A new class of Carleson measures and integral operators on Bergman spaces

Abstract

Let $n$ be a positive integer and $\mathbf{g}=(g_0,g_1,\cdots ,g_{n-1})$, with $g_k\in H(\mathbb{D})$ for $k=0,1,\cdots ,n-1$. Let $I_{\mathbf{g}}^{(n)}$ be the generalized Volterra-type operators on $H(\mathbb{C})$, which is represented as $$I_{\mathbf{g}}^{(n)}f=I^n\left(f{g}_0+f^{\prime }{g}_1+\cdots +f^{(n-1)}{g}_{n-1}\right),$$ where $I$ denotes the integration operator $$(If)(z)={\int }_0^{z}f(w)dw,$$ and $I^n$ is the $n$th iteration of $I$. This operator is a generalization of the operator that was introduced by Chalmoukis in \cite{Cn}. In this paper, we study the boundedness and compactness of the operator $I_{\mathbf{g}}^{(n)}$ acting on Bergman spaces to another. As a consequence of these characterizations, we obtain conditions for certain linear differential equations to have solutions in Bergman spaces. Moreover, we study the boundedness, compactness and Hilbert-Schmidtness of the following sums of generalized weighted composition operators: Let $\mathbf{u}=(u_0,u_1,\cdots ,u_n)$ with $u_k\in H(\mathbb{D})$ for $0\le k\le n$ and $\phi$ be an analytic self-map of $\mathbb{D}.$ The sums of generalized weighted composition operators is defined by $$L_{\mathbf{u},\phi }^{(n)}=\sum _{k=0}^n{W}_{u_k,\phi }^{(k)},$$ where $$W_{u_k,\phi }^{(k)}f=u_k\cdot f^{(k)}\circ \phi .$$ Our approach involves the study of new class of Sobolev-Carleson measures for classical Bergman spaces on unit disk which appears in the first main Theorems \ref{Theorem1.1} and \ref{Theorem1.2}.