Harmonic Besov spaces with small exponents

TL;DR

This paper investigates harmonic Besov spaces with small exponents on the unit ball, providing characterizations, duality relations, and boundary growth properties, advancing understanding of these function spaces.

Contribution

It introduces new characterizations and duality results for harmonic Besov spaces with exponents less than one, including growth and atomic decomposition insights.

Findings

Dual of $b^p_eta$ is weighted Bloch space under certain pairings

Characterizations via derivatives and differential operators

Results on boundary growth and atomic decomposition

Abstract

We study harmonic Besov spaces $b^p_\alpha$ on the unit ball of $\mathbb{R}^n$, where $0<p<1$ and $\alpha\in\mathbb{R}$. 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 show that the dual of harmonic Besov space $b^p_\alpha$ is weighted Bloch space $b_\beta^{\infty}$ under certain volume integral pairing for $0<p<1$ and $\alpha,\beta\in\mathbb{R}$. Our other results are about growth at the boundary and atomic decomposition.