Walsh, David (2006) Radial variation of functions in Besov Spaces. Publicacions Matemàtiques, 50 (2). pp. 371399. ISSN 02141493
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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: 
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