Low frecuency resolvent estimates for long range perturbations of the euclidean Laplacian

• Autores: Jean-François Bony, Dietrich Häfner
• Localización: Mathematical research letters, ISSN 1073-2780, Vol. 17, Nº 2-3, 2010, págs. 301-306
• Idioma: inglés
• DOI: 10.4310/mrl.2010.v17.n2.a9
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• Resumen

  • Let $P$ be a long range metric perturbation of the Euclidean Laplacian on $\R^d,\, d\ge 3$. We prove that the following resolvent estimate holds: \begin{equation*} \Vert\^{-\alpha}(P-z)^{-1}\^{-\beta}\Vert\lesssim 1\quad \forall z\in \C\setminus \R,\,\vert z\vert<1, \end{equation*} if $\alpha,\beta> 1/2$ and $\alpha+\beta> 2$. The above estimate is false for the Euclidean Laplacian in dimension $3$ if $\alpha\le 1/2$ or $\beta\le 1/2$ or $\alpha+\beta<2$.