Ju n 20 09 A WEAK NOTION OF STRICT PSEUDO - CONVEXITY . APPLICATIONS AND EXAMPLES

Let Ω be a bounded C∞-smoothly bounded domain in C. 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 ∂Ω is small with respect to Minkowski dimension: near each point in the boundary ∂Ω, 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 ∂Ω 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(Ω), then the previous measure is Carleson. Examples of such pseudo-convex domains are finite type domains in C, finite type convex domains in C, finite type domains which have locally diagonalizable Lévi form, domains with real analytic boundary and of course, stricly pseudo-convex domains in C. Domains like |z1| 2 +exp{1−|z2| −2} < 1, which are not of finite type are nevertheless strongly pseudo-convex, in this sense.