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

Domelevo, Komla; Petermichl, Stefanie

doi:10.1016/j.crma.2014.03.022

Harmonic analysis

Sharp

L^{p}

estimates for discrete second-order Riesz transforms

Présenté par : Gilles Pisier

Domelevo, Komla ; Petermichl, Stefanie

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

Résumé
Abstract

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.

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.

Reçu le : 2014-03-06
Accepté le : 2014-03-31
Publié le : 2014-04-26

DOI : 10.1016/j.crma.2014.03.022

@article{CRMATH_2014__352_6_503_0,
     author = {Domelevo, Komla and Petermichl, Stefanie},
     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},
     url = {http://www.numdam.org/articles/10.1016/j.crma.2014.03.022/}
}

TY  - JOUR
AU  - Domelevo, Komla
AU  - Petermichl, Stefanie
TI  - Sharp $ {L}^{p}$ estimates for discrete second-order Riesz transforms
JO  - Comptes Rendus. Mathématique
PY  - 2014
SP  - 503
EP  - 506
VL  - 352
IS  - 6
PB  - Elsevier
UR  - http://www.numdam.org/articles/10.1016/j.crma.2014.03.022/
DO  - 10.1016/j.crma.2014.03.022
LA  - en
ID  - CRMATH_2014__352_6_503_0
ER  -

%0 Journal Article
%A Domelevo, Komla
%A Petermichl, Stefanie
%T Sharp $ {L}^{p}$ estimates for discrete second-order Riesz transforms
%J Comptes Rendus. Mathématique
%D 2014
%P 503-506
%V 352
%N 6
%I Elsevier
%U http://www.numdam.org/articles/10.1016/j.crma.2014.03.022/
%R 10.1016/j.crma.2014.03.022
%G en
%F CRMATH_2014__352_6_503_0

Domelevo, Komla; Petermichl, Stefanie. Sharp $ {L}^{p}$ estimates for discrete second-order Riesz transforms. Comptes Rendus. Mathématique, Tome 352 (2014) no. 6, pp. 503-506. doi : 10.1016/j.crma.2014.03.022. http://www.numdam.org/articles/10.1016/j.crma.2014.03.022/

Bibliographie
Cité par

[1] Burkholder, Donald L. Sharp inequalities for martingales and stochastic integrals, Palaiseau, 1987 (Astérisque), Volume 157–158 (1988), pp. 75-94

[2] Choi, K.P. A sharp inequality for martingale transforms and the unconditional basis constant of a monotone basis in $L^{p} (0, 1)$ , Trans. Amer. Math. Soc., Volume 330 (1992) no. 2, pp. 509-529

[3] Lust-Piquard, F. Dimension free estimates for discrete Riesz transforms on products of Abelian groups, Adv. Math., Volume 185 (2004) no. 2, pp. 289-327

[4] Meyer, P.-A. Transformations de Riesz pour les lois gaussiennes, Seminar on Probability, XVIII, Lecture Notes in Mathematics, vol. 1059, Springer, Berlin, 1984, pp. 179-193

[5] Pisier, G. Riesz transforms: a simpler analytic proof of P.-A. Meyer's inequality, Séminaire de probabilités, XXII, Lecture Notes in Mathematics, vol. 1321, Springer, Berlin, 1988, pp. 485-501

[6] Stein, E.M. Some results in harmonic analysis in $R^{n}$ , for $n \to \infty$ , Bull. Amer. Math. Soc. (N.S.), Volume 9 (1983) no. 1, pp. 71-73

[7] Volberg, A.; Nazarov, F. Heat extension of the Beurling operator and estimates for its norm, St. Petersburg Math. J., Volume 15 (2004) no. 4, pp. 563-573

Cité par Sources :