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

Abstract

The weighted Kohn Laplacian ${\square }_{\phi }$ is a natural second order elliptic operator associated to a weight $\phi :{\mathbb{C}}^n\to \mathbb{R}$ 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., $\phi (z,w):={\sum }_{(\alpha ,\beta )\in \Gamma }|z^{\alpha }{w}^{\beta }{|}^2,$ where $\Gamma \subseteq {\mathbb{N}}^2$ is finite. Our goal is to prove coercivity estimates of the form ${\square }_{\phi }\ge {\mu }^2$, where $\mu :{\mathbb{C}}^n\to \mathbb{R}$ acts by pointwise multiplication on $(0,1)$-forms, and the inequality is in the sense of self-adjoint operators. We recently proved (arxiv.org:1502.00865) how to derive from $\mu$-coercivity estimates for ${\square }_{\phi }$ pointwise bounds for the weighted Bergman kernel associated to $\phi$. Here we introduce a technique to establish $\mu$-coercivity with $\mu (z,w)=c(1+|z{|}^a+|w{|}^b)(a,b\ge 0),$ where $a,b\ge 0$ depend (and are easily computable from) $\Gamma$. As a corollary we also prove that, for a wide class of model monomial weights, the spectrum of ${\square }_{\phi }$ is discrete if and only if the weight is not decoupled, i.e. $\Gamma$ contains at least a point $(\alpha ,\beta )$ with $\alpha \ne 0\ne \beta$. Our methods comprise a new holomorphic uncertainty principle and linear optimization arguments.

22 pages. arXiv admin note: text overlap with arXiv:1501.06311

References in corpus (1)

• Matrix Schrödinger Operators and Weighted Bergman Kernels