It is proved that for every continuous lattice there is a unique semiuniform structure generating both the order and the Lawson topology. The way below relation can be characterized with this uniform structure. These results are used to extend many of the analytical properties of real-valued l.s.cḟunctions to l.s.cḟunctions with values in a continuous lattice. The results of this paper have some applications in potential theory.

Mathematics Subject Classification: 47B38, 31B10, 42B20, 42B15.We obtain the Lp → Lq - estimates for the fractional acoustic potentials in R^n, which are known to be negative powers of the Helmholtz operator, and some related operators. Some applications of these estimates are also given.* This paper has been supported by Russian Fond of Fundamental Investigations under Grant No. 40–01–008632 a.

A compact set $K\subset {ℂ}^N$ satisfies Łojasiewicz-Siciak condition if it is polynomially convex and there exist constants B,β > 0 such that $V_K\left(z\right)\ge B{\left(dist(z,K)\right)}^{\beta }$ if dist(z,K) ≤ 1. (LS) Here $V_K$ denotes the pluricomplex Green function of the set K. We cite theorems where this condition is necessary in the assumptions and list known facts about sets satisfying inequality (LS).

Soit $\Omega$, ouvert de ${\mathbf{R}}^n$ et $f:\Omega \to \mathbf{R}$, continue. On dit qu’une majorante surharmonique de $f$ dans $\Omega$ est minimale si cette majorante surharmonique est harmonique dans l’ensemble (ouvert) où elle diffère de $f$. Beaucoup de propriétés de ces fonctions sont semblables à celles des fonctions harmoniques $\ge 0$ (lesquelles correspondent à $f=0$) ; par exemple la famille entière est uniformément équicontinue dans chaque partie compacte de $\Omega$, relativement à la structure uniforme de $\bar{\mathbf{R}}$. On traite le problème de Dirichlet : détermination...

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.