Harmonic Bloch Space on the Real Hyperbolic Ball

TL;DR

This paper investigates the properties of harmonic Bloch and little Bloch spaces on the real hyperbolic ball, establishing projection onto these spaces, duality relations, and atomic decompositions.

Contribution

It demonstrates that Bergman projections are onto these spaces, characterizes dual and predual spaces, and provides an atomic decomposition of Bloch functions.

Findings

Bergman projections from $L^(\u00b7)$ to  are onto.

Dual space of _ is , predual is _0.

Atomic decomposition of Bloch functions via Bergman kernels.

Abstract

We study the Bloch and the little Bloch spaces of harmonic functions on the real hyperbolic ball. We show that the Bergman projections from $L^\infty(\mathbb B)$ to $\mathcal B$, and from $C_0(\mathbb B)$ to $\mathcal B_0$ are onto. We verify that the dual space of the hyperbolic harmonic Bergman space $\mathcal B^1_\alpha$ is $\mathcal B$ and its predual is $\mathcal B_0$. Finally, we obtain an atomic decomposition of Bloch functions as a series of Bergman reproducing kernels.