Gundy-Varopoulos martingale transforms and their projection operators on manifolds and vector bundles

TL;DR

This paper establishes the $L^p$ boundedness of generalized Riesz transforms derived from martingale transforms on various manifolds and vector bundles, with applications to Lie groups, the Heisenberg group, and spinor forms.

Contribution

It proves the $L^p$ boundedness of generalized first order Riesz transforms on manifolds and vector bundles, extending previous results to more general diffusions and settings.

Findings

$L^p$ boundedness of Riesz transforms on manifolds and bundles

Applications to Lie groups, Heisenberg group, and spinor forms

Examples demonstrating the theory's broad applicability

Abstract

This paper proves the $L^p$ boundedness of generalized first order Riesz transforms obtained as conditional expectations of martingale transforms \`a la Gundy-Varopoulos for quite general diffusions on manifolds and vector bundles. Several specific examples and applications are presented: Lie groups of compact type, the Heisenberg group, SU(2), and Riesz transforms on forms and spinors.