Sums of weighted differentiation composition operators from weighted Bergman spaces to weighted Zygmund and Bloch-type spaces

Let H(D) be the space of analytic functions on the unit disc D and let S(D) denote the set of all analytic self maps of the unit disc D. Let Ψ=(ψj)j=0k be such that ψ_j∈ H(D) and φ∈ S(D). To treat the Stević–Sharma type operators and the products of composition operators, multiplication operators, differentiation operators in a unified manner, Wang et al. considered the following sum operator: TΨ,φkf=∑j=0kψj·f(j)∘φ=∑j=0kDψj,φjf,f∈H(D).We characterize the boundedness and compactness of the operators TΨ,φk from the weighted Bergman spaces A_v,p to the weighted Zygmund-type spaces Z_w and the weighted Bloch-type spaces B_w. Besides, giving examples of bounded, unbounded, compact and non-compact operators TΨ,φk, we give an example of two unbounded weighted differentiation composition operators Dψ0,φ0,Dψ1,φ1:Av,p⟶Zw(Bw) such that their sum operator Dψ0,φ0+Dψ1,φ1=TΨ,φ1:Av,p⟶Zw(Bw) is bounded.