Exceptional sets for holomorphic Sobolev functions
This paper is concerned with the holomorphic function theory of the unit ball \(B^ n\) in \({\mathbb{C}}^ n\). One can define holomorphic Sobolev spaces \(H^ p_{\beta}(B^ n)\) for \(p,\beta >0\) (recently introduced by Beatrous and Burbea). For example, when \(\beta =1\) the space consists of functions in the usual Hardy space \(H^ p(B^ n)\) whose radial derivative (the result of applying the Euler operator) is also in \(H^ p(B^ n)\). The author studies such functions on admissible approach regions towards a boundary point. He demonstrates control over the corresponding maximal function and hence the existence of almost everyhwere admissible limits with respect to any suitable boundary measure. The paper contains careful analysis and a good discussion of the conditions and consequences of the various constructions.
existence of almost everyhwere admissible limits
scientific article; zbMATH DE number 4093870