Tangential boundary limits and exceptional sets for holomorphic functions in Dirichlet-type spaces

Abstract

According to the well-known theorem of Fatou, the harmonic extension u to the upper half-space R~_ +x of a function f in LP(Rn), 1 1. In this paper we study the analogous question in the unit ball B of C'. Recall that the correct analogue of Fatou's theorem for B is the theorem of Kor/myi [K] stating that invariant harmonic extensions (Poisson-Szeg6 integrals) of functions in LP(S), 1 < p < oo, have "admissible" limits at almost every point of the unit sphere S. In particular, this is true for holomorphic functions in the Hardy space HP(B). As is well known, Korfinyi's admissible regions allow parabolic tangential approach along the complex-tangential directions. These admissible regions are best possible as far as the order of tangency to the boundary is concerned, even if only bounded holomorphic functions are considered, as shown in [H-S] (however, they are not best possible in a stronger sense [Su 1]). By analogy with the real-variable situation, it seems reasonable to expect Kor/myi's regions not to be optimal for holomorphic functions whose (admissible) boundary values are, in some sense, Bessel-type potentials of LP(S) functions. Thus we consider holomorphic functions in the unit ball which are given by "fractional Cauchy integrals" of LP(S) functions. These are obtained by modifying the exponent in the denominator of the Cauchy kernel of B, a procedure suggested by the