On the p-Bergman theory

Abstract

In this paper we attempt to develop a general p-Bergman theory on bounded domains in Cn. To indicate the basic difference between Lp and L2 cases, we show that the p-Bergman kernel Kp(z) is not real-analytic on some bounded complete Reinhardt domains when p>4 is 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 Kp(z,⋅) is Hölder continuous of order 12 for p>1 and of order 12(n+2) for p=1. We also show that the p-Bergman metric Bp(z;X) tends to the Carathéodory metric C(z;X) as p→∞ and the generalized Levi form i∂∂¯log⁡Kp(z;X) is no less than Bp(z;X)2 for p≥2 and C(z;X)2 for p≤2. Stability of Kp(z,w) or Bp(z;X) as p varies, boundary behavior of Kp(z), as well as basic facts on the p-Bergman projection, are also investigated.