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...

We study spaces of Hölder type functions harmonic in the unit ball and half space with some smoothness conditions up to the boundary. The first type is the Hölder type space of harmonic functions with prescribed modulus of continuity $\omega =\omega \left(h\right)$ and the second is the variable exponent harmonic Hölder space with the continuity modulus ${\left|h\right|}^{\lambda (·)}$. We give a characterization of functions in these spaces in terms of the behavior of their derivatives near the boundary.

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.