Products of differentiation and weighted composition operators from hardy spaces to weighted-type spaces

Abstract

Let $\mathcal{H}(\mathbb{D})$ be the space of all analytic functions on the open unit disk $\mathbb{D}$. Let $\psi_1$ and $\psi_2$ be analytic functions on $\mathbb{D}$, and $\phi$ be an analytic self-map of $\mathbb{D}$. We consider the operator $T_{\psi_1,\psi_2,\phi}$ that is defined on $\mathcal{H}(\mathbb{D})$ by $$\left(T_{\psi_1,\psi_2,\phi}f\right)(z)= \psi_1(z)f(\phi(z))+\psi_2(z)f^{\prime}(\phi(z)).$$ In this paper, we characterize the boundedness and compactness of the operator $T_{\psi_1,\psi_2,\phi}$ that act from the Hardy spaces $H^p$ into the weighted-type space $H^{\infty}_{\mu}$ and the little weighted-type space $H^{\infty}_{\mu, 0}$.