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Exponential decay of Bergman kernels on complete Hermitian manifolds with Ricci curvature bounded from below. (English) Zbl 1462.32007

This interesting paper gives a pointwise estimate for the Bergman kernel of weighted spaces \(A^2(M, h, \mu )\), \(\mu= e^{-2 \psi} {\text{Vol}}\), of holomorphic, square integrable functions on a complete Hermitian manifold with Ricci curvature bounded from below. The estimates are of the form \(K_\mu (z,w) e^{-\psi (z) - \psi (w)} \le C \frac{e^{-\gamma d(z,w)}}{\sqrt{{\text{Vol}}(z,1){\text{Vol}}(w,1)}}\). In addition, the authors establish an interesting exponential decay of canonical solutions of the \(\overline \partial\)-equation: \(|f(w)|^2 \, e^{-2\psi (w)} \le \frac{C}{{\text{Vol}}(w,R)} \, e^{-\gamma d(z,w)} \, \int_{B(z,R)} |u|_h^2 \, d\mu.\) In order to study coercivity of the corresponding Kohn Laplacian \(\Box_{h,\mu}\) they prove an interesting version of the basic identity of the Kohn Laplacian, where the torsion tensor of the Chern connection is involved. They mention that their version of the basic identity follows from [P. A. Griffiths, Am. J. Math. 88, 366–446 (1966; Zbl 0147.07502)]. It is also indicated that the exponential decay of Bergman kernels has already been established in the literature for various special cases, see for instance [M. Christ, J. Geom. Anal. 1, No. 3, 193–230 (1991; Zbl 0737.35011)] for the one-dimensional case, or [H. Delin, Ann. Inst. Fourier 48, No. 4, 967–997 (1998; Zbl 0918.32007); the second author, Adv. Math. 285, 1706–1740 (2015; Zbl 1329.32022); A. P. Schuster and D. Varolin, J. Reine Angew. Math. 691, 173–201 (2014; Zbl 1309.32002)], where the point of view is mainly complex analytic; in [X. Ma and G. Marinescu, Math. Ann. 362, No. 3–4, 1327–1347 (2015; Zbl 1337.32011)] a similar estimate is proved in the more general setting of Hermitian line bundles over symplectic manifolds.

MSC:

32A25 Integral representations; canonical kernels (Szegő, Bergman, etc.)
35N15 \(\overline\partial\)-Neumann problems and formal complexes in context of PDEs
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[1] Bergman, S.The kernel function and conformal mapping. Revised ed. Providence (RI): American Mathematical Society; 1970. (Mathematical Surveys; V). · Zbl 0208.34302
[2] Krantz, SG., Geometric analysis of the Bergman kernel and metric, Vol. 268 (2013), New York: Springer, New York · Zbl 1281.32004
[3] Ma, X.; Marinescu, G., Holomorphic Morse inequalities and Bergman kernels, Vol. 254 (2007), Basel: Birkhäuser, Basel · Zbl 1135.32001
[4] Christ, M., On the \(####\) equation in weighted \(####\) norms in \(####\), J Geom Anal, 1, 3, 193-230 (1991) · Zbl 0737.35011 · doi:10.1007/BF02921303
[5] Maria Dall’Ara, G., Pointwise estimates of weighted Bergman kernels in several complex variables, Adv Math, 285, 1706-1740 (2015) · Zbl 1329.32022 · doi:10.1016/j.aim.2015.06.024
[6] Delin, H., Pointwise estimates for the weighted Bergman projection kernel in \(####\), using a weighted \(####\) estimate for the \(####\) equation, Ann Inst Fourier (Grenoble), 48, 4, 967-997 (1998) · Zbl 0918.32007 · doi:10.5802/aif.1645
[7] Lindholm, N., Sampling in weighted \(####\) spaces of entire functions in \(####\) and estimates of the Bergman kernel, J Funct Anal, 182, 2, 390-426 (2001) · Zbl 1013.32008 · doi:10.1006/jfan.2000.3733
[8] Ma, X.; Marinescu, G., Exponential estimate for the asymptotics of Bergman kernels, Math Ann, 362, 34, 1327-1347 (2015) · Zbl 1337.32011 · doi:10.1007/s00208-014-1137-0
[9] Marzo, J.; Ortega-Cerdà, J., Pointwise estimates for the Bergman kernel of the weighted Fock space, J Geom Anal, 19, 4, 890-910 (2009) · Zbl 1183.30058 · doi:10.1007/s12220-009-9083-x
[10] Schuster, AP; Varolin, D., New estimates for the minimal \(####\) solution of \(####\) and applications to geometric function theory in weighted Bergman spaces, J Reine Angew Math, 691, 173-201 (2014) · Zbl 1309.32002
[11] Lu, Z.; Zelditch, S., Szegő kernels and Poincaré series, J Anal Math, 130, 167-184 (2016) · Zbl 1365.53066 · doi:10.1007/s11854-016-0033-9
[12] Asserda, S., Pointwise estimate for the bergman kernel of holomorphic line bundles, Palestine J Math, 6, 1, 6-14 (2017) · Zbl 1354.32001
[13] Gallot, S.; Hulin, D.; Lafontaine, J., Riemannian geometry (2004), Berlin: Springer, Berlin · Zbl 1068.53001
[14] Seto, S.On the asymptotic expansion of the bergman kernel [PhD thesis]. Irvine: University of California; 2015.
[15] Agmon, S.Lectures on exponential decay of solutions of second-order elliptic equations: bounds on eigenfunctions of N-body Schrödinger operators. Mathematical Notes. Vol. 29. Princeton (NJ): Princeton University Press; 1982. · Zbl 0503.35001
[16] Li, P.; Tam, L-F., The heat equation and harmonic maps of complete manifolds, Invent Math, 105, 1, 1-46 (1991) · Zbl 0748.58006 · doi:10.1007/BF01232256
[17] Li, P.; Schoen, R., \(####\) and mean value properties of subharmonic functions on Riemannian manifolds, Acta Math, 153, 3-4, 279-301 (1984) · Zbl 0556.31005 · doi:10.1007/BF02392380
[18] McNeal, JD; Varolin, D., \(####\) estimates for the \(####\) operator, Bull Math Sci, 5, 2, 179-249 (2015) · Zbl 1320.32045 · doi:10.1007/s13373-015-0068-8
[19] Maria Dall’Ara, G., Coercivity of weighted Kohn Laplacians: the case of model monomial weights in \(####\), Trans Amer Math Soc, 369, 7, 4763-4786 (2017) · Zbl 1368.32027
[20] Maz’ya, V.; Shubin, M., Discreteness of spectrum and positivity criteria for Schrödinger operators, Ann Math, 162, 2, 919-942 (2005) · Zbl 1106.35043 · doi:10.4007/annals.2005.162.919
[21] Devyver, B., A Gaussian estimate for the heat kernel on differential forms and application to the Riesz transform, Math Ann, 358, 12, 25-68 (2014) · Zbl 1287.58012 · doi:10.1007/s00208-013-0949-7
[22] Pasternak-Winiarski, Z., On the dependence of the reproducing kernel on the weight of integration, J Funct Anal, 94, 1, 110-134 (1990) · Zbl 0739.46010 · doi:10.1016/0022-1236(90)90030-O
[23] Zeytuncu, YE., \(####\) regularity of weighted Bergman projections, Trans Amer Math Soc, 365, 6, 2959-2976 (2013) · Zbl 1278.32007 · doi:10.1090/S0002-9947-2012-05686-8
[24] Brüning, J.; Lesch, M., Hilbert complexes, J Funct Anal, 108, 1, 88-132 (1992) · Zbl 0826.46065 · doi:10.1016/0022-1236(92)90147-B
[25] Ohsawa, T., L \(####\) approaches in several complex variables (2015), Tokyo: Springer Monographs in Mathematics, Tokyo · Zbl 1355.32001
[26] Berger, F., Essential spectra of tensor product Hilbert complexes and the \(####\)-Neumann problem on product manifolds, J Funct Anal, 271, 6, 1434-1461 (2016) · Zbl 1350.58009 · doi:10.1016/j.jfa.2016.06.004
[27] Straube, EJ.Lectures on the \(####\)-Sobolev theory of the \(####\)-Neumann problem. ESI Lectures in Mathematics and Physics. Zürich: European Mathematical Society (EMS); 2010. · Zbl 1247.32003
[28] Simon, B., Semiclassical analysis of low lying eigenvalues. I. Nondegenerate minima: asymptotic expansions, Ann Inst H. Poincaré Sect. A (NS), 38, 3, 295-308 (1983) · Zbl 0526.35027
[29] Teschl, G.Mathematical methods in quantum mechanics. 2nd ed., Graduate Studies in Mathematics. Vol. 157. Providence (RI): American Mathematical Society; 2014. With applications to Schrödinger operators. · Zbl 1342.81003
[30] Ji, L, Li, P, Schoen, R, et al. Handbook of geometric analysis. No. 1. Advanced Lectures in Mathematics (ALM). Vol. 7. Somerville (MA): International Press; 2008. · Zbl 1144.53004
[31] Gauduchon, P., La 1-forme de torsion d’une variété hermitienne compacte, Math Ann, 267, 4, 495-518 (1984) · Zbl 0523.53059 · doi:10.1007/BF01455968
[32] Griffiths, PA., The extension problem in complex analysis. II. Embeddings with positive normal bundle, Amer J Math, 88, 366-446 (1966) · Zbl 0147.07502 · doi:10.2307/2373200
[33] Klembeck, PF., Kähler metrics of negative curvature, the Bergmann metric near the boundary, and the Kobayashi metric on smooth bounded strictly pseudoconvex sets, Indiana Univ Math J, 27, 2, 275-282 (1978) · Zbl 0422.53032 · doi:10.1512/iumj.1978.27.27020
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