Semiclassical resolvent estimates for trapping perturbations

We study the semiclassical estimates of the resolvent R(lambda + i tau), lambda is an element of J subset of subset of R+, tau is an element of]0, 1] of a self-adjoint operator L(h) in the space of bounded operators L(H-0,H-s , H-0,H--s), s > 1/2. In the general case of long-range trapping black-box perturbations we prove that the estimate of the cut-off resolvent \\chi(x)R(lambda + i0)chi(x)\\(-->H) less than or equal to C exp(Ch(-p)), chi(x) is an element of C-0(infinity)(R-n), p greater than or equal to 1 implies the estimate \\R(lambda + i tau)\\(s,-s) less than or equal to C-1 exp(C(1)h(-p)).