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

Komla Domelevo; Stefanie Petermichl

doi:10.1016/j.crma.2014.03.022

Harmonic analysis

Sharp

L^{p}

estimates for discrete second-order Riesz transforms

Presented by: Gilles Pisier

Komla Domelevo; Stefanie Petermichl

Comptes Rendus. Mathématique, Volume 352 (2014) no. 6, pp. 503-506.

Abstract
Résumé

We show that multipliers of second-order Riesz transforms on products of discrete Abelian groups enjoy the $L^{p}$ estimate $(p^{⁎} - 1)$ , where $p^{⁎} = \max {p, q}$ , $1 / p + 1 / q = 1$ . This estimate is sharp for certain multipliers such as $R_{1} R_{1}^{⁎} - R_{2} R_{2}^{⁎}$ on products of infinite groups. For other multipliers such as $R_{1} R_{1}^{⁎}$ , the best possible estimate is given by the Choi constant. Those are the first known sharp $L^{p}$ estimates of discrete Calderón–Zygmund operators.

Nous montrons que les carrés des transformations de Riesz sur des produits de groupes abéliens discrets ont une norme $L^{p}$ bornée par la constante $(p^{⁎} - 1)$ , avec $p^{⁎} = \max {p, q}$ , $1 / p + 1 / q = 1$ . Cette constante est optimale dans le cas de goupes infinis pour certains opérateurs, parmi lesquels $R_{1} R_{1}^{⁎} - R_{2} R_{2}^{⁎}$ . Pour d'autres opérateurs, parmi lesquels $R_{1} R_{1}^{⁎}$ , la constante optimale est donnée par la constante de Choi. Il s'agit des premières estimations $L^{p}$ optimales connues d'opérateurs discrets de type Calderón–Zygmund.

Received: 2014-03-06
Accepted: 2014-03-31
Published online: 2014-04-26

DOI: 10.1016/j.crma.2014.03.022

@article{CRMATH_2014__352_6_503_0,
     author = {Komla Domelevo and Stefanie Petermichl},
     title = {Sharp $ {L}^{p}$ estimates for discrete second-order {Riesz} transforms},
     journal = {Comptes Rendus. Math\'ematique},
     pages = {503--506},
     publisher = {Elsevier},
     volume = {352},
     number = {6},
     year = {2014},
     doi = {10.1016/j.crma.2014.03.022},
     language = {en},
}

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Komla Domelevo; Stefanie Petermichl. Sharp $ {L}^{p}$ estimates for discrete second-order Riesz transforms. Comptes Rendus. Mathématique, Volume 352 (2014) no. 6, pp. 503-506. doi : 10.1016/j.crma.2014.03.022. https://comptes-rendus.academie-sciences.fr/mathematique/articles/10.1016/j.crma.2014.03.022/

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Cited by

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