Coercivity of weighted Kohn Laplacians: the case of model monomial weights in C²

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The weighted Kohn Laplacian φ is a natural second order elliptic operator associated to a weight φ:Cⁿ 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 C², i.e., φ(z,w):=∑₍α,ᵦ₎∈ᵧ|zαwβ|², where Γ N² is finite. Our goal is to prove coercivity estimates of the form φ≥ μ², where μ:Cⁿ 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 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) (a,b≥0), where a,b≥0 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 φ is discrete if and only if the weight is not decoupled, i.e. Γ contains at least a point (α,β) with α≠0≠β. Our methods comprise a new holomorphic uncertainty principle and linear optimization arguments.