On boundary behaviour of the Bergman projection on pseudoconvex domains

Abstract

Abstract. It is shown that on strongly pseudoconvex domains the Bergman projection maps a space Lv k of functions growing near the boundary like some power of the Bergman distance from a fixed point into a space of functions which can be estimated by the consecutive power of the Bergman distance. This property has a local character.Let Ω be a bounded, pseudoconvex set with C 3 boundary. We show that if the Bergman projection is continuous on a space E ⊃ L ∞ (Ω) defined by weighted-sup seminorms and equipped with the topology given by these seminorms, then E must contain the spaces Lv k for each natural k. As a result, in the case of strongly pseudoconvex domains the inductive limit of this sequence of spaces is the smallest extension of L ∞ in the class of spaces defined by weighted-sup seminorms on which the Bergman projection is continuous. This is a generalization of the results of J. Taskinen in the case of the unit disc as well as of the previous research of the author concerning the unit ball.

Atomic Decomposition of a Weighted Inductive Limit in $$\mathbb{C}^{n} $$

“…Jasiczak extended the result to the case of the unit ball of C n (see [5]) and in [3] we proved the same result for H ∞ V (Ω), which is defined on an arbitrary bounded smooth pseudoconvex domain Ω of C N (see also the reference [6]). …”

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

“…Part (i) of Theorem 2 was proved in a weaker form for strongly pseudoconvex domains with smooth boundary in [8]. The proof is based on expression of the Bergman kernel due to Fefferman, the Boutet de Movel and Sjöstrand.…”

“…Namely, it is the smallest space in the class of weighted-sup spaces LW ⊃ L ∞ , on which the Bergman projection is continuous. This point of view was the motivation to investigating such spaces in [22] in the case of the unit disk and in [7] and [8] for the unit ball and strongly pseudoconvex domains with smooth boundary. Similar results for strongly pseudoconvex domains were also obtained independently in [5].…”

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.…”