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

Let n be a positive integer and g= (g₀, g₁, , g₍-₁), with gₖ H (D) for k=0, 1, , n-1. Let I₆^ (n) be the generalized Volterra-type operators on H (C), which is represented as I₆^ (n) f=Iⁿ (fg₀+f'g₁++f^ (n-1) g₍-₁), where I denotes the integration operator (If) (z) =₀ᶻf (w) dw, and Iⁿ is the nth iteration of I. This operator is a generalization of the operator that was introduced by Chalmoukis in Cn. In this paper, we study the boundedness and compactness of the operator I₆^ (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 u= (u₀, u₁, , uₙ) with uₖ H (D) for 0 k n and be an analytic self-map of D. The sums of generalized weighted composition operators is defined by Lₔ, ^ (n) =₊=₀ⁿWₔ䂵, ^ (k), where Wₔ䂵, ^ (k) f=uₖ f^ (k). 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 Theorem1. 1 and Theorem1. 2.