Given a Hörmander system $X=\{X_1,\cdots ,X_m\}$ on a domain $\Omega \subseteq {𝐑}^n$ we show that any subelliptic harmonic morphism $\phi$ from $\Omega$ into a $\nu$-dimensional riemannian manifold $N$ is a (smooth) subelliptic harmonic map (in the sense of J. Jost & C-J. Xu, [9]). Also $\phi$ is a submersion provided that $\nu \le m$ and $X$ has rank $m$. If $\Omega ={𝐇}_n$ (the Heisenberg group) and $X=\left\{\frac{1}{2}\left(L_{\alpha }+L_{\overline{\alpha }}\right),\frac{1}{2i}\left(L_{\alpha }-L_{\overline{\alpha }}\right)\right\}$, where $L_{\overline{\alpha }}=\partial /\partial {\overline{z}}^{\alpha }-i{z}^{\alpha }\partial /\partial t$ is the Lewy operator, then a smooth map $\phi :\Omega \to N$ is a subelliptic harmonic morphism if and only if $\phi \circ \pi :(C\left({𝐇}_n\right),F_{{\theta }_0})\to N$ is a harmonic morphism, where $S^1\to C\left({𝐇}_n\right)\stackrel{\pi }{\to }\to {𝐇}_n$ is the canonical circle bundle and $F_{{\theta }_0}$...

The Martin compactification of a bounded Lipschitz domain $D\subset {\mathbf{R}}^n$ is shown to be $\bar{D}$ for a large class of uniformly elliptic second order partial differential operators on $D$. Let $X$ be an open Riemannian manifold and let $M\subset X$ be open relatively compact, connected, with Lipschitz boundary. Then $\bar{M}$ is the Martin compactification of $M$ associated with the restriction to $M$ of the Laplace-Beltrami operator on $X$. Consequently an open Riemannian manifold $X$ has at most one compactification which...

Let $(X,\mathscr{H})$ and $(X^{\text{'}},{\mathscr{H}}^{\text{'}})$ be two strong biharmonic spaces in the sense of Smyrnelis whose associated harmonic spaces are Brelot spaces. A biharmonic morphism from $(X,\mathscr{H})$ to $(X^{\text{'}},{\mathscr{H}}^{\text{'}})$ is a continuous map from $X$ to $X^{\text{'}}$ which preserves the biharmonic structures of $X$ and $X^{\text{'}}$. In the present work we study this notion and characterize in some cases the biharmonic morphisms between $X$ and $X^{\text{'}}$ in terms of harmonic morphisms between the harmonic spaces associated with $(X,\mathscr{H})$ and $(X^{\text{'}},{\mathscr{H}}^{\text{'}})$ and the coupling kernels of them.

We study the duals of the spaces $A^{p\alpha }\left(X\right)$ of harmonic functions in the unit ball of ${ℝ}^n$ with values in a Banach space X, belonging to the Bochner $L^p$ space with weight ${(1-|x\left|\right)}^{\alpha }$, denoted by $L^{p\alpha }\left(X\right)$. For 0 < α < p-1 we construct continuous projections onto $A^{p\alpha }\left(X\right)$ providing a decomposition $L^{p\alpha }\left(X\right)=A^{p\alpha }\left(X\right)+M^{p\alpha }\left(X\right)$. We discuss the conditions on p, α and X for which $A^{p\alpha }\left(X\right)*=A^{q\alpha }(X*)$ and $M^{p\alpha }\left(X\right)*=M^{q\alpha }(X*)$, 1/p+1/q = 1. The last equality is equivalent to the Radon-Nikodým property of X*.

The main result of the present paper is : every separately-subharmonic function $u(x,y)$, which is harmonic in $y$, can be represented locally as a sum two functions, $u=u^{*}+U$, where $U$ is subharmonic and $u^{*}$ is harmonic in $y$ , subharmonic in $x$ and harmonic in $(x,y)$ outside of some nowhere dense set $S$.