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

Let n be a positive integer and u=(u0,u1,& mldr;,un) with uk is an element of H(D) for 0 <= k <= n, where H(D) is the space of analytic functions in the unit disk D. For 0<p,q0 such that integral(D)|u(0)(z)f(z)+u1(z)f '(z)+& ctdot;+un(z)f(n)(z)|(q)d mu(z) <= C & Vert;f & Vert;(q)(p) for all f is an element of Ap. Using Sobolev Carleson measures, we characterized the boundedness and compactness of the generalized Volterra-type operator Ig(n) acting on Bergman space to another, which is represented as I(g)((n))f=I-n(f(g0)+f '(g1)+& ctdot;+f((n-1))g(n-1)), here g=(g0,& ctdot;,g(n-1)) with g(k)is an element of H(D) for 0 <= k <= n-1 and (If)(z)=integral(z)(0)f(w)dw is the usual integration operator. This operator is a generalization of the operator introduced by Chalmoukis in [5]. As a consequence, 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: Lu,phi(n)=& sum;(n)(k=0) W-uk,phi((k)), Where phi is an analytic self-map of D and W-uk,phi(f)=u(k)& sdot;f((k))degrees phi. (c) 2024 Elsevier Masson SAS. All rights are reserved, including those for text and data mining, AI training, and similar technologies.