The main result of this paper is the following: if a compact subset E of ${ℝ}^n$ is UPC in the direction of a vector $v\in S^{n-1}$ then E has the Markov property in the direction of v. We present a method which permits us to generalize as well as to improve an earlier result of Pawłucki and Pleśniak [PP1].

Consider the normed space $(\mathbb{P}\left({ℂ}^N\right),|\left|·\right|\left|\right)$ of all polynomials of N complex variables, where || || a norm is such that the mapping $L_g:(\mathbb{P}\left({ℂ}^N\right),\left|\right|·\left|\right|\left)\ni f\mapsto gf\in \right(\mathbb{P}\left({ℂ}^N\right),\left|\right|·\left|\right|)$ is continuous, with g being a fixed polynomial. It is shown that the Markov type inequality $|\partial /\partial z_j{P\left|\right|\le M\left(degP\right)}^m\left|\right|P\left|\right|$, j = 1,...,N, $P\in \mathbb{P}\left({ℂ}^N\right)$, with positive constants M and m is equivalent to the inequality $\left|\right|{\partial }^N/\partial z₁...\partial z_{N}P\left|\right|\le M^{\text{'}}{\left(degP\right)}^{m^{\text{'}}}\left|\right|P\left|\right|$, $P\in \mathbb{P}\left({ℂ}^N\right)$, with some positive constants M’ and m’. A similar equivalence result is obtained for derivatives of a fixed order k ≥ 2, which can be more specifically formulated in the language of normed algebras. In...

In this paper, we study the Martin boundary associated with a harmonic structure given by a coupled partial differential equations system. We give an integral representation for non negative harmonic functions of this structure. In particular, we obtain such results for biharmonic functions (i.e. ${▵}^2\varphi =0$) and for non negative solutions of the equation ${▵}^2\varphi =\varphi$.

In this paper we treat noncoercive operators on simply connected homogeneous manifolds of negative curvature.

We study two known theorems regarding Hermitian matrices: Bellman's principle and Hadamard's theorem. Then we apply them to problems for the complex Monge-Ampère operator. We use Bellman's principle and the theory for plurisubharmonic functions of finite energy to prove a version of subadditivity for the complex Monge-Ampère operator. Then we show how Hadamard's theorem can be extended to polyradial plurisubharmonic functions.

Soit $K$ un compact polynomialement convexe de ${\mathbf{C}}^n$ et $V_K$ son “potentiel logarithmique extrémal” dans ${\mathbf{C}}^n$. Supposons que $K$ est régulier (i.e. $V_K$ continue) et soit $f$ une fonction holomorphe sur un voisinage de $K$. On construit alors une suite $\{{\mathbf{P}}_{\ell }{\}}_{\ell \ge 1}$ de polynôme de $n$ variables complexes avec deg$({\mathbf{P}}_{)}\le \ell$ pour $\ell \ge 1$, telle que l’erreur d’approximation ${\max }_{z\in K}|f\left(z\right)-P_{\ell }\left(z\right)|$ soit contrôlée de façon assez précise en fonction du “pseudorayon de convergence” de $f$ par rapport à $K$ et du degré de convergence $\ell$. Ce résultat est ensuite utilisé pour étendre...

We study minimal thinness in the half-space $H:=\{x=(\tilde{x},x_d):\tilde{x}\in {ℝ}^{d-1},x_d\>0\}$ for a large class of subordinate Brownian motions. We show that the same test for the minimal thinness of a subset of $H$ below the graph of a nonnegative Lipschitz function is valid for all processes in the considered class. In the classical case of Brownian motion this test was proved by Burdzy.