Sharp $L^p$ estimates for discrete second order {R}iesz transforms

Abstract: We show that multipliers of second order Riesz transforms on products of discrete abelian groups enjoy the $L^{p} $ estimate $p^{\ast} -1$, where $p^{\ast} = \max { p,q }$ and $p$ and $q$ are conjugate exponents. This estimate is sharp if one considers all multipliers of the form $\sum_i \sigma_{i} R_{i} R^{\ast}_{i}$ with $| \sigma_{i} | \leqslant 1$ and infinite groups. In the real valued case, we obtain better sharp estimates for some specific multipliers, such as $\sum_{i} \sigma_{i} R_{i} R^{\ast}_{i}$ with $0 \leqslant \sigma_{i} \leqslant 1$. These are the first known precise $L^{p} $ estimates for discrete Calder\'on-Zygmund operators.