Weighted composition operators from the logarithmic weighted-type space to the weighted Bergman space in CnCn

Let ΩΩ be a bounded, circular, strictly convex domain in CnCn with C2C2 boundary and H(Ω)H(Ω) the space of all analytic functions on ΩΩ. Let u∈H(Ω)u∈H(Ω) and φφ be a holomorphic self-map of ΩΩ. The weighted composition operator uCφuCφ on H(Ω)H(Ω) is defined by (uCφ)(f)(z)=u(z)f(φ(z))(uCφ)(f)(z)=u(z)f(φ(z)), where f∈H(Ω)f∈H(Ω) and z∈Ωz∈Ω. Let Hlogγβ(Ω),β>0,γ∈R+, be the logarithmic weighted-type space on ΩΩ, and Aαp(Ω),p∈(0,∞),α∈(-1,∞), the weighted Bergman space on ΩΩ. Here we characterize the boundedness and compactness of the weighted composition operator uCφ:Hlogγβ(Ω)→Aαp(Ω).