On the p-Bergman theory
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...
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Published in | Advances in mathematics (New York. 1965) Vol. 405; p. 108516 |
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Main Authors | , |
Format | Journal Article |
Language | English |
Published |
Elsevier Inc
27.08.2022
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Subjects | |
Online Access | Get full text |
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Summary: | 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∂∂¯logKp(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. |
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ISSN: | 0001-8708 1090-2082 |
DOI: | 10.1016/j.aim.2022.108516 |