Second order Riesz transforms on multiply-connected Lie groups and processes with jumps

TL;DR

This paper establishes sharp $L^{p}$ bounds for second order Riesz transforms on multiply-connected Lie groups with discrete and compact parts, using stochastic integrals with jumps and novel representations.

Contribution

It extends previous results by providing sharp estimates and stochastic integral representations for Riesz transforms on semi-discrete Lie groups with jumps.

Findings

Sharp $L^{p}$ estimates for the operators.

Stochastic integral representations involving jump processes.

Discrete tangent plane dimension doubling due to non-local derivatives.

Abstract

We study a class of combinations of second order Riesz transforms on Lie groups that are multiply connected, composed of a discrete abelian component and a compact connected component. We prove sharp $L^{p}$ estimates for these operators, therefore generalising previous results. We construct stochastic integrals with jump components adapted to functions defined on our semi-discrete set. We show that these second order Riesz transforms applied to a function may be written as conditional expectation of a simple transformation of a stochastic integral associated with the function. The analysis shows that Ito integrals for the discrete component must be written in an augmented discrete tangent plane of dimension twice larger than expected, and in a suitably chosen discrete coordinate system. Those artifacts are related to the difficulties that arise due to the discrete component, where derivatives of functions are no longer local. Previous representations of Riesz transforms through stochastic integrals in this direction do not consider discrete components and jump processes.