Walsh, David
(2006)
Radial variation of functions in Besov Spaces.
Publicacions Matemàtiques, 50 (2).
pp. 371399.
ISSN 02141493
Abstract
This paper considers the radial variation function F(r, t) of an an
alytic function f(z) on the disc D. We examine F(r, t) when f be
longs to a Besov space As
pq and look for ways in which F imitates
the behaviour of f. Regarded as a function of position (r, t) in D,
we show that F obeys a certain integral growth condition which
is the real variable analogue of that satisfied by f. We consider
also the radial limit F(t) of F as a function on the circle. Again,
F 2 Bs
pq whenever f 2 As
pq, where Bs
pq is the corresponding real
Besov space. Some properties of F are pointed out along the way,
in particular that F(r, t) is real analytic in D except on a small
set. The exceptional set E on the circle at which limr!1 f(reit)
fails to exist, is also considered; it is shown to have capacity zero
in the appropriate sense. Equivalent descriptions of E are also
given for certain restricted values of p, q, s.
Item Type: 
Article

Keywords: 
Radial variation; Besov space; Radial limit; 
Subjects: 
Science & Engineering > Mathematics 
Item ID: 
3670 
Depositing User: 
Dr. David Walsh

Date Deposited: 
22 May 2012 11:38 
Journal or Publication Title: 
Publicacions Matemàtiques 
Publisher: 
Universitat Autònoma de Barcelona 
Refereed: 
Yes 
URI: 

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