Nous montrons que les carrés des transformations de Riesz sur des produits de groupes abéliens discrets ont une norme bornée par la constante , avec , . Cette constante est optimale dans le cas de goupes infinis pour certains opérateurs, parmi lesquels . Pour d'autres opérateurs, parmi lesquels , la constante optimale est donnée par la constante de Choi. Il s'agit des premières estimations 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 estimate , where , . This estimate is sharp for certain multipliers such as on products of infinite groups. For other multipliers such as , the best possible estimate is given by the Choi constant. Those are the first known sharp estimates of discrete Calderón–Zygmund operators.
@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}, }
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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