A Weyl pseudodifferential calculus associated with exponential weights on Rd

Abstract

We construct a Weyl pseudodifferential calculus tailored to studying boundedness of operators on weighted $L^p$ spaces over ${\mathbb{R}}^d$ with weights of the form $exp(-\mathit{\varphi }(x))$, for ϕ a $C^2$ function, a setting in which the operator associated to the weighted Dirichlet form typically has only holomorphic functional calculus. A symbol class giving rise to bounded operators on $L^p$ is determined, and its properties are analyzed. This theory is used to calculate an upper bounded on the $H^{\mathrm{\infty }}$ angle of relevant operators and deduces known optimal results in some cases. Finally, the symbol class is enriched and studied under an algebraic viewpoint.