A weak notion of strict pseudo-convexity. Applications and examples

TL;DR

The paper introduces a new notion called 'strong pseudo-convexity' for bounded domains in complex space, which lies between strict pseudo-convexity and pseudo-convexity, and explores its implications for sequences and measures in complex analysis.

Contribution

It defines strong pseudo-convexity based on Minkowski dimension of weakly pseudo-convex boundary points and proves results on boundedness and Carleson measures for sequences in such domains.

Findings

Boundedness of canonical measures for separated sequences in strong pseudo-convex domains.

Carleson measure property for dual bounded sequences in p-regular strong pseudo-convex domains.

Application to interpolating sequences in finite type convex domains.

Abstract

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. As an application we prove a theorem on interpolating sequences in bounded convex domains of finte type in ${\mathbb{C}}^{n}.$ 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.