Radial variation of functions in Besov Spaces

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