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

Komla Domelevo (1) , Stefanie Petermichl (1)

1 IMT - Institut de Mathématiques de Toulouse UMR5219

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 $\sum_{i} \sigma_{i} R_{i} R^{\ast}_{i} \nocomma$ with $| \sigma_{i} | \leqslant 1$ and infinite groups. In the real valued case, we obtain better sharp estimates for some specific multipliers, such as $\sum_{i} \sigma_{i} R_{i} R^{\ast}_{i} \nocomma$ with $0 \leqslant \sigma_{i} \leqslant 1$. These are the first known precise $L^{p} $ estimates for discrete Calder{ó}n-Zygmund operators.

Mots clés: Discrete groups; Second order Riesz transforms; Sharp $L^{p}$ estimates; Sharp $L^p$ estimates

Domaines: Analyse classique [math.CA]; Probabilités [math.PR]

Komla Domelevo, Stefanie Petermichl. Sharp $L^{p}$ estimates for discrete second order Riesz transforms. Advances in Mathematics, 2014, 262, pp.932--952. ⟨10.1016/j.aim.2014.06.003⟩. ⟨hal-00971440⟩