Harmonic Besov spaces with small exponents

Abstract

We study harmonic Besov spaces $b_{\alpha }^p$ 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 $b_{\alpha }^p$ 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.