Radial Variation of Functions in Besov Spaces

Abstract This paper considers the radial variation function F (r, t) of an analytic function f(z) on the disc D. We examine F (r, t) when f belongs to a Besov space Apq 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 ∈ B pq whenever f ∈ A s pq, where B s 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(re) 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.