Necessary condition for measures which are $(L^{q},L^{p})$ multipliers

Bérenger Akon Kpata; Ibrahim Fofana; Konin Koua

doi:10.5802/ambp.271

Necessary condition for measures which are

(L^{q}, L^{p})

multipliers

Bérenger Akon Kpata ¹ ; Ibrahim Fofana ¹ ; Konin Koua ¹

¹ UFR Mathématiques et Informatique Université de Cocody 22 BP 582 Abidjan 22 Côte d’Ivoire

Annales mathématiques Blaise Pascal, Tome 16 (2009) no. 2, pp. 339-353.

Résumé
Abstract

Soit $G$ un groupe localement compact et $ρ$ la mesure de Haar à gauche sur $G$ . Etant donné une mesure de Radon positive $μ$ , nous établissons une condition nécessaire sur les couples $(q, p)$ pour lesquels $μ$ est un multiplicateur de $L^{q} (G, ρ)$ dans $L^{p} (G, ρ)$ . Appliqué à $ℝ^{n}$ , notre résultat est plus fort que la condition nécessaire établie par Oberlin dans [14] et est très lié à une classe de mesures définie par Fofana dans [7].

Lorsque $G$ est le tore, nous obtenons une généralisation d’une condition énoncée par Oberlin [15] et l’améliorons dans certains cas.

Let $G$ be a locally compact group and $ρ$ the left Haar measure on $G$ . Given a non-negative Radon measure $μ$ , we establish a necessary condition on the pairs $(q, p)$ for which $μ$ is a multiplier from $L^{q} (G, ρ)$ to $L^{p} (G, ρ)$ . Applied to $ℝ^{n}$ , our result is stronger than the necessary condition established by Oberlin in [14] and is closely related to a class of measures defined by Fofana in [7].

When $G$ is the circle group, we obtain a generalization of a condition stated by Oberlin [15] and improve on it in some cases.

MR Zbl

DOI : 10.5802/ambp.271

Classification : 43A05, 43A15
Keywords: Cantor-Lebesgue measure, $L^{q}$-improving measure, non-negative Radon measure
Mots clés : Mesure de Cantor-Lebesgue, mesure $L^{q}$-improving, mesure de Radon positive

Affiliations des auteurs :

Bérenger Akon Kpata ¹ ; Ibrahim Fofana ¹ ; Konin Koua ¹

¹ UFR Mathématiques et Informatique Université de Cocody 22 BP 582 Abidjan 22 Côte d’Ivoire

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     title = {Necessary condition for measures which are $(L^{q},L^{p})$ multipliers},
     journal = {Annales math\'ematiques Blaise Pascal},
     pages = {339--353},
     publisher = {Annales math\'ematiques Blaise Pascal},
     volume = {16},
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     zbl = {1178.43001},
     language = {en},
     url = {https://ambp.centre-mersenne.org/articles/10.5802/ambp.271/}
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Bérenger Akon Kpata; Ibrahim Fofana; Konin Koua. Necessary condition for measures which are $(L^{q},L^{p})$ multipliers. Annales mathématiques Blaise Pascal, Tome 16 (2009) no. 2, pp. 339-353. doi : 10.5802/ambp.271. https://ambp.centre-mersenne.org/articles/10.5802/ambp.271/

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