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

TL;DR

This paper investigates the generic properties of Hardy spaces in strictly pseudoconvex domains, showing that most functions are unbounded and do not belong to higher-order Hardy spaces, with results extended from the unit ball to more general domains.

Contribution

It establishes new generic results about the unboundedness and non-inclusion of certain functions in Hardy spaces within strictly pseudoconvex domains, extending known results from the unit ball.

Findings

Most functions in Hardy spaces are totally unbounded.

Generically, functions do not belong to higher-order Hardy spaces.

Results are extended from the unit ball to strictly pseudoconvex domains.

Abstract

In this paper we prove generic results concerning Hardy spaces in one or several complex variables. More precisely, we show that the generic function in certain Hardy type spaces is totally unbounded and hence non-extentable, despite the fact that these functions have non tangential limits at the boundary of the domain. We also consider local Hardy spaces and show that generically these functions do not belong, not even locally, to Hardy spaces of higher order. We work first 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.