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

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., \[ \varphi (z,w):=\sum _{(\alpha ,\beta )\in \Gamma }|z^\alpha w^\beta |^2, \] where $\Gamma \subseteq \mathbb {N}^2$ is finite. Our goal is to prove coercivity estimates of the form \[ (*)\hspace {10pc} \Box _\varphi \geq \mu ^2,\hspace {10pc} \] 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 \[ \mu (z,w)=c(1+|z|^a+|w|^b) \qquad (a,b\geq 0),\] 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.