Harmonic Besov spaces with small exponents

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.