Weyl Pseudodifferential Calculus and the Heisenberg Group in New Settings

Abstract

We examine both Weyl pseudodifferential calculus and the Heisenberg group in new settings, through two published papers and a chapter representing work towards a construction of Haar bases of unimodular LC groups. We finish with some remarks about an extension of both Heisenberg group and pseudodifferential calculus techniques to non-Abelian settings, detailing possible links with representation theory and the Langlands program.

The first of the two papers achieves the following

"We give a simple proof of the fact that the classical Ornstein-Uhlenbeck operator L is R-sectorial of angle arcsin|1-2/p| on $L^p(\mathbb{R}^n, \exp(-|x|^2/2)dx)$ (for $1<p<\infty$). Applying the abstract holomorphic functional calculus theory of Kalton and Weis, this immediately gives a new proof of the fact that L has a bounded $H^\infty$ functional calculus with this optimal angle."

In the second paper,

"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 $\phi(x)$ 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."

The construction of Haar bases on unimodular LC groups proceeds via the tools of fractal tilings. After a review of these concepts we prove the key results required to obtain a "good" Haar basis, namely that the boundary of the relevant tilings is of measure $0$. We explain how the constructed Haar bases can be used to study Fourier multipliers in such settings, detailing future work.