Almost strictly pseudo-convex domains. Examples and applications

Abstract

In this work we introduce a class of smoothly bounded domains Ω in Cn with few non strictly pseudo-convex points in @ Ω with respect to a certain Minkowski dimension. We call them almost strictly pseudo-convex, aspc. For these domains we prove that a canonical measure associated to a separated se- quence of points in Ω which projects on the set of weakly pseudo-convex points is automatically a geometric Carleson measure. This class of aspc domains con- tains of course strictly pseudo-convex domains but also pseudo-convex domains of finite type in C2 , domains locally diagonalizable, convex domains of finite type 2 in Cn , domains with real analytic boundary and domains like |z1 | |z2 | 2 } < 1, which are not of finite type. + exp{1 As an application we study interpolating sequences for convex domains of finite type in Cn . After proving a Carleson-type embedding theorem, we get that if Ω is a convex domain of finite type in Cn and if S ⇢ Ω is a dual bounded sequence of points in H p (Ω), if p = 1 then for any q < 1, S is H q (Ω) interpolating with the linear extension property and if p < 1 then S is H q () interpolating with the linear extension property, provided that q < min( p, 2).