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

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 ${\sum }_i{\sigma }_i{R}_i{R}_i^{\ast }$ with $0 \leqslant \sigma_{i} \leqslant 1$\mathrm{.}Thesearethefirstknownprecise$L^{p} $ estimates for discrete Calder'on-Zygmund operators.