Riesz potentials and amalgams

Cowling, Michael; Meda, Stefano; Pasquale, Roberta

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

Soit $(M,d)$ un espace métrique, muni d’une mesure borélienne $\mu$ telle que la mesure $\mu \left(B\right(x,\rho \left)\right)$ de la boule $B(x,\rho )$ de centre $x$ et de rayon $\rho$ soit polynomiale en $\rho$. Un amalgame $A_p^q\left(M\right)$ est un espace de fonctions qui ressemble localement à $L^p\left(M\right)$ et globalement à $L^q\left(M\right)$. 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.

Let $(M,d)$ be a metric space, equipped with a Borel measure $\mu$ satisfying suitable compatibility conditions. An amalgam $A_p^q\left(M\right)$ is a space which looks locally like $L^p\left(M\right)$ but globally like $L^q\left(M\right)$. We consider the case where the measure $\mu \left(B\right(x,\rho )$ 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 $kerK(x,y)$ 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.

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