A generalization of Riesz's uniqueness theorem

Abstract

There have been, over the last 8 years, a number of far reaching extensions of the famous original F. and M. Riesz's uniqueness theorem that states that if a bounded analytic function in the unit disc of the complex plane $\Bbb C$ has the same radial limit in a set of positive Lebesgue measure on its boundary, then the function has to be constant. First Beurling [B], considering the case of non-constant meromorphic functions mapping the unit disc on a Riemann surface of finite spherical area, was able to prove that if such a function showed an appropriate behavior in the neighborhood of the limit value where the function maps a set on the boundary of the unit disc, then those sets have logarithmic capacity zero. The author of the present note, in [V], was able to weaken Beurling's condition on the limit value. Those results where quite restrictive in a two folded way, namely, they were in dimension $n=2$ and the regularity requirements on the treated functions were quite strong. Koskela in [K], was able to remove those two restrictions by proving a uniqueness result for functions in $ACL^p(\Bbb B^n)$ for values of $p$ in the interval $(1,n]$. Koskela also shows in his paper that his result is sharp..