Riesz potentials and amalgams
Annales de l'Institut Fourier, Volume 49 (1999) no. 4, pp. 1345-1367.

Let (M,d) be a metric space, equipped with a Borel measure μ satisfying suitable compatibility conditions. An amalgam A p q (M) is a space which looks locally like L p (M) but globally like L q (M). We consider the case where the measure μ(B(x,ρ) of the ball B(x,ρ) with centre x and radius ρ behaves like a polynomial in ρ, and consider the mapping properties between amalgams of kernel operators where the kernel kerK(x,y) behaves like d(x,y) -a when d(x,y)1 and like d(x,y) -b when d(x,y)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.

Soit (M,d) un espace métrique, muni d’une mesure borélienne μ telle que la mesure μ(B(x,ρ)) de la boule B(x,ρ) de centre x et de rayon ρ soit polynomiale en ρ. Un amalgame A p q (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)1 et comme d(x,y) -b quand d(x,y)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.

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     title = {Riesz potentials and amalgams},
     journal = {Annales de l'Institut Fourier},
     pages = {1345--1367},
     publisher = {Association des Annales de l{\textquoteright}institut Fourier},
     volume = {49},
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}
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Cowling, Michael; Meda, Stefano; Pasquale, Roberta. Riesz potentials and amalgams. Annales de l'Institut Fourier, Volume 49 (1999) no. 4, pp. 1345-1367. doi : 10.5802/aif.1720. http://archive.numdam.org/articles/10.5802/aif.1720/

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