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Comptes Rendus Mathématique
Volume 347, n° 23-24
pages 1385-1388 (décembre 2009)
Doi : 10.1016/j.crma.2009.10.003
Received : 20 September 2009 ;  accepted : 6 October 2009
Norm inequalities for convolution operators
Inégalités de normes pour les opérateurs de convolution
 

Erlan Nursultanov a , Sergey Tikhonov b , Nazerke Tleukhanova c
a Kazakh Branch of Moscow State University, Munatpasova, 7, 010010 Astana, Kazakhstan 
b ICREA and Centre de Recerca Matemàtica, Apartat 50, 08193 Bellaterra, Barcelona, Spain 
c Gumilyov Eurasian National University, Munatpasova, 5, 010008 Astana, Kazakhstan 

Abstract

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.,  . Finally, similar estimates in the weighted Lorentz spaces are presented. To cite this article: E. Nursultanov et al., C. R. Acad. Sci. Paris, Ser. I 347 (2009).

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Résumé

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  . Enfin, nous présentons des estimations similaires dans les espaces de Lorentz à poids. Pour citer cet article : E. Nursultanov et al., C. R. Acad. Sci. Paris, Ser. I 347 (2009).

The full text of this article is available in PDF format.


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