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

The weighted Kohn Laplacian $\Box_φ$ is a natural second order elliptic operator associated to a weight $φ:\mathbb{C}^n\rightarrow\mathbb{R}$ 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., $ φ(z,w):=\sum_{(α,β)\inΓ}|z^αw^β|^2, $ where $Γ\subseteq \mathbb{N}^2$ is finite. Our goal is to prove coercivity estimates of the form $\Box_φ\geq μ^2$, where $μ:\mathbb{C}^n\rightarrow\mathbb{R}$ acts by pointwise multiplication on $(0,1)$-forms, and the inequality is in the sense of self-adjoint operators. We recently proved (arxiv.org:1502.00865) how to derive from $μ$-coercivity estimates for $\Box_φ$ pointwise bounds for the weighted Bergman kernel associated to $φ$. Here we introduce a technique to establish $μ$-coercivity with $ μ(z,w)=c(1+|z|^a+|w|^b) \qquad(a,b\geq0),$ where $a,b\geq0$ depend (and are easily computable from) $Γ$. As a corollary we also prove that, for a wide class of model monomial weights, the spectrum of $\Box_φ$ is discrete if and only if the weight is not decoupled, i.e. $Γ$ contains at least a point $(α,β)$ with $α\neq0\neqβ$. Our methods comprise a new holomorphic uncertainty principle and linear optimization arguments.