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

TL;DR

This paper establishes sharp $L^p$ bounds for second order Riesz transforms on discrete abelian groups, providing the first precise estimates for such discrete Calderón-Zygmund operators and their multipliers.

Contribution

It proves sharp $L^p$ estimates for second order Riesz transform multipliers on discrete groups, advancing understanding of discrete Calderón-Zygmund operators.

Findings

Sharp $L^p$ estimate $p^*-1$ for second order Riesz transform multipliers.

Optimal bounds for specific multipliers with real values.

First precise $L^p$ estimates for discrete Calderón-Zygmund operators.

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.