Holomorphic Q Classes
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Jie Xiao
Holomorphic Q Classes
123
Author Jie XIAO Department of Mathematics and Statistics Concordia University 1455 de Maisonneuve Blvd. West H3G 1M8 Montreal, Quebec, Canada E-mail: [email protected]
Cataloging-in-Publication Data applied for
Mathematics Subject Classification (2000): 30D55, 30H05, 31A20, 32A37, 41A15, 46E15, 46G10, 47B33, 47B38 ISSN 0075-8434 ISBN 3-540-42625-6 Springer-Verlag Berlin Heidelberg New York This work is subject to copyright. All rights are reserved, whether the whole or part 5-of the material is concerned, specifically the rights of translation, reprinting, re-use of illustrations, recitation, broadcasting, reproduction on microfilms or in any other way, and storage in data banks. Duplication of this publication or parts thereof is permitted only under the provisions of the German Copyright Law of September 9, 1965, in its current version, and permission for use must always be obtained from Springer-Verlag. Violations are liable for prosecution under the German Copyright Law. Springer-Verlag Berlin Heidelberg New York a member of BertelsmannSpringer Science+Business Media GmbH http://www.springer.de © Springer-Verlag Berlin Heidelberg 2001 Printed in Germany The use of general descriptive names, registered names, trademarks, etc. in this publication does not imply, even in the absence of a specific statement, that such names are exempt from the relevant protective laws and regulations and therefore free for general use. Typesetting: Camera-ready TEX output by the author SPIN: 10852603
41/3142-543210/du - Printed on acid-free paper
Preface
problems in any area of mathematics is to determine an object under consideration. As for complex-functional analysis, one is interested, for example, in studying the equivalent representations of the conformally invariant classes of holomorphic functions. This problem is addressed here for the holomorphic Q classes. the element of the two dimensional Lebesgue meaFor P E [0, oo) and dm that a holomorphic function in the unit disk D, is of the class f, sure, we say Qp provided One of the fundamental
the distinct variants of
-
Ep(f)
sup
=
t (JD
I ff (Z) 12 log
11-iv-zl
I
W-Z
1) Pdm(z)
1/2
:wED
k E N. all for such that c 1 c > nk+l/nk =
Theorem 1.2.1. Let p, E (i) f E Qp if and only
[0, 1] if f
E
f (z) E'k=o akZnlDp if and only if
and
=
is in
a
HG,
constant
be in HG. Then
00
E 2 k(1-p) k=O
(ii) f
E B
jaj 12 2k p and so, can limit space of Qp as q \, p. Now, we consider the special. cases stated as above. First, let p E f
Then
E
B
:5
11 f 1113
(ID
(1
_
IZ12)-2(l
_
jo'w(Z)j)pdM(Z))
(1, oo)
1/2
obviously implies f E Qp and so, B C Qp. On the other hand, if f (1.3) infers f E=- B, and consequently, Qp 9 B. Thus (i) is true.
which
Secondly, the
gap series
fl(z)
that
f3(z) E0k0=0 (iii) is proved.
=
2 k/2 z
Z2 Ek' =0 k= 2k
k
shows
(ii)
(1.5) E
Q"
via Theorem 1. 2. 1.
applied to show 0 if < Furthe
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