Various sharp estimates for semi-discrete Riesz transforms of the second order

We give several sharp estimates for 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 connected component ${G}_{y}$ endowed with a biinvariant measure. These estimates include new sharp $L^p$ estimates via Choi type constants, depending upon the multipliers of the operator. They also include weak-type, logarithmic and exponential estimates. We give an optimal $L^q \to L^p$ estimate as well. It was shown recently by Arcozzi, Domelevo and Petermichl that such 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 proofs of our theorems combine this stochastic integral representation with a number of deep estimates for pairs of martingales under strong differential subordination by Choi, Banuelos and Osekowski. When two continuous directions are available, sharpness is shown via the laminates technique. We show that sharpness is preserved in the discrete case using Lax-Richtmyer theorem.