Weighted Bergman kernel, directional Lelong number and John-Nirenberg exponent. (English) Zbl 1436.32013
Summary: Let \(\psi\) be a plurisubharmonic function on the closed unit ball and \(K_{t\psi}(z)\) the Bergman kernel on the unit ball with respect to the weight \(t\psi\). We show that the boundary behavior of \(K_{t\psi}(z)\) is determined by certain directional Lelong number of \(\psi\) for all \(t\) smaller than the John-Nirenberg exponent of \(\psi\) associated to certain family of nonisotropic balls, which is always positive.
Keywords:
weighted Bergman kernel; plurisubharmonic function; directional Lelong number; John-Nirenberg exponentReferences:
| [1] | Benelkourchi, S.; Jennane, B.; Zeriahi, A., Polya’s inequalities, global uniform integrability and the size of plurisubharmonic lemniscates, Ark. Mat., 43, 85-112 (2005) · Zbl 1092.31005 · doi:10.1007/BF02383612 |
| [2] | Berndtsson, B., The extension theorem of Oshawa-Takegoshi and the theorem of Donnelly-Fefferman, Ann. Inst. Fourier, 46, 1083-1094 (1996) · Zbl 0853.32024 · doi:10.5802/aif.1541 |
| [3] | Blocki, Z., A note on the Hörmander, Donnelly-Fefferman, and Berndtsson \(L^2\)-estimate for the \({\bar{\partial }}-\) operator, Ann. Pol. Math., 84, 87-91 (2004) · Zbl 1108.32019 · doi:10.4064/ap84-1-9 |
| [4] | Brudnyi, A., Local inequalities for plurisubharmonic functions, Ann. Math., 149, 511-533 (1999) · Zbl 0927.31001 · doi:10.2307/120973 |
| [5] | Calderón, AP, Inequalities for the maximal function relative to a metric, Stud. Math., 57, 297-306 (1976) · Zbl 0341.44007 · doi:10.4064/sm-57-3-297-306 |
| [6] | Chen, B-Y, Completeness of the Bergman metric on non-smooth pseudoconvex domains, Ann. Pol. Math., 71, 241-251 (1999) · Zbl 0937.32014 · doi:10.4064/ap-71-3-241-251 |
| [7] | Demailly, J-P, Regularization of closed positive currents and intersection theory, J. Algebr. Geom., 1, 361-409 (1992) · Zbl 0777.32016 |
| [8] | Demailly, J-P, Complex Analytic and Differential Geometry (1997), Grenoble: University of Grenoble I, Grenoble |
| [9] | Fefferman, C.; Stein, EM, \(H^p\) spaces of several variables, Acta Math., 129, 137-193 (1972) · Zbl 0257.46078 · doi:10.1007/BF02392215 |
| [10] | Hörmander, L., An Introduction to Complex Analysis in Several Variables (1990), North Holland: Elsevier, North Holland · Zbl 0685.32001 |
| [11] | Hörmander, L., Notions of Convexity, Progress in Mathematics (1994), Basel: Birkhäuser, Basel · Zbl 0835.32001 |
| [12] | John, F.; Nirenberg, L., On functions of bounded mean oscillation, Commun. Pure Appl. Math., 14, 415-426 (1961) · Zbl 0102.04302 · doi:10.1002/cpa.3160140317 |
| [13] | Stein, EM, Harmonic Analysis: Real-Variable Methods, Orthogonality, and Oscillatory Integrals (1993), Princeton: Princeton University Press, Princeton · Zbl 0821.42001 |
| [14] | Zhu, K., Spaces of Holomorphic Functions in the Unit Ball, GTM (2005), New York: Springer, New York · Zbl 1067.32005 |
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