On a conjecture about generalized integration operators on Hardy spaces

A conjecture posed by Chalmoukis in 2020 states that if $T_{g,a}:H^p\to H^q(0<q<p<\mathrm{\infty })$ 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 $T_{g,a}:H^p\to H^q(0<q<p<\mathrm{\infty })$.