Abstract
We study the composition operators on Banach spaces of harmonic mappings that extend several well-known Banach spaces of analytic functions on the open unit disk in the complex plane, including the α-Bloch spaces, the growth spaces, the Zygmund space, the analytic Besov spaces, and the space BMOA.
Highlights
Given a connected region Ω in the complex plane C, a harmonic mapping with domain Ω is a complex-valued function h defined on Ω satisfying the Laplace equation: Δh ≔ 4hzz ≡ 0, on Ω, (1)where hzz is the mixed complex second partial derivative of h
We introduce the harmonic Besov space B1H as the collection of harmonic mappings h on D for which b1(h) ≔ hzz(z) +hzz(z)dA(z) < ∞
Let φ be an analytic self-map of D, X a Banach space of analytic functions with norm ‖ · ‖ satisfying the following conditions: (a) X contains the constants
Summary
Given a connected region Ω in the complex plane C, a harmonic mapping with domain Ω is a complex-valued function h defined on Ω satisfying the Laplace equation: Δh ≔ 4hzz ≡ 0, on Ω,. Journal of Function Spaces below and identify the isometries for most of them For each of these spaces, we examine the eigenfunctions of the composition operators. Let us denote XH as a harmonic extension of the Banach space X of analytic functions whose corresponding norms coincide on the elements of X. For the spaces treated in this work, due to simple estimates connecting the seminorm of a harmonic mapping in XH to the seminorm of the associated analytic and antianalytic (i.e., conjugate of analytic) components in X, it turns out that the composition operator Cφ acting on XH is bounded (respectively, compact, bounded below, closed range) if and only if Cφ acting on X is bounded (respectively, compact, bounded below, closed range).
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