On a conjecture about generalized integration operators on Hardy spaces

TL;DR

This paper confirms a conjecture that bounded generalized integration operators between Hardy spaces imply the symbol function belongs to a specific Hardy space, and also explores their compactness properties.

Contribution

It proves a conjecture relating boundedness of generalized integration operators to the membership of the symbol in a Hardy space, and analyzes their compactness.

Findings

Confirmed the conjecture that bounded operators imply $g \

Analyzed the compactness conditions of the operators.

Abstract

A conjecture posed by Chalmoukis in 2020 states that if $T_{g,a}:H^p\to H^q(0<q<p<\infty)$ is bounded, then $g$ must be in $H^{\frac{pq}{p-q}}$. In this article, we provide a positive answer to the aforementioned conjecture. We also consider the compactness of $T_{g,a}:H^p\to H^q(0<q<p<\infty)$.