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