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
URI:
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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