Subelliptic pseudo-differential operators and Fourier integral operators on compact Lie groups

TL;DR

This paper develops a comprehensive subelliptic pseudo-differential calculus on compact Lie groups, extending classical theories to sub-Riemannian structures and establishing boundedness, ellipticity, and functional calculus results.

Contribution

It introduces a new subelliptic pseudo-differential calculus on compact Lie groups based on matrix-valued quantisation, extending classical results to sub-Riemannian settings.

Findings

Established boundedness on Sobolev and Besov spaces

Proved subelliptic versions of Fefferman and Calderón-Vaillancourt theorems

Constructed parametrices and analyzed heat traces for subelliptic operators

Abstract

In this memoir we extend the theory of global pseudo-differential operators to the setting of arbitrary sub-Riemannian structures on a compact Lie group. More precisely, given a compact Lie group $G$, and the sub-Laplacian $\mathcal{L}$ associated to a system of vector fields $X=\{X_1,\cdots,X_k\}$ satisfying the H\"ormander condition, we introduce a (subelliptic) pseudo-differential calculus associated to $\mathcal{L},$ based on the matrix-valued quantisation process developed in [138]. This theory will be developed as follows. First, we will investigate the singular kernels of this calculus, estimates of $L^p$-$L^p$, $H^1$-$L^1$, $L^\infty$-$BMO$ type and also the weak (1,1) boundedness of these subelliptic H\"ormander classes. Between the obtained estimates we prove subelliptic versions of the celebrated sharp Fefferman $L^p$-theorem and the Calder\'on-Vaillancourt theorem. The obtained estimates will be used to establish the boundedness of subelliptic operators on subelliptic Sobolev and Besov spaces. We will investigate the ellipticity, the construction of parametrices, the heat traces and the regularisation of traces for the developed subelliptic calculus. A subelliptic global functional calculus will be established as well as a subelliptic version of Hulanicki theorem. The approach established in characterising our subelliptic H\"ormander classes (by proving that the definition of these classes is independent of certain parameters) will be also applied in order to characterise the global H\"ormander classes on arbitrary graded Lie groups developed in [90].