97-428 Boutet de Monvel A., Georgescu V., Sahbani J.

Higher Order Estimates in the Conjugate Operator Theory (251K, LaTex 2e) Aug 4, 97

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Abstract. Let $H$ be a self-adjoint operator which admits a locally conjugate operator $A$. Set $R(z)=(H-z)^{-1}$, let $\Pi_{\pm}$ be the spectral projection of $A$ associated to the interval $\pm \lbrack 0,\infty )$ and let $\C{H}_{s,p}$ ($s\in \D{R}, 1\leq p\leq \infty $) be the Besov scale associated to the operator $A$. We study the regularity properties of the maps $\lambda \mapsto R(\lambda \pm i0)$, $\lambda \mapsto \Pi_{\mp}R(\lambda \pm i0)$ and $\lambda \mapsto \Pi_{\mp}R(\lambda \pm i0)\Pi_{\pm}$ when considered with values in a space of the form $B(\C{H}_{s,p};\C{H}_{t,q})$. Our results imply optimal local decay and propagation properties of $H$ with respect to $A$, in particular estimates of the form $\Vert \langle A\rangle^t\Pi_{\mp}\exp (\mp i\tau H)\langle A\rangle^{-s}\Vert\leq c\tau^{-\alpha }$ for $\tau \geq 1$.

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