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

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.

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.

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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},
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Cowling, Michael; Meda, Stefano; Pasquale, Roberta. Riesz potentials and amalgams. Annales de l'Institut Fourier, Tome 49 (1999) no. 4, pp. 1345-1367. doi : 10.5802/aif.1720. https://aif.centre-mersenne.org/articles/10.5802/aif.1720/

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