Stability and localization of the Lp Bergman kernel

Abstract

The aims of this paper are twofold. First, we generalize the classical Ramadanov theorem and Skwarczyński theorem for the [Formula: see text] Bergman kernels to the [Formula: see text] case, which are concerned with the compact convergence of [Formula: see text] Bergman kernels on an increasing or decreasing sequence of domains in [Formula: see text]. Second, we prove a localization principle for the [Formula: see text] Bergman kernel on bounded strongly pseudoconvex domains with smooth boundary. Utilize the localization, we study the asymptotic behavior of the [Formula: see text] Bergman kernel near strongly pseudoconvex boundary points and show that the [Formula: see text]-Schwarz content is identical to [Formula: see text] on a neighborhood of a strongly pseudoconvex boundary point.