Walsh, David (2006) Radial variation of functions in Besov Spaces. Publicacions Matemàtiques, 50 (2). pp. 371-399. ISSN 0214-1493
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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; | 
| Academic Unit: | Faculty of Science and Engineering > Mathematics and Statistics | 
| 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 | 
| Use Licence: | This item is available under a Creative Commons Attribution Non Commercial Share Alike Licence (CC BY-NC-SA). Details of this licence are available here | 
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