767 articles – 1379 references  [version française]
 HAL: hal-00394573, version 1
 arXiv: 0906.1956
 A weak notion of strict pseudo-convexity. Applications and examples
 (2009-06-10)
 Let $\Omega$ be a bounded ${\mathcal{C}}^{\infty}$-smoothly bounded domain in ${\mathbb{C}}^{n}.$ For such a domain we define a new notion between strict pseudo-convexity and pseudo-convexity: the size of the set $W$ of weakly pseudo-convex points on $\partial \Omega$ is small with respect to Minkowski dimension: near each point in the boundary $\partial \Omega ,$ there is at least one complex tangent direction in which the slices of $W$ has a upper Minkowski dimension strictly smaller than $2.$ We propose to call this notion "strong pseudo-convexity"; this word is free since "strict pseudo-convexity" gets the precedence in the case where all the points in $\partial \Omega$ are stricly pseudo-convex. For such domains we prove that if $S$ is a separated sequence of points contained in the support of a divisor in the Blaschke class, then a canonical measure associated to $S$ is bounded. If moreover the domain is $p$-regular, and the sequence $S$ is dual bounded in the Hardy space $H^{p}(\Omega),$ then the previous measure is Carleson. Examples of such pseudo-convex domains are finite type domains in ${\mathbb{C}}^{2},$ finite type convex domains in ${\mathbb{C}}^{n},$ finite type domains which have locally diagonalizable Levi form, domains with real analytic boundary and of course, stricly pseudo-convex domains in ${\mathbb{C}}^{n}.$ Domains like $|{z_{1}}| ^{2}+\exp \{1-|{z_{2}}| ^{-2}\}<1,$ which are not of finite type are nevertheless strongly pseudo-convex, in this sense.
 1: Institut de Mathématiques de Bordeaux (IMB) CNRS : UMR5251 – Université Sciences et Technologies - Bordeaux I – Université Victor Segalen - Bordeaux II
 Subject : Mathematics/Complex VariablesMathematics/Functional Analysis
 Keyword(s): Interpolating sequence – Hardy space – Carleson measures