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

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

Annales mathématiques Blaise Pascal, Tome 16 (2009), p. 339-353 / Harvested from Numdam

Résumé

Soit $G$ un groupe localement compact et $\rho $ la mesure de Haar à gauche sur $G$. Etant donné une mesure de Radon positive $\mu $, nous établissons une condition nécessaire sur les couples $\left( q,\ p\right) $ pour lesquels $\mu $ est un multiplicateur de $L^{q}\left( G,\ \rho \right) $ dans $L^{p}\left( G,\ \rho \right) $. Appliqué à $\mathbb{R}^{n}$, 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.

Abstract

Let $G$ be a locally compact group and $\rho $ the left Haar measure on $G$. Given a non-negative Radon measure $\mu $, we establish a necessary condition on the pairs $\left( q,\ p\right) $ for which $\mu $ is a multiplier from $L^{q}\left( G,\ \rho \right) $ to $L^{p}\left( G,\ \rho \right) $. Applied to $\mathbb{R}^{n}$, 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.

Détail

Publié le : 2009-01-01

MR 2568870 | Zbl 1178.43001

DOI : https://doi.org/10.5802/ambp.271
Classification:  43A05,  43A15
Mots clés: Mesure de Cantor-Lebesgue, mesure $L^{q}$-improving, mesure de Radon positive

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