Sharp $L^p$ estimates for second order Riesz transforms on multiply–connected Lie groups

Abstract

We study a class of combinations of second order Riesz transforms on Lie groups $G = G_x \times G_y$ that are multiply connected, composed of a discrete abelian component $G_x$ and a compact connected component $G_y$ . We prove sharp $L^p$ estimates for these operators, therefore generalizing previous results [13][4]. The proof uses stochastic integrals with jump components adapted to functions defined on the semi-discrete set $G = G_x \times G_y$ . 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