Weighted Harmonic Bloch Spaces on the Ball

Abstract

We study the family of weighted harmonic Bloch spaces \(b_\alpha , \alpha \in {\mathbb {R}}\), on the unit ball of \({\mathbb {R}}^n\). We provide characterizations in terms of partial and radial derivatives and certain radial differential operators that are more compatible with reproducing kernels of harmonic Bergman–Besov spaces. We consider a class of integral operators related to harmonic Bergman projection and determine precisely when they are bounded on \(L^\infty _\alpha \). We define projections from \(L^\infty _\alpha \) to \(b_\alpha \) and as a consequence obtain integral representations. We solve the Gleason problem and provide atomic decomposition for all \(b_\alpha , \alpha \in {\mathbb {R}}\). Finally we give an oscillatory characterization of \(b_\alpha \) when \(\alpha >-1\).