We consider harmonic Bergman-Besov spaces $b_{\alpha }^p$ and weighted Bloch spaces $b_{\alpha }^{\infty }$ on the unit ball of ${ℝ}^n$ for the full ranges of parameters $0<p<\infty$, $\alpha \in ℝ$, and determine the precise inclusion relations among them. To verify these relations we use Carleson measures and suitable radial differential operators. For harmonic Bergman spaces various characterizations of Carleson measures are known. For weighted Bloch spaces we provide a characterization when $\alpha >0$.

Let $\mathscr{H}$ denote the class of positive harmonic functions on a bounded domain $\Omega$ in ${ℝ}^N$. Let $S$ be a sphere contained in $\overline{\Omega }$, and let $\sigma$ denote the $(N-1)$-dimensional measure. We give a condition on $\Omega$ which guarantees that there exists a constant $K$, depending only on $\Omega$ and $S$, such that ${\int }_{S}ud\sigma \le K{\int }_{\partial \Omega }ud\sigma$ for every $u\in \mathscr{H}\cap C\left(\overline{\Omega }\right)$. If this inequality holds for every such $u$, then it also holds for a large class of non-negative subharmonic functions. For certain types of domains explicit values for $K$ are given. In particular the classical value...

Let ${\Omega }_1,...,{\Omega }_n$ be harmonic spaces of Brelot with countable base of completely determining domains. The elements of a subcone $\mathbf{C}$ of the cone of positive $n$-superharmonic functions in ${\Omega }_1\times ...\times {\Omega }_n$ is shown to have an integral representation with the aid of Radon measures on the extreme elements belonging to a compact base of $\mathbf{C}$. The extreme elements are shown to be the product of extreme superharmonic functions on the component spaces and the measure representing each element is shown to be unique. Necessary and sufficient...