On Carleson-type embeddings for Bergman spaces of harmonic functions (Q1715047)

Let $\Omega \subset \mathbb{R}^{n}$ be a bounded domain with $C^{1}$ boundary, and consider the embedding $$ A_{\alpha}^{p}(\Omega) \hookrightarrow L^{p}(\mu,\Omega),\;\; (1)$$ where $A_{\alpha}^{p}(\Omega)$ denotes the weighted harmonic Bergman space on $\Omega$ for $0 < p < +\infty$ and $\alpha > -1.$\par The main result is that (1) holds for any $p > 0$ (and for any bounded domain $\Omega$), whenever the measure $\mu$ satisfies $$\frac{\mu(\Delta_{k})}{\vert \Delta_{k} \vert^{1+\frac{\alpha}{n}}} \leq C, \quad k \geq 1,\;\; (2) $$ where the sequence of cubes $\Delta_{k}$ is a Whitney-type covering of $\Omega$; conversely, if (1) holds for some $p > 1 + \frac{\alpha + 2}{n-2},$ then $\mu$ satisfies (2).