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

Let n be a positive integer and g=(g₀,g₁,⋯,gₙ₋₁), with gₖ∈ H(D) for k=0,1,⋯,n-1. Let Ig⁽ⁿ⁾ be the generalized Volterra-type operators on H(C), which is represented as Ig⁽ⁿ⁾f=Iⁿ(fg₀+f'g₁+⋯+f⁽ⁿ⁻¹⁾gₙ₋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 Ig⁽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 φ be an analytic self-map of D. The sums of generalized weighted composition operators is defined by Lu,φ⁽ⁿ⁾=∑ₖ₌₀ⁿWuₖ,φ⁽ᵏ⁾, where Wuₖ,φ⁽ᵏ⁾f=uₖ· f⁽ᵏ⁾∘φ. 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}.