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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
Published electronically: December 22, 2016
MathSciNet review: 3632549
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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.

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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

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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