Propagation of low regularity for solutions of nonlinear PDEs on a Riemannian manifold with a sub-Laplacian structure

TL;DR

This paper develops a para-product framework on Riemannian manifolds with sub-Laplacian structures, enabling the analysis of regularity propagation in nonlinear PDEs without relying on Fourier transforms.

Contribution

It introduces a para-product concept on manifolds with sub-Laplacian structures and proves a paralinearization theorem applicable to nonlinear PDEs in this setting.

Findings

Established a para-product framework on non-Euclidean manifolds.

Proved a paralinearization theorem for nonlinear PDEs on such manifolds.

Applied the theory to demonstrate regularity propagation in specific PDEs.

Abstract

Following \cite{B2}, we introduce a notion of para-products associated to a semi-group. We do not use Fourier transform arguments and the background manifold is doubling, endowed with a sub-laplacian structure. Our main result is a paralinearization theorem in a non-euclidean framework, with an application to the propagation of regularity for some nonlinear PDEs.