BOUNDARY-VALUES OF REGULAR RESOLVENT FAMILIES

We study properties of the boundary values (H - lambda +/- i0)(-1) of the resolvent of a selfadjoint operator H for lambda in a real open se t Omega on which H admits a locally strictly conjugate operator A tin the sense of E. Mourre, i.e. phi(H)[H, iA]phi(H) greater than or equa l to a/phi(H)\(2) for some real a > 0 if phi is an element of C-0(infi nity)(Omega)). In particular, we determine the Holder-Zygmund class of the B(E; F)-valued maps lambda --> (H - lambda +/- i0)(-1) and lambda --> Pi(+/-) (H - lambda +/- i0))(-1) in terms of the regularity prope rties of the map tau --> e(-iA tau) He-iA tau. Here E, F are spaces fr om the Besov scale associated to A and Pi(+/-) are the spectral projec tions of A associated to the half-lines +/-x > 0.