Continuity of Bergman and Szegö projections on weighted-sup function spaces on pseudoconvex domains

Abstract

We study regularity of Bergman and Szegö projections on Sobolev type weightedsup spaces. The paper covers the case of strongly pseudoconvex domains with C 4 boundary and, partially, domains of finite type in the sense of D'Angelo. Introduction.Let be a bounded domain in C n defined by a differentiable, nondegenerate function r. A fundamental problem in several complex variables is to prove sharp kernel and mapping properties for Bergman and Szegö projections. The problem of determining such estimates is closely connected with regularity of canonical solutions of ∂ and ∂ b , respectively.It is an immediate consequence of the definition that the Bergman projection is bounded on L 2 . However, the situation on other L p , Lipschitz or Sobolev spaces is more complicated. Importantly, regularity of the Bergman projection on some spaces is of interest, because of its application to problems of extension of biholomorphic maps up to the boundary.It is a classic fact that even in the case of the unit disc the Bergman projection B does not map L ∞ into itself. However, B maps boundedly the space L ∞ into BMO. This property is often recognized as a substitute of continuity on L ∞ .The aim of this paper is to study mapping properties of the Bergman projection B on weighted-sup (Sobolev) spaces. We point out a space, which is another subsitute of BMO, as far as continuity of the Bergman projection is concerned. This space is in some sense minimal.Let us start with necessary definitions. Assume that W is a family of continuous functions on with values in R + . We will refer to W as the weight family, provided:

The Bézout equation for functions of log-type growth in convex domains of finite type

“…Importantly, B preserves the property of log-type growth (cf. [16], [17]). The problem which we consider bears resemblance to the celebrated corona problem, which is unsolved at the time of writing this paper, at least for many standard domains in C n , n > 1, including the unit ball and the unit polidisk.…”

Toeplitz operators on the space of analytic functions with logarithmic growth

“…This description is better adapted to the study of operators defined on the space, and is the one we outline in Section 2. The research of Taskinen [15] in this direction was continued by several authors [6,8,9,16].…”