Generic versions of a result in the theory of Hardy spaces

TL;DR

This paper investigates the behavior of primitives of functions in Hardy spaces for 0<p<1, revealing that generically these primitives do not belong to smaller Hardy spaces than the expected order, even locally.

Contribution

It demonstrates that, contrary to previous assumptions, the primitive of a Hardy space function generally does not belong to any smaller Hardy space than the one predicted by the classical relation.

Findings

Primitives of Hardy space functions typically do not belong to smaller Hardy spaces.

The result holds even locally, indicating a strong generic property.

Challenges assumptions about the regularity of primitives in Hardy spaces.

Abstract

For 0<p<1 and f a function in the Hardy space of order p its primitive belongs to the Hardy space q=p/1-p. We show that generically the primitive does not belong, even not locally, in any Hardy space smaller than the Hardy space of order q.