Semi-classical functional calculus on manifolds with ends and weighted $L^p$ estimates

Bouclet, Jean-Marc

Annales de l'Institut Fourier, Tome 61 (2011) no. 3, pp. 1181-1223.

Pour une classe de variétés riemanniennes à bouts, nous donnons des développements semi-classiques de fonctions bornées du Laplacien. Nous étudions ensuite des propriétés de continuité $L^p$ de ces opérateurs et montrons en particulier que, bien qu’ils ne soient en général pas bornés sur $L^p$, ils le sont toujours sur des espaces $L^p$ à poids convenables.

For a class of non compact Riemannian manifolds with ends, we give semi-classical expansions of bounded functions of the Laplacian. We then study related $L^p$ boundedness properties of these operators and show in particular that, although they are not bounded on $L^p$ in general, they are always bounded on suitable weighted $L^p$ spaces.

Classification : 58J40
Keywords: Manifold with ends, $L^p$ estimates, $h$-pseudodifferential operators
Mot clés : variété à bouts, estimations $L^p$, opérateurs $h$-pseudodifférentiels

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