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Transactions of the American Mathematical Society

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Coercivity of weighted Kohn Laplacians: The case of model monomial weights in $ \mathbb{C}^2$


Author: Gian Maria Dall’Ara
Journal: Trans. Amer. Math. Soc. 369 (2017), 4763-4786
MSC (2010): Primary 32W99; Secondary 90C05
DOI: https://doi.org/10.1090/tran/6843
Published electronically: December 22, 2016
MathSciNet review: 3632549
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Abstract | References | Similar Articles | Additional Information

Abstract: The weighted Kohn Laplacian $ \Box _\varphi $ is a natural second order elliptic operator associated to a weight $ \varphi :\mathbb{C}^n\rightarrow \mathbb{R}$ and acting on $ (0,1)$-forms, which plays a key role in several questions of complex analysis.

We consider here the case of model monomial weights in $ \mathbb{C}^2$, i.e.,

$\displaystyle \varphi (z,w):=\sum _{(\alpha ,\beta )\in \Gamma }\vert z^\alpha w^\beta \vert^2, $

where $ \Gamma \subseteq \mathbb{N}^2$ is finite. Our goal is to prove coercivity estimates of the form

$\displaystyle (*)\hspace {10pc} \Box _\varphi \geq \mu ^2,\hspace {10pc} $

where $ \mu :\mathbb{C}^n\rightarrow \mathbb{R}$ acts by pointwise multiplication on $ (0,1)$-forms, and the inequality is in the sense of self-adjoint operators. We proved in 2015 how to derive from $ (*)$ new pointwise bounds for the weighted Bergman kernel associated to $ \varphi $. Here we introduce a technique to establish $ (*)$ with

$\displaystyle \mu (z,w)=c(1+\vert z\vert^a+\vert w\vert^b) \qquad (a,b\geq 0),$

where $ a,b\geq 0$ depend on (and are easily computable from) $ \Gamma $. As a corollary we also prove that, for a wide class of model monomial weights, the spectrum of $ \Box _\varphi $ is discrete if and only if the weight is not decoupled, i.e., $ \Gamma $ contains at least a point $ (\alpha ,\beta )$ with $ \alpha \neq 0\neq \beta $.

Our methods comprise a new holomorphic uncertainty principle and linear optimization arguments.


References [Enhancements On Off] (What's this?)

  • [CD14] Ph. Charpentier and Y. Dupain, Extremal bases, geometrically separated domains and applications, Algebra i Analiz 26 (2014), no. 1, 196-269; English transl., St. Petersburg Math. J. 26 (2015), no. 1, 139-191. MR 3234809, https://doi.org/10.1090/s1061-0022-2014-01335-0
  • [Chr91] Michael Christ, On the $ \overline \partial $ equation in weighted $ L^2$ norms in $ {\bf C}^1$, J. Geom. Anal. 1 (1991), no. 3, 193-230. MR 1120680 (92j:32066), https://doi.org/10.1007/BF02921303
  • [CS01] So-Chin Chen and Mei-Chi Shaw, Partial differential equations in several complex variables, AMS/IP Studies in Advanced Mathematics, vol. 19, American Mathematical Society, Providence, RI; International Press, Boston, MA, 2001. MR 1800297 (2001m:32071)
  • [Dal14] Gian Maria Dall'Ara, Matrix Schrödinger operators and weighted Bergman kernels, Ph.D. thesis, arXiv:1501.06311 (2014).
  • [Dal15] Gian Maria Dall'Ara, Pointwise estimates of weighted Bergman kernels in several complex variables, Adv. Math. 285 (2015), 1706-1740. MR 3406539, https://doi.org/10.1016/j.aim.2015.06.024
  • [Fef83] Charles L. Fefferman, The uncertainty principle, Bull. Amer. Math. Soc. (N.S.) 9 (1983), no. 2, 129-206. MR 707957 (85f:35001), https://doi.org/10.1090/S0273-0979-1983-15154-6
  • [Has98] Friedrich Haslinger, The Bergman kernel functions of certain unbounded domains, Complex analysis and applications (Warsaw, 1997), Ann. Polon. Math. 70 (1998), 109-115. MR 1668719 (2000a:32006)
  • [Has11] Friedrich Haslinger, Compactness for the $ \overline \partial $-Neumann problem: a functional analysis approach, Collect. Math. 62 (2011), no. 2, 121-129. MR 2792515 (2012b:32048), https://doi.org/10.1007/s13348-010-0013-9
  • [Has14] Friedrich Haslinger, The $ \overline {\partial }$-Neumann problem and Schrödinger operators, de Gruyter Expositions in Mathematics, vol. 59, De Gruyter, Berlin, 2014. MR 3222570
  • [HH07] Friedrich Haslinger and Bernard Helffer, Compactness of the solution operator to $ \overline \partial $ in weighted $ L^2$-spaces, J. Funct. Anal. 243 (2007), no. 2, 679-697. MR 2289700 (2007m:32022), https://doi.org/10.1016/j.jfa.2006.09.004
  • [Hör65] Lars Hörmander, $ L^{2}$ estimates and existence theorems for the $ \bar \partial $ operator, Acta Math. 113 (1965), 89-152. MR 0179443 (31 #3691)
  • [Iwa86] Akira Iwatsuka, Magnetic Schrödinger operators with compact resolvent, J. Math. Kyoto Univ. 26 (1986), no. 3, 357-374. MR 857223 (87j:35287)
  • [Koe02] Kenneth D. Koenig, On maximal Sobolev and Hölder estimates for the tangential Cauchy-Riemann operator and boundary Laplacian, Amer. J. Math. 124 (2002), no. 1, 129-197. MR 1879002 (2002m:32061)
  • [MS94] J. D. McNeal and E. M. Stein, Mapping properties of the Bergman projection on convex domains of finite type, Duke Math. J. 73 (1994), no. 1, 177-199. MR 1257282 (94k:32037), https://doi.org/10.1215/S0012-7094-94-07307-9
  • [MS97] J. D. McNeal and E. M. Stein, The Szegő projection on convex domains, Math. Z. 224 (1997), no. 4, 519-553. MR 1452048 (98f:32023), https://doi.org/10.1007/PL00004593
  • [NP] Alexander Nagel and Malabika Pramanik, Diagonal estimates for the Bergman kernel on certain domains in $ \mathbb{C}^n$, preprint.
  • [NRSW89] A. Nagel, J.-P. Rosay, E. M. Stein, and S. Wainger, Estimates for the Bergman and Szegő kernels in $ {\bf C}^2$, Ann. of Math. (2) 129 (1989), no. 1, 113-149. MR 979602 (90g:32028), https://doi.org/10.2307/1971487
  • [NS06] Alexander Nagel and Elias M. Stein, The $ \overline {\partial }_b$-complex on decoupled boundaries in $ \mathbb{C}^n$, Ann. of Math. (2) 164 (2006), no. 2, 649-713. MR 2247970 (2007d:32036), https://doi.org/10.4007/annals.2006.164.649
  • [Rai06] Andrew S. Raich, Heat equations in $ \mathbb{R}\times \mathbb{C}$, J. Funct. Anal. 240 (2006), no. 1, 1-35. MR 2259891 (2007h:32059), https://doi.org/10.1016/j.jfa.2006.06.016
  • [Rai07] Andrew Raich, Pointwise estimates for relative fundamental solutions of heat equations in $ \mathbb{R}\times \mathbb{C}$, Math. Z. 256 (2007), no. 1, 193-220. MR 2282265 (2008b:32030), https://doi.org/10.1007/s00209-006-0065-4
  • [She99] Zhongwei Shen, On fundamental solutions of generalized Schrödinger operators, J. Funct. Anal. 167 (1999), no. 2, 521-564. MR 1716207 (2000j:35055), https://doi.org/10.1006/jfan.1999.3455

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Additional Information

Gian Maria Dall’Ara
Affiliation: Scuola Normale Superiore, 56126 Pisa, Italy
Address at time of publication: Fakultät für Mathematik, Oskar-Morgenstern-Platz 1, 1090 Vienna, Austria
Email: gianmaria.dallara@univie.ac.at

DOI: https://doi.org/10.1090/tran/6843
Received by editor(s): February 10, 2015
Received by editor(s) in revised form: July 15, 2015
Published electronically: December 22, 2016
Article copyright: © Copyright 2016 American Mathematical Society

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