We prove the equivalence of various capacitary strong type estimates. Some of them appear in the characterization of the measures $\mu$ that are admissible data for the existence of solutions to semilinear elliptic problems with power growth. Other estimates are known to characterize the measures $\mu$ for which the Sobolev space $W^{2,p}$ can be imbedded into $L^p\left(\mu \right)$. The motivation comes from the semilinear problems: simpler descriptions of admissible data are given. The proof surprisingly involves the theory of singular...

On définit les capacités de Choquet dans le cas fini en utilisant une forme bilinéaire non dégénérée associée à la base de Choquet. On montre que, dans le cas fini, une capacité de Choquet est la donnée d’un convexe de mesure qu’on caractérise. Le cas profini, issu des arbres, est obtenu par passage à la limite projective du cas fini. Sur les capacités profinies, on définit une forme bilinéaire dont le rapport avec l’intégration, dans des cas simples, est étudié.

On étudie les espaces de Sobolev $W^{r,p}(E,\mu )$ construits sur un espace localement convexe $E$ muni d’une mesure gaussienne centree $\mu$. Si $\mu$ est de Radon, on démontre que les capacités naturelles $c_{r,p}$ sont tendues sur les compacts. Cela résulte d’un principe général relatif aux quasi-normes.On s’intéresse également aux fonctions quasi-continues a valeurs banachiques, ce qui est utile pour les propriétés de Nikodym, et à des applications à la continuité des trajectoires des intégrales stochastiques.

We apply the Cauchy-Poisson transform to prove some multivariate polynomial inequalities. In particular, we show that if the pluricomplex Green function of a fat compact set E in ${ℝ}^N$ is Hölder continuous then E admits a Szegö type inequality with weight function $dist{(x,\partial E)}^{-(1-\kappa )}$ with a positive κ. This can be viewed as a (nontrivial) generalization of the classical result for the interval E = [-1,1] ⊂ ℝ.

For algebraic surfaces, several global Phragmén-Lindelöf conditions are characterized in terms of conditions on their limit varieties. This shows that the hyperbolicity conditions that appeared in earlier geometric characterizations are redundant. The result is applied to the problem of existence of a continuous linear right inverse for constant coefficient partial differential operators in three variables in Beurling classes of ultradifferentiable functions.

We show the equivalence of some different definitions of p-superharmonic functions given in the literature. We also provide several other characterizations of p-superharmonicity. This is done in complete metric spaces equipped with a doubling measure and supporting a Poincaré inequality. There are many examples of such spaces. A new one given here is the union of a line (with the one-dimensional Lebesgue measure) and a triangle (with a two-dimensional weighted Lebesgue measure). Our results also...

This is a survey of various applications of the notion of the Choquet integral to questions in Potential Theory, i.e. the integral of a function with respect to a non-additive set function on subsets of Euclidean n-space, capacity. The Choquet integral is, in a sense, a nonlinear extension of the standard Lebesgue integral with respect to the linear set function, measure. Applications include an integration principle for potentials, inequalities for maximal functions, stability for solutions to...

Let $X$ be a metric space with a doubling measure, $Y$ be a boundedly compact metric space and $u:X\to Y$ be a Lebesgue precise mapping whose upper gradient $g$ belongs to the Lorentz space $L_{m,1}$, $m\ge 1$. Let $E\subset X$ be a set of measure zero. Then ${\widehat{\mathscr{H}}}_m(E\cap u^{-1}\left(y\right))=0$ for ${\mathscr{H}}_m$-a.e. $y\in Y$, where ${\mathscr{H}}_m$ is the $m$-dimensional Hausdorff measure and ${\widehat{\mathscr{H}}}_m$ is the $m$-codimensional Hausdorff measure. This property is closely related to the coarea formula and implies a version of the Eilenberg inequality. The result relies on estimates of Hausdorff content of level sets...

We prove that two Toeplitz operators acting on the pluriharmonic Bergman space with radial symbol and pluriharmonic symbol respectively commute only in an obvious case.

Traitant la série de Poincaré d’un groupe discret d’isométries en courbure négative comme un noyau de Green, on établit une théorie du potentiel assez comparable à la théorie classique pour affirmer un parallèle entre densités conformes à la Patterson-Sullivan et densités harmoniques, et notamment définir une frontière de Martin où les densités ergodiques forment la partie minimale, et enfin l’identifier géométriquement sous hypothèse d’hyperbolicité.