Hankel operators on doubling Fock spaces

Abstract

Suppose ϕ is a real-valued subharmonic function on C with ΔϕdA is a doubling measure. The doubling Fock space Fϕp is the family of holomorphic functions on C such that f(⋅)e−ϕ(⋅)∈Lp. We introduce the function space IDArs,q,α and discuss the decomposition theorem for this space. We use it to characterize the boundedness and compactness of Hankel operators from a doubling Fock space Fϕp to a weighted Lesbegue space Lϕq for all possible 1≤p,q<∞, which extends the results of [9] from the special case ρ≍1. We also obtain the relationship between the solution operators to ∂‾-equation and Hankel operator. As some applications, we obtain the characterizations on f for which Hankel operators Hf and Hf‾ are both bounded (or compact) from Fϕp to Lϕq.