Using recent characterizations of the compactness of composition operators on HardyOrlicz and Bergman-Orlicz spaces on the ball ([2, 3]), we first show that a composition operator which is compact on every Hardy-Orlicz (or Bergman-Orlicz) space has to be compact on H∞. Then, although it is well-known that a map whose range is contained in some nice Korányi approach region induces a compact comp...

In this inequality, ∇ denotes the full gradient in R + : ∇ = (∂/∂x1, . . . , ∂/∂xd, ∂/∂y); R + is the usual upper half space Rd×(0,∞); μ is a positive Borel measure defined on R + ; and v is a non-negative function in Lloc(R d). We studied this inequality primarily for p and q in the range 1 < p ≤ q < ∞. For the case in which q ≥ 2, we proved sufficient conditions on μ and v (depending on p, q,...

We show that under mild conditions, a Gaussian analytic function F that a.s. does not belong to a given weighted Bergman space or Bargmann–Fock space has the property that a.s. no non-zero function in that space vanishes where F does. This establishes a conjecture of Shapiro (1979) on Bergman spaces and allows us to resolve a question of Zhu (1993) on Bargmann–Fock spaces. We also give a simila...

Let Π = {z ∈ C : Imz > 0} be the upper half-plane in the complex plane. This paper characterizes the bounded products of differentiation operator and composition operator acting from the weighted Bergman space Aα(Π) to the weighted-type space A∞(Π) and the Bloch-type space B∞(Π). Mathematics Subject Classification: Primary 47B38; Secondary 47B33, 47B37

Let ϕ be a holomorphic self-map and let ψ be a holomorphic function on the unit ball B. The boundedness and compactness of the weighted composition operator ψC ϕ from the generalized weighted Bergman space into a class of weighted-type spaces are studied in this paper.