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*.

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 $F_{{\theta }_0}$...

This paper deals with tangential boundary behaviors of harmonic functions with gradient in Lebesgue classes. Our aim is to extend a recent result of Cruzeiro (C.R.A.S., Paris, 294 (1982), 71–74), concerning tangential boundary limits of harmonic functions with gradient in $L^n({\mathbf{R}}_{+}^n)$, ${\mathbf{R}}_{+}^n$ denoting the upper half space of the $n$-dimensional euclidean space ${\mathbf{R}}^n$. Our method used here is different from that of Nagel, Rudin and Shapiro (Ann. of Math., 116 (1982), 331–360); in fact, we use the integral representation...

Soit $\Phi$ une fonction non négative réelle ; une fonction $u$ harmonique sur une surface de Riemann $R$ est dite $\Phi$-bornée si $\Phi \left(\right|u\left|\right)$ admet une majorante harmonique. On étudie la classe $H\Phi \left(R\right)$ des fonctions $\Phi$-bornées sur $R$ et on montre, en particulier, que chaque $u$ de $H\Phi \left(R\right)$ est essentiellement positive pour toute $R$, si et seulement si ${\inf }_{t\to \infty }\Phi \left(t\right)/t\&gt;0$.

Let $\Delta u={\Sigma }_i\frac{{\partial }^2}{{\partial }_{x_i}^2}$, $Lu={\Sigma }_{i,j}{a}_{ij}\frac{{\partial }^2}{\partial x_i\partial x_j}u+{\Sigma }_i{b}_i\frac{\partial }{\partial x_i}u+cu$ be elliptic operators with Hölder continuous coefficients on a bounded domain $\Omega \subset {\mathbf{R}}^n$ of class $C^{1,1}$. There is a constant $c\&gt;0$ depending only on the Hölder norms of the coefficients of $L$ and its constant of ellipticity such that $$c^{-1}{G}_{\Delta }^{\Omega }\le G_L^{\Omega }\le c{G}_{\Delta }^{\Omega }\text{on}\Omega \times \Omega ,$$ where ${\gamma }_{\Delta }^{\Omega }$ (resp. $G_L^{\Omega }$) are the Green functions of $\Delta$ (resp. $L$) on $\Omega$.

Some properties of the functions of the form $v\left(x\right)={\sum }_{i=0}^m{\left|x\right|}^i{h}_i\left(x\right)$ in ℝⁿ, n ≥ 2, where each $h_i$ is a harmonic function defined outside a compact set, are obtained using the harmonic measures.

Let $F$ be a relatively closed subset of a Euclidean domain $\Omega$. We investigate when solutions $u$ to certain elliptic equations on $\Omega \setminus F$ are restrictions of solutions on all of $\Omega$. Specifically, we show that if $\partial F$ is not too large, and $u$ has a suitable decay rate near $F$, then $u$ can be so extended.