Sharp LP estimates for discrete second-order Riesz transforms

We show that multipliers of second-order Riesz transforms on products of discrete Abelian groups enjoy the L-P estimate (p* - 1), where p* = max{p, q}, 1/p + 1/q = 1. This estimate is sharp for certain multipliers such as R1R1* - R2R2* on products of infinite groups. For other multipliers such as R1R1*, the best possible estimate is given by the Choi constant. Those are the first known sharp L-P estimates of discrete Calderon-Zygmund operators. (C) 2014 Academie des sciences. Published by Elsevier Masson SAS. All rights reserved.