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The weighted Kohn Laplacian $\Box_\varphi$ is a natural second order ellipticoperator associated to a weight $\varphi:\mathbb{C}^n\rightarrow\mathbb{R}$ andacting on $(0,1)$-forms, which plays a key role in several questions of complexanalysis. We consider here the case of model monomial weights in$\mathbb{C}^2$, i.e., $ \varphi(z,w):=\sum_{(\alpha,\beta)\in\Gamma}|z^\alphaw^\beta|^2, $ where $\Gamma\subseteq \mathbb{N}^2$ is finite. Our goal is toprove coercivity estimates of the form $\Box_\varphi\geq \mu^2$, 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. Werecently proved (arxiv.org:1502.00865) how to derive from $\mu$-coercivityestimates for $\Box_\varphi$ pointwise bounds for the weighted Bergman kernelassociated to $\varphi$. Here we introduce a technique to establish$\mu$-coercivity with $ \mu(z,w)=c(1+|z|^a+|w|^b) \qquad(a,b\geq0),$ where$a,b\geq0$ depend (and are easily computable from) $\Gamma$. As a corollary wealso 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$\alpha\neq0\neq\beta$. Our methods comprise a new holomorphic uncertaintyprinciple and linear optimization arguments.

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