On a conjecture about generalized integration operators on Hardy spaces

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 $T_{g,a}:H^p\to H^q(0<q<p<\infty )$.

This paper was finished and submitted to manuscripta mathematica on April 24, 2024. In May 24, we found that Nikolaos Chalmoukis and Georgios Nikolaidis have also independently proven this conjecture on arXiv. See arXiv:2405.13920. arXiv admin note: substantial text overlap with arXiv:2405.16278