Sharp LpLp estimates for discrete second order Riesz transforms

We show that multipliers of second order Riesz transforms on products of discrete abelian groups enjoy the LpLp estimate p⁎−1p⁎−1, where p⁎=max{p,q}p⁎=max{p,q} and p and q   are conjugate exponents. This estimate is sharp if one considers all multipliers of the form ∑iσiRiRi⁎ with |σi|⩽1|σi|⩽1 and infinite groups. In the real valued case, we obtain better sharp estimates for some specific multipliers, such as ∑iσiRiRi⁎ with 0⩽σi⩽10⩽σi⩽1. These are the first known precise LpLp estimates for discrete Calderón–Zygmund operators.