---
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 $\exp(-\phi(x))$, for $\phi$ 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 analysed. This theory is used to calculate an upper bounded on the $H^\infty$ angle of relevant operators, and deduces known optimal results in some cases. Finally, the symbol class is enriched and studied under an algebraic viewpoint.'
address: 'Hanna Neumann Building \#145, Science Road The Australian National University Canberra ACT 2601.'
author:
- Sean Harris
bibliography:
- 'mybib.bib'
title: 'A Weyl Pseudodifferential Calculus associated with exponential weights on $\mathbb{R}^d$'
---

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Introduction
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We construct a Weyl pseudodifferential calculus on typically non-doubling measure spaces over $\mathbb{R}^d$, with an aim to study the $L^p$ behaviour of the natural analogue of the Laplacian in such contexts. Typically, this analogue of the Laplacian is such that bounded spectral multipliers on $L^p$ have to be of holomorphic type for $p \neq 2$. From a PDE point of view, the operators 