Boutet de Monvel A., Georgescu V., Sahbani J.
Boundary Values of Regular Resolvent Families.
(120K, TeX)
ABSTRACT. We study properties of the boundary values $(H-\gl\pm i0)^{-1}$
of the resolvent of a self-adjoint operator $H$ for $\gl$ in a
real open set $\gW$ on which $H$ admits a locally strictly
conjugate operator $A$ (in the sense of E.~Mourre, i.e.\
$\gf(H)^*[H,iA]\gf(H)\geq a|\gf(H)|^2$ for some real $a>0$ if
$\gf\in C_0^\infty(\gW)$). In particular, we determine the
H\"older-Zygmund class of the $B(\C{E};\C{F})$-valued maps $\gl
\mapsto(H-\gl\pm i0)^{-1}$ and
$\gl\mapsto\gP_\pm(H-\gl\pm i0))^{-1}$ in terms of the regularity
\mapsto(H-\gl\pm i0)^{-1}$ and
$\gl\mapsto\gP_\pm(H-\gl\pm i0))^{-1}$ in terms of the regularity
properties of the map $\gt \mapsto e^{-iA\gt}He^{iA\gt}$. Here
$\C{E}$,
$\C{F}$ are spaces from the Besov scale associated to $A$ and
$\gP_\pm$ are the spectral projections of $A$ associated to the
half-lines $\pm x>0$.