Second Order Riesz Transforms on Multiply–Connected Lie Groups and Processes with Jumps

We study a class of combinations of second order Riesz transforms on Lie groups G= Gx× Gy that are multiply connected, composed of a discrete abelian component Gx and a compact connected component Gy. We prove sharp Lp estimates for these operators, therefore generalizing previous results (Volberg, A., Nazarov, F.: St Petersburg Math J. 15 (14), 563–573 2004; Domelevo, K., Petermichl, S.: Adv. Math. 262, 932–952 2014; Bañuelos, R., Baudoin, F.: Potential Anal. 38(4), 1071–1089 2013). We construct stochastic integrals with jump components adapted to functions defined on the semi-discrete set Gx× Gy. 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 Itô 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.