Norm inequalities for convolution operators

Nursultanov, Erlan; Tikhonov, Sergey; Tleukhanova, Nazerke

doi:10.1016/j.crma.2009.10.003

Harmonic Analysis

Norm inequalities for convolution operators
[Inégalités de normes pour les opérateurs de convolution]

Présenté par : Jean-Pierre Kahane

Nursultanov, Erlan ¹ ; Tikhonov, Sergey ² ; Tleukhanova, Nazerke ³

¹ Kazakh Branch of Moscow State University, Munatpasova, 7, 010010 Astana, Kazakhstan
² ICREA and Centre de Recerca Matemàtica, Apartat 50, 08193 Bellaterra, Barcelona, Spain
³ Gumilyov Eurasian National University, Munatpasova, 5, 010008 Astana, Kazakhstan

Comptes Rendus. Mathématique, Tome 347 (2009) no. 23-24, pp. 1385-1388.

Résumé
Abstract

Nous étudions des inégalités de normes de convolutions dans les espaces de Lebesgue et de Lorentz. En premier lieu, nous améliorons l'inégalité bien connue de O'Neil sur les opérateurs de convolution et nous établissons une minoration. En second lieu, nous donnons une estimation du type de Young–O'Neil dans les espaces de Lorentz, à savoir $‖ K * f ‖_{L (p, h_{1}) \to L (p, h_{2})}$ . Enfin, nous présentons des estimations similaires dans les espaces de Lorentz à poids.

We study norm convolution inequalities in Lebesgue and Lorentz spaces. First, we improve the well-known O'Neil's inequality for the convolution operators and prove corresponding estimate from below. Second, we obtain Young–O'Neil-type estimate in the Lorentz spaces for the limit value parameters, i.e., $‖ K * f ‖_{L (p, h_{1}) \to L (p, h_{2})}$ . Finally, similar estimates in the weighted Lorentz spaces are presented.

Reçu le : 2009-09-20
Accepté le : 2009-10-06
Publié le : 2009-10-29

DOI : 10.1016/j.crma.2009.10.003

Affiliations des auteurs :

Nursultanov, Erlan ¹ ; Tikhonov, Sergey ² ; Tleukhanova, Nazerke ³

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     title = {Norm inequalities for convolution operators},
     journal = {Comptes Rendus. Math\'ematique},
     pages = {1385--1388},
     publisher = {Elsevier},
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     doi = {10.1016/j.crma.2009.10.003},
     language = {en},
     url = {http://www.numdam.org/articles/10.1016/j.crma.2009.10.003/}
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Nursultanov, Erlan; Tikhonov, Sergey; Tleukhanova, Nazerke. Norm inequalities for convolution operators. Comptes Rendus. Mathématique, Tome 347 (2009) no. 23-24, pp. 1385-1388. doi : 10.1016/j.crma.2009.10.003. http://www.numdam.org/articles/10.1016/j.crma.2009.10.003/

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