A weyl pseudodifferential calculus associated with exponential weights on ℝ^d

We construct a Weyl pseudodifferential calculus tailored to studying boundedness of operators on weighted L^p spaces over ℝ^d with weights of the form exp. Φ.x//,for Φ a C² 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 calcu-late an upper bounded on the H¹ angle of relevant operators and deduces known optimal results in some cases. Finally, the symbol class is enriched and studied under an algebraic viewpoint.