Riesz potentials and amalgams

Cowling, Michael ; Meda, Stefano ; Pasquale, Roberta

Annales de l'Institut Fourier, Tome 49 (1999), p. 1345-1367 / Harvested from Numdam

Résumé

Soit $(M,d)$ un espace métrique, muni d’une mesure borélienne $\mu $ telle que la mesure $\mu (B(x,\rho ))$ de la boule $B(x,\rho )$ de centre $x$ et de rayon $\rho $ soit polynomiale en $\rho $. Un amalgame $A^q_p(M)$ est un espace de fonctions qui ressemble localement à $L^p(M)$ et globalement à $L^q (M)$. On étudie les applications linéaires entre amalgames dont les noyaux se comportent comme $d(x,y)^{-a}$ quand $d(x,y) \le 1$ et comme $d(x,y)^{-b}$ quand $d(x,y) \ge 1$. On démontre un théorème de régularité du type Hardy–Littlewood–Sobolev pour l’opérateur de Laplace–Beltrami sur certaines variétés riemanniennes et pour certains opérateurs sous-elliptiques sur les groupes de Lie à croissance polynomiale.

Abstract

Let $(M,d)$ be a metric space, equipped with a Borel measure $\mu $ satisfying suitable compatibility conditions. An amalgam $A^q_p (M) $ is a space which looks locally like $L^p(M)$ but globally like $L^q (M)$. We consider the case where the measure $\mu (B(x,\rho )$ of the ball $B(x,\rho )$ with centre $x$ and radius $\rho $ behaves like a polynomial in $\rho $, and consider the mapping properties between amalgams of kernel operators where the kernel $\ker \,K (x,y)$ behaves like $d(x,y)^{-a}$ when $d(x,y) \le 1$ and like $d(x,y)^{-b}$ when $d(x,y) \ge 1$. As an application, we describe Hardy–Littlewood–Sobolev type regularity theorems for Laplace–Beltrami operators on Riemannian manifolds and for certain subelliptic operators on Lie groups of polynomial growth.

Détail

Publié le : 1999-01-01

MR 2000i:47058 | Zbl 0938.47027 | 1 citation dans Numdam

DOI : https://doi.org/10.5802/aif.1720

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