On the p-Bergman theory

In this paper we attempt to develop a general p-Bergman theory on bounded domains in C-n. To indicate the basic difference between L-p and L-2 cases, we show that the p-Bergman kernel K-p(z) is not real-analytic on some bounded complete Reinhardt domains when p > 4is an even number. By the calculus of variations we get a fundamental reproducing formula. This together with certain techniques from nonlinear analysis of the p-Laplacian yield a number of results, e.g., the off-diagonal p-Bergman kernel K-p(z, center dot) is Holder continuous of order 1/2 for p > 1and of order 1/2(n+2) for p = 1. We also show that the p-Bergman metric Bp(z; X) tends to the Caratheodory metric C(z; X) as p -> infinity and the generalized Levi form i partial derivative(partial derivative) over bar logK(p)(z; X) is no less than B-p(z; X)(2) for p >= 2 and C(z; X)(2) for p <= 2. Stability of K-p( z, w) or B-p(z; X) as pvaries, boundary behavior of K-p( z), as well as basic facts on the p-Bergman projection, are also investigated. (c) 2022 Elsevier Inc. All rights reserved.

Automated summary

In this paper, the authors attempt to develop a general p-Bergman theory on bounded domains in C-n. By using calculus of variations and techniques from nonlinear analysis, they obtain several results and investigate related properties and stability.