1~11.
Thus fez) cannot be regular at any point on
Izl =1.
Theorem 1. 3. 3. Let (f.(z)} be a family of functinns that are 7neTQ/1UJT'phic in a certain domain. Then at any of lis ab7wrrMl point zo, 0.( {f} ,zo) =w=
;.
CHAPTER
42 ~VofR: Thus
1
a family of merom orphic functions at any of its abnormal point has not
only w>O but also is impossible to have O<wp>r>
ro, with suitably fixed ro> 0, m(r,f)r>mllx(ro ,R-1),
1
m(r,f)~Mo+2{2+Log2+2Log R-r
+ [{
J
(00, R, ro)
+ log , I' ~ 0) , }.
(1.4.4)
Note in which M is a constant that satisfies
1
x-12log+x~2x, x~M.
It follows that if
II'
(0)
I~ :~,
then
m(r,f)~Mo+2{2+Log2+2Log --R 1 +[{J(oo,R,ro) + r However, if P.
II'
II' (z)
(0)
I ~ :~,
I RH if n>max(R-l ,ro), then RA+r 2 m(r,g)~Mo+ {2+log2+2log r(r-A) (R-r) 00 2r2R } +A1(z,R1,ro)+log 00(r2-A2) . T
But from 0.4.7), we have r+A
m(r,J)~-Am(r,g)
r-
hence, for
r>max(R~A ,R-l ,ro), 4R
7)
,
we obtain
3R
m(r,f)~R_r {Mo+2(2+log2+2log (R-r)
Hence, for
45
THEORY OF NORMAL FAMILIES
+[{)(~ ,R),ro)+log oo(:~r)J}.
(1. 4. 8)
Noting that m(r,f) is an increasing function of r, we have for
R>r>max(A,R-l ,ro), R+r
12R
8R
m(r,f)~m(-2-,f)~R_r {M o+2(2+log2+2Iog (R-r)2
• 00 2R2} +A)(2,R,r o)+log oo(R-r)J •
Thus when Izl~A,
Hence, for r>max(R-l ,ro), we have
1 00 m(r,f)00•
Since f.,(z) is regular and#O,I, we have, for Iz-zol
t8
Proof. We may assume without any loss of generality that F(O)=O· max If' (z)
a constant uuJependent of fez).
f (0) = O.
Set
I, Izl~l-O.
Then F(O) is continuous on the interval O~O~l with F(O)=O,F(l)=1. It follows that there exists some 00 with O-plane draws a circle of radius larger than log ( f t
~
I
+ 1 ) + ~i I (particularly a circle of radius 1) that will contain some points from
the set Z. We calculate g' (z) the derivative of g(z). If for some 0 such that g' (~)#O in I~
I
and hence Ks(f(O),(})
most general form of Schottky theorem. Theorem 1.4. 3.(Schottky). If fez) is regular with If(O) I<M and omits vahles 0,1 in 1 Izl ~
Iz
I
0, sm z takes every finite value an infinity of times in IArg z I <e. In general a meromorphic function f (z) with a period w has Arg z=w,Arg w+:n- as its Julia directions. By Picard theorem fez) takes every value, with two possible exceptions at most, an infinity of times; moreover, if f (zo) =
a, then f(zo±nw) =a. Hence, to any given e>O, when n is sufficiently large, z±nw will fall into IArg z-Arg
wi <e, IArg
z-Arg w-:n-I <e respectively.
3. Every direction is a Julia direction for the
Weierstrass elliptic function:9(z).
Since, to any given rp and e>O, the sector:rp-eb 2 , · · · . If limN(n)=O, then the set: n boundary of F(z) ;
Izl=l
becomes the natural
IV I . (Polya). Let N(n) denote the number of nonzero terms in the first n terms of
. N(n) the sequence: ao, al , a2 , .... If l~m - - = 0, then every direction is a Julia direction for f 11_00
n
(z). Theorem 1. 5. 3 can also be rephrased as follows. Let fez) be a transcendental entire function. Then given e> 0, there exists a sequence of points {z.} with z.-oo such that in
Iz-z.l<elz.1
O. 5. 1)
fez) takes every value, with two possible exceptions at most, an infinity of times.
Condition 0.5. 1) can also be replaced by assuming
THEORY OF NORMAL FAMILIES
57
Iz-z.I<e.Iz.I,
(1. 5. 2)
e.-O
where {€.} is a sequence properly chosen. Moreover, it is known (Valiron-Milloux) that the rate of decrease of {e.} is related to the increasing property of T (1' ,f). The same conclusion holds for a function fez) which is regular in a neighborhood of an isolated essential singularity zo, or f may not be regular but omits at least one value in a neighborhood of zoo Generally speaking, if there exists a continuous curve L which tends to the isolated essential singularity
Zo
such that
f(z)-a as
Z-Zo
along L, then
Theorem 1. 5. 1 holds.
Theorem 1. 5. 4. Let z= 0 be an isolated essential singularity of the functllJll f(z), which is me1'amorphic
HI
0< 1z 1< 41'0.
0 such 1'40< 1z 1< 41'0, ii
If there exist a value a arul a continuous CU1've L COII1X!T'gmg to
that f(z)-a as z-O along L, then (f.(z)}
=
(f( ;. )} is not normal
UI
follows that there exists at least one di1'ectwn: Arg z = 'Po such that, for any gwen e> 0 , f (z) takes
every value an mfiniiy of times, with two possible exceptwns at 1nost, in -e:::::: -
mal value on
1z 1= 1- ~.
f~Z)
But since
0 an d 2'
" cannot be zero, w h'ICh ta k es t h e mml-
is equal to
1 at
the origin, If(zo)
1~ 1Zo 1
p(e),Laop
lat I)
1_,1O) can be decom-
posed as
fez) =B(z)g(z), where B (z) is tke Blaschke product, g Cz)
E H' and hns
1W
zeros in
Iz I< 1.
Similarly, to every fEN, it can be decomposed as f=Bg, where g beWngs to N and does not vanish in
Iz I< 1.
Proof. We may assume that f has infinitely many zeros (otherwise, the conclusion is obvi-
ous). Let
B.(z)=z'"
ir~ t=1
Then for a fixed nand e>O with -e. Hence
at
Izl
a.-z
1-iit z'
fez) g.(z)= B.(z)'
being sufficiently close to 1, we have IB.(z)
1>1
85
M
~(1-E)'· The value of the integration is thus bounded by a quantity which is independent of rand decreases monotonically with r. From the above, by letting e---O, we obtain
f:" Ig.(re
~M
l'dO
08 )
(2.2.1)
(n,r«l) arbitrary).
By virtue of Theorem 2. 2. 4 we have by letting n-oo, ..,f.....I'
g.(z) -
fez) g(z)=8(z)
This shows that g E H' and has no zeros in
(lzl=R 0, we can choose a set QoC Q of finite measure such that J(Qo)<e.
Now choose 0'>0 such that to every measurable set QCQo it will satisfy J(Q)<e. whenever m(Q)<e. By Egorov' s theorem, there exists a set QCQo(with m(Q)O), then lim r-l
f
2"
1log +
0
IfCre oll ) 1-l.og+ IfCe oll ) lido = O.
The above corollary can be derived directly from Theorem 2. 2. 6 and Lemma 2 be low.
Lemma
2. If
a~O, b~O,O0, we can, by
Theorem 2.2.6, choose pO, b>O, OO there exists an 0>0 such that III(t' ) - f(t")
-t" IO such that VaT {cpC!Cl) }~Mllcpll.
CHAPTER
98
3
As a consequence
lis"
1
-S"
2
11= 0.,11=1 $UP Icp(s"
1
-s" ) I(J(z)). We write
f(z;a,p)=~p{~(J(z+a)- f(z))a-
a
pI (J(z+P)- f(z))}.
The Lemma 3. 1. 5 then asserts that 19>(f(z;a,p))I~A1(g>,f,8)
for every choice of z, z+ a and z+ P in 8. By the theorem of uniform boundedness. This implies the existence of a finite A1 (J , s) such that Iif(z;a,p) 1I~A1(J ,s). If We take the limit as
P- 0, we obtain
1I~(J(z+a)-f(z))-f' (z)II~lalA1(f,s) a for all z and z+a in 8.
Theorem
Q.E.D.
3. 1. 10.( Cauchy). If f (z) is a regulnr vectar-vabJ.ed function on the aomam
D with
values Ln the B-space E, then
J(,f(Z)dZ=O far ev(!T'Y simple closed rectifW/k contour C in D such that the interior of C belongs to D.
CHAPTER
104
3
Proof. For any linear bounded functional rpE E' we have rp( tf(Z)dZ)= L,rp(f(Z))dZ=O,
hence
t
f(z)dz=(}.
Theorem 3. 1. 11. (Cauchy integral formula). ut f (z) be a reqular functwn on the domain D wah values in E. Let C be a closed path in D, the interim of which
/,8
in D, and let Z be such that
arg (t - z) increase,s by 2n when t de,scrWe,s C (positit'f! oriRntatwn). Then
J
f(t)dt f (')(z)=~ 2ni c (t- Z),+1
for n=0,1,2,···.
(3 1 10) • •
Theorem 3.1. 12.(Cauchy-Hadamard). Given the power series 2:,; a, (z- zo)', a, E E.
(3. 1. 11)
,-0
Set
~=lim P
suplla, 11+. Then the series is absolutely COlll'f!T'gent for
Iz-zo I
p. The
series cont'erges to a requiar function on Iz - Zo I
O by the compactness of S we get ,
}dIS (Zt
1
J
,o(T e) )-:::>S,
where S(Ztj,o(; e)) is the disc with the center Zt J and the radius 0(; e). Since {f•.• (Zt)} is convergent, J
For arbitrary zES there exists S(ZtJ,o(+e)) such that zES(Z."o(; e)). Since A is sequentially compact, A is bounded. It follows that {f} is uniformly bounded on S. Thus {f} is equicontinuous on S. We infer that 1 IIf•.• (z) - f •.• (zt,) 11N= max Nt. Therefore ,
1(z)=~
c
21J"'
[{(z,t)dt t-z
(2)1>(z)EVH(II-)
UI
the closed
[{ E VH (11-). Then
is regular when zisin D,
as zED,
(3)1>+(t)=~[(t,t)+~ 2
2m
J c
[{(t,-r)d-r -r-t
for tEL.
Proof. By methods similar to those of complex analysis we can obtain that 1>+ (t) exists.
For any rpE E' we have that rp
(1)(z))=~ 2m
J c
rp([{(z,t))dt t-z
is regular in D, thus (/J(z) is a vector-valued regular function. On the other hand, (rp( 1>(t)) + = rp( 1>+ (t))
=
1. 21 rp([{(t,t)) +-2 1J"~
J c
rp([{(t't-r)) d-r (tE e). -r-
Therefore (3) holds. (1) and (2) are obvious.
Note. For each rpE E' , if (rp(1>(t))+ is defined, we may not get that 1>+ (t) is defined.
For example, we take 1>(z) = (1 ,Z,Z2, ••• ,z',"·) in l2' where (aO,al""
,a.'''')El2=l~,
we have rp«(/J(z)) =
~a.z·. Hence
zE S(O, 1). For any (rp(1>(e olJ )))+=
..... 0
rp=
~a.e·oII
11=0
is defined, but 1>+ (eolJ) can not be defined.
2.2. VECfOR-VALUED SINGULAR INTEGRAL EQUATIONS
In this section, we shall deal with vector- valued singular integral equations with range in a Banach algebra Eo.
CHAPTER
120 A(t)y(t) +~
1n
J
K(t, 'f) y( 'r)d'r=j(t) as tE C, 'r-t
3
(3.2.2)
where C consists of a finite number of arc-wise smooth closed curves which do not cross each other; D+ is the inner domain which is on the left of C traced along a positive direction; A(z) is a reguar vector-valued function which is on D+ with values in Eo and satisfies a vector-valued Holder condition on
D. For each fixed wE 15,
K(z,w) is a regular vec-
tor-valued function in D+ and for each fixed zED+, K(z,w) is a regular vector-valued function in D+ and satisfies a vector-valued Holder condition for both z and won
D.
The
function f(t) satisfies a vector-valued Holder condition on C. Under the conditions just mentioned, we shall study the representation of the solution yet) of (3.2.2).
Let B(z)=K(z,z), then B(z) is regular in D+. In fact, for arbitrary rp
qJ
(B(z»
rp
is regular by Hartogs theorem (see[2]). CDThus B(z) is regular in
(B(z» =rp (K(z,z» D+. Since
EE' ,
E H(p,) on 15,
SupposethatA(t)±B(t)#8
B(z) E VH(p,) by Theorem 3. 1. 6. fortEC, thatA(t)+B(t) haszerosal,···,a.. , with
multiplicities A. for k= 1, "',171 respectively, that A(z) -B(z) has zeros Pl'" ,P. with multiplicities
p,j
for j=l,"',n, that (A(t)±B(t»-1 exists for z in
15
except at the ze-
ros of A(z)±B(z), and that both of
dA. -d 1 (A(a.)+B(at»
(3.2.3)
z•
have inverses. Under the above conditions we obtain yet)
CD
=
(A(t)
+B(t) )-1 A(t) (A(t) -
B(t) )-If(t)
A function F(ZI,Z2) defined in a region D of the space O. is said to be a regular analytic function if the
partial derivatives
aaf
%1
and
aaf
exist at every (interior) point of D.
Z2
F. Hartogs was the first to prove that if F(ZI,Z2) is a regular analytic function in a bi-cylinder
1ZI 1~rl'
1z.1 ~r.
that converges absolu tely and uniformly in every closed subregion of (
(
* ).
*)
VECTOR-VALUED ANALYSIS
121
•
+4(A(r)+B(r»-I,6
"'j-I
,6 Bjr(t)Cp •
(3.2.4)
j= 1 r=O
where
(3. 2. 5) for r
= O. 1 ••••• Ilj - 1 and j = 1 ••••• n.
arul Cjr are defined by the following linear equa-
tions: •
,uj-l
~ ~ :~ «A(z) 1 =-4 nt
f aaa ( ~
B(z» (A(z)
+ B(z»
- I Bjr(z)
),=a. ,Cjr
(A(z)-B(z») (A(z)+B(z») -IK(z.r) ),=a i
C
for
(7=
0, I ••••• A'_I and k= 1 •••• • m.
(3.2.6)
The formula (3.2.6) has a solution exactly when (3.9.1) does. We shall only give some of the main steps needed in the proof the result. If equation (3.2.6) has solutions C p
•
then we can let
ji(z) = (A(z) - B(z» (A(z)
{2If K(z.r) _ (A( r) 1n c
r
i. e. y(z) is defined on
• 2,6 ,6 Bjr(z)Cjr } • "j-l
B( r) )-If( r)dr-
z
D.
+ B(Z»-I (3.2.7)
j= 1 r=O
In fact. by (3.2.3) and Theorem 3. 1. 16 we can conclude
that (Ae;;) ±B(z) )-1 has poles of order Ilj at points a, and Pj. and is regular except for points a, and pj. Because (A(t) - B(t) )-Iexists on C. (A(t) - B(t» -I. K(z.t) (A(t)B(t) )-If(t) and K(z.t) (A(t) - B(t) )-1 (t- pj)' E VH(Il) for each fixed z on D+
It is clear that the first term and second term in {
U C.
} in (3.2.7) are regular. Thus ji(z)
is defined for zE D. By Lemma 3. 2. 1 we show that ji+ (t) exists. On the other hand from (3. 2. 3). Theorem 3. 1. 15 and Theorem 3. 1. 16. we get
1
I}
{ I«z.r) r-z (A(r)-B(r»- (r-pj)r
Bp(z)=:;:(,es
T=P )
for r=O, L.···.llj-1 and j=I.···.n, where Djl(t)=(t-pj)~J(A(t)-B(t»-1
and Cp=y(r)(pj)'
CHAPTER
122
3
Considering a,o:/=-pj and setting z= Pi ,j= 1, "',n in
+ B(z) )-1
y(z) = (A(z) - B(z)) (A(z)
(3.2.8) "1- 1
we see that (A(z)+B(z))y(z)
and-2(A(z)-B(z))
~Bj,(z)y(')(P) have identical r=O
expansions to order jlj-1 at points z= Pj' Letting H/z) =B(z)Djl (z),
we compare values between FI (z) =Hj(z)y(z)
and
for the corresponding cases. Since (3.2.3) holds, we get that Hj(z) are regular in a neighborhood U(P) of the points pj for j= 1, ••. ,no From Theorem 3. 2. 2 we infer that
~
2:rn
J [{(z,-r) (A(-r)-B(-r))-If(-r)d-r T-Z ('
is regular in D+. From the condition (3.2.6) we can verify that y(z) is regular in lI(/l,) for j=l,···,n, as is F1(z). Now the expansion to order jlj - 1 at z= pj of F 2 is F 2 (z) =Hj(P)Y(pj)
+ (Hj(z)y(z)) , (z- pj) + ... fJ
Hence (3.2.8) holds. Let z-t, where zED+ and tEL, from Theorem 3.8.2 and f(t) - [A(t) - BCt) J[A(t) = [A(t) - B(t) J[A(t)
+ B(t) J-I B(t) [A(t)-B(t) J-I f(t)
+ B(t) J-I A(t) [A(t) -
B(t) J-I f(t)
we can obtain through simplification yet) = [A(t) - B(t) J-I [f(t) - 2y+ (t) J.
(3.2.9)
Since A(t)-B(t), f(t)-2y+(t)EVHCjl), by Theorem 3.3.2 and Theorem 3.3.3 we obtain that yet) E VH(jl). From(3. 2. 8)we infer
1. [ACz)+B(z)][A(z)-B(z)]-ly(z) =-2 !ITt
J
[(Cz,-r) [AC-r)-B(-r)]-lfCT)dT
('
-r-z
VECTOR-VALUED ANALYSIS
123
Using (3.2.9), Theorem 3. 1. 14 and Theorem 3. 1. 16 yield
1. y(z) =-2 :Tn
-~ :Tn
J o
J
[{t(z,t) [A(t) - B(t) ]-If(t)dt
-z
o
[{(z,t)[A(t)-B(t)]-ly+(t)dt t-z
=_1_ 2:Trt
J
[{(z,t) (t)dt. t-z y
(3.2.10)
C
Now Lemma 3.2.1 gives
1 y+(t)= 21 B(t)y(t) +-2
:Tn
J
[{(t,T)y(-r)d-rk. T-t
o
(3.2.11)
When used in (3.2.9), (3.2.10) and (3.2.11) imply that yet) satisfies equation (3.
2.
n.
2.3. VECTOR-VALUED DOUBLY-PERIODIC RIEMANN BAUNDARY VALUE PROBLEMS
I'DF is the set of all vector-valued doubly-periodic continuous functions on Po (see
Sec. 7) with periods
WI
and
~,
the range of which are in E.
Suppose an operator [{ is defined by the formula [{f=
-b~t) Jf(T)(~(T-t)+~(t))dT' :Tn o
where fCt) ,bCt) E VDFn VH(p) , and ~(t) is the Weierstrass ~-function.
Theorem
3.2.5.
If f,bEVDFnVHCp), then [{fEVDFnVHCp) and II[{fll~~qllbll~lIf
II", where q is a constant and Ilfll.=11Illxllf(t) ~ 'EC
11+
sup '."2 EO o
IIf(t l )-f(t2 )1I
It l -
t21 #
her C . e o!'s a
,w
0t (t) =1>1 (t)G, (t)
(3.2.20)
We consider the following formal expression as a particular solution of (3.2. 20) += 1>o(z)=I+
J
tz+l --rp(t)dt, t-z
(3. 2. 21)
where rp(t) is an unknown function with rp(t) E VH(p,), and 1 is the unit element. From (3.2.14) we obtain the boundary values (3. 2. 22) where 1>t (t) and 1>0 (t) are boundary values of 1>o(z) as z tends to a point t on the real axis from the upper half plane and the lower half plane respectively. If z=w(z) =
-+zi, t
z
then w maps the real axis onto the circle C. Let us denote points
on the circle by t, and let
- -
tt
-
-
-
zt
rp(t)=!(-t+i) and 1>o(z) =1>0(- zti)'
From (3. 2. 22) we deduce -Ii .- - 2il-l 1>0" (I+i) =I±mrp(t) et+i)2+
JIi+Ti-1 q;(l) dT-I (T+i)2
'T.
(3.2.23)
c
Let - il G, (t) =G, (-I+i)'
(]jo" (l) =1>0" ( - I ! i ) ' - . 2il-l - a(t)=m ct+i)2(I+G(t)), Ii+ :ri-l - . (~+i)2 (I-G(t) )m,
let) =0, (t) - I ,
(3.2.24)
VECTOR-VALUED ANALYSIS
129 bet) =K(t,t).
We replace (3.2.20) by (3.2.24). Thus
- - - -
1
f
a(t)q>(t)+---=-
1He
K(t,T)- - - - '" t q>(t)dT=f(t) on G.
(3.2.25)
T-
Clearly, --
--
-
1--
--
--
-
1--
(3.2.26)
aCt) +b(t) =t- 2iS(t) ,
(3.2.27)
aCt) -bet) =t- 2iD(t), - -
where Set) =
4rrI
-
4rrOl (t) (t+i)2.
- -
(t+i)2 and D(t) =
(ii(t)±b(t»-1 exists, because (G 1(t»-1 exists. 0Iel;) is regular inside G ,since
- - - is G1 (z) is regular in the upper half plane. The only zero of a(z)-b(z)
1 p= 2i.
We write down some obvious results for the solution of the integral equation (3. 2. 25) as follows:
en
(:z(ii(P)±b(P»)-1 exists.
(ii)If q>(t)
E VH(p,) , then --
IJK(t'T)--- t q>(T)dTEVH(p,).
W(t)=-2'
rr,
(iii) If q>(t)
e
T-
E VH(p,) , then :::i.(-) 1 'f' z = 2rri
J ;-t
K(z,t)-(-)dq> T T
(3.2.28)
C !,s
regular insule C and) J;+
(-t)
'f'
=J...
'-(-t) (2ti-l! (I -G 1 (t) (t+i)2
2 mq>
+_1_
2rri
J
Kct, T) -(-)dT-e q> T T.
(3.2.29)
e
Form (3. 2. 25) and (3. 2. 28) we get Q;(t)=C.f(t)-2q;+(t)(ii(t)-b(t)-1 for tEG.
Let K(t,P)lJ- 1(P)
P-;
b' (P)=[{' .CP,P)+k' iCP,P)=k' I+k' 2,
where K' j = 4rr(I -
0 1 CP»
for j = 1 ,2.
(3. 2. 30)
CHAPTER
130
3
Here there are two cases: (j) If
G1 (/1)=-1 and (I-G 1(/1))-1 exist, then g;(t) =a (t)
1 (1)
(a 2 (t)-b 2 (t)
)-1
J[(St'~)l(T)(a)(T)-b(T)-lctr
(a(t)+~(t))-I m
c
T-t
(3.2.31)
+C Bo(t) (a(t) +b(t)) - I ,
where c= 2D(~) (S(/1) _ 2ki/ )-1 m
J
k(/1 ,:;)l(T) ~a(T)) -beT) )-1 dT. T-/1
c
Proof. we replace (3. 2. 28)by (3. 2. 30) . Then
J
¢(z)=~ 21n
_~ 1n
[((z'T)j(r~(a~T-b(T))-ldT T-Z
c
J
[((z, T)(aST) ;-b(:;)) - I dT T-Z
J
c
=~
[((z,T)l(T)_(a(!)-b(T))-lli-
2m
T-Z
c
We let
then
(a(z) +b(z) )a(z) -b(z))) -I¢(Z) -F(-)-2 -
Z
res
([{(z,t)(a(t)-b(t))_ t-~
;;'(/1)
T-PV-·
Hence (3.2.32) where
7J (:)_
o '" -res
([((Z,rf)(a(i)-b(T))-I)_ t-z T=P
Obviously both of the following two conditions should be satisfied in order that the equa tion (3.2.25) has a solution:
VECTOR-VALUED ANALYSIS
131
Condition (i)(3. 2. 32) holds as z=p Condition (ii) the right side of (3.2.32) is regular in the interior of C. Under condition(1) , we let (3.2.32) with
z=p,
we can get
c=,¢(P)
be the constant element. From the equation
c, then let z in the interior tend to t and replace z
by
t in (3. 2. 32). Therefore we obtain the representation of ;jj+ (0 by substituting this into Q. E. D.
(3.2.30) we derive(3. 2. 31). Under condition (ii) if
GI(P)=-I,
we have the following results:
( 1) If
J
K(P, T)J (i)4(a CT) -beT) )-1 dT=O,
-r-p
c
thell the gellRT'al sowtion of the equation (3.2.25) is (3.2.31), where
c is any constant element
(2)lf
then the equation (3. 2. 25) has no sohdion.
From
Ja)
we get rp(t), hence (/>o(z) is derived.
Theorem 3.2.8.lf «(/>O(Z))-Iand «(/>o±(t))-Iexistfar -ooo(z)P(z) , where P(z) is any vector-valued polynomUll w -
z
i.z + .. t
Proof. The Mobius transformation
t= - t~i maps the real axis onto the circle C ,and prob-
lem(3. 2. 20) becomes: (1)1 + (l) =(1)1 (t)G I (t) on C.
Suppose
(3.2.33)
; (I) , thus
(3.2.43) Similarly. 1IcJ>~.-cJ>tllp-O
as k-oo.
(3.2.44)
Equation (3.2.39) and the vector-valued Plemelj formula give
(3.2.45) Therefore 1. cJ>.(z) =-2 Jrt
f
«G(t)+Ll.(t))cJ>; (t)+g.(t)) «(;(t-z)+(;(z))dt
Co
satisfies the boundary value problem (3.2.36). We therefore obtain the following conclusion.
Proposition
3.2. 2. If qIIG+Ll.llp: (0 - rr>; (0 = y. (0,
(3. 2. 50) The integral equation (3.2.46) becomes rr>: (t) - rr>; (t) =rr>; (t) (G(t)
+.1. (t)) +g. (t).
The remainder of the proof is easy, by using the basic properties of the t;-function.
Q.E.D. Suppose operators A., A are defined by the formula A.y=a.(t)y(t)
J +-. J + b.(O 1I't
Ay=a(Oy(O
bet) 1I't
y( 1') (t;( 1'-0 +t;(t) )d1',
Co
y( 1') (t;( 1'-0 +t;(t) )d1',
Co
where a"a,b.,b,yEDVH(p,). Obviously A., A are bounded linear operators on the space DVH(p,). For an arbitrarily small positive number r, set H.= {(a. ,b.): IIA.II~r}.
Proposition
3.2. 4. If IIa.-all~-+O, IIb.-bll~-+O, IIf.- fll~-+O
as n-+oo
(3.2.51)
and there extst r, N such thnt (a.,b.)EH.for n>N, then the following results hold:
(3. 2. 52)
CHAPTER
136
3
(i)The integral equation (3.2.46) has a unique sohdion y•• (ii ) The integral equation a(t)y(t)
bet) +-. 1n
f
y( r) (~( r-t) +~(t) )dr= f(t)
Co
has a unUJUe solution y. (iii)
Ily.-yllp-O as n-oo.
Proof. Since (0 and (ii) are obvious by Theorem 2. 11. 6 in [11 J, we only prove (iii). We have that
II(A.-A)yll
= II (a.(t) -a(t) )y(t) + b.(t) ~b(t) 1n
f
y( r)
(~( r-t) -~(t))dtll
Co
~ Ila. -allpllyllp+cllb. -blip IIYII~, IIA.-AII~lIa.-allp+cllb.-bllp
(3.2.53)
where c is a constant. On the other hand,
A;l- A-I =A;l (A- A.)A- l
,
and IIA;11l is bounded. By (3.2.51) and (3.2.53) we have
IIA;I_A-111-0 as n-oo, i. e.
Proposition
3.2.5. If
conditIOnS (3.2.11), (3.2.52), (3.2.50) and (3.2.53) hold and 112b.(a+b.)-I+GII~-0
as n-oo,
(3.2.54)
then lim t1>.(z)=t1>(z) = hm ~ 11-00
11_00
1ft
f
('0
Y.( r)
(~( r-z) +~(z) )dt
is the solution of the baundary value '[I1'oblem (3. 2. 55) and
t1>(z)=~ 1n
f y(r)(~(r-z)+~(z))dr.
(3.2.55)
Co
Proof. From Proposition 3. 2. 4, we obtain that the integral equation (3. 2. 46) has unique solution. By Proposition 3. 2. 3, the disturbance problem (3. 2. 36) has a unique solution t1>.(z). From Proposition 3.2.4 and (3.2.49) we conclude that (3.2.55) holds
VECfOR-VALUED ANALYSIS
137
and satisfies the boundary value problem (3.2.40).
Q.E.D.
2.6. THE VECfOR-VALUED BOUNDARY VALUE PROBLEM IN 1p
In 1979, J. K. b.J solved the aaubly- perUxlic Riemann baunrIary value '[1T'oblem wiih the rlomain ana the range in a complex plane '6" (see [25J). In 1988,C. G. Hu extended above problem to vector valued cases with the domain in '6" and the range in a Banach space and got some unique solution of this problem under stronger conditions (see [19J). In this section, under weaker conditions we obtain general sol utions (of above problem) belonging to a Banach space U. Let S = {w: Iw I
< l} and S = {w: Iwi:::;;; l}. Set
U = {f:f is regular in S and continuous on S}
with the norm IIfllu=sup{ If(w) I :wES} and the multiplication f
* g=fog. Obviously,
U is a complete Banach algebra.
Suppose that VDF is the set of all vector-valued doubly periodic continuous functions with periods
WI
and
~,
and ranges are elements in U, that DVH(p,) is the set of all vector
valued doubly- periodic functions which satisfy the Holder condition, that 10 is a closed path and has a non-empty interior, that the track 10 of 10 is in the fundamental parallelogram
and the point z= 0 is included in the interior of 10 , that 1= {t+mwl +nw2 :tE 10 , m and n are arbitrary integers} ,
and that the inner domain of 10 is denoted by S+ and S- =Po- (S+
Uzo).
Functions tP(z) in VDF are found to be piecewise regular and with poles of order at most
a at
z= 0 and Z=11iW1 +nw2 and which satisfy the boundary condition tP+ (t) =tP- (t)
* G(t) +g(t) as
where tP±(t),g(t)EDVH(p,) and G(t)EDVH(l). Let
tE 1,
(3.2.56)
CHAPTER
138
3
00
G(t,w)=
~a.(t)w· as
.-0
(3.2.57)
(t,w)Elri XS,
and
Lemma
3. 2. 2. Suppose
thnt 00
~ lIa.III1(l)here
m and n are integers and
711 2 +n 2 ::;i:
O.
(a)Suppose k+d>O, then a general solution of (3.2.56) can be expressed as
142
CHAPTER
cJ>(z,w)=X(z,~,) 2m
3
f
g(t,w) (s(t-z)+s(z))dt ,.X+(t,w)
+ X(z,w) (ao(w) +al (w)s' (z)
+ ... +aHl-1 (W)S(Hl-1) (z)) ,
where aj(w)(j=O,l,···,k+d-1) are arburary in U mul cJ>(z,w)EU far each fw:ed
zE
st. (b)Suppose k+d=O, then the general sohdinns of (3.2.56) uruier the canduion _1_ 21fi
are m the farm: cJ>(z,w)=
f
X(z,w) 2' 1ft
'.
f
g(t,w) dt= 0 ,.X+(t,w)
(3.2.71)
g(t,w) X+(t )s(t-z)dt+a(w)X(z,w) , ,w
(3.2.72)
where a ( w) is arburary in U. (c) Suppose k+d=-l arui (3.2.71)
nique arui cJ>(z,w)=
f
X(z,w) 2'
1.8
satisfied, then the solution of (3.2.56) is u-
g(t,w) X+(t ) (s(T-z)-s(t))dT.
,w (d) Suppose that k+dO, then a general solutlDl! of (3.2.56) can be expressed
f'0 ;;.~~w))
cJ>(z,w)=X
