On Hardy type spaces in strictly pseudoconvex domains and the density, in these spaces, of certain classes of singular functions

Abstract

In this paper we study some Hardy type spaces in one or several complex variables and we prove that the set of the holomorphic functions which are totally unbounded in certain domains is dense and Gδ in these spaces. These totally unbounded functions are non-extendable, despite the fact that they have non-tangential limits at the boundary of the domain. Similarly we show that the set of the holomorphic functions in these spaces which are non-extendable is dense and Gδ in these spaces. We also consider local Hardy spaces and show that the set of the functions in these Hardy type spaces which do not belong – not even locally – to Hardy spaces of higher order is dense and Gδ. We first work in the case of the unit ball of Cn where the calculations are easier and the results are somehow better, and then we extend them to the case of strictly pseudoconvex domains.