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) no. 2, pp. 339-353.

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 ${ℝ}^n$, 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 $\rho$ 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 ${ℝ}^n$, 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.

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

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