Analyticity properties and estimates of resolvent kernels near thresholds

Resolvent estimates are derived for the family of ordinary differential ope rators {-c(2)(y)[rho(y)d/dy (1/rho(y) d/dy) - p(2)]}, p is an element of [0, infin ity), y is an element of R. It is assumed that c(y) = c(+/-) > 0, rho(y) = rho(+/-) for +/-y > y(c), an d the kernels are studied in neighborhoods of the points {c(+/-)(2)p(2)}, u niformly in compact intervals of p. This family arises in the direct integr al decomposition of the acoustic propagator in layered media, -c(2)(y) rho( y) del(x,y) (1/rho(y) del(x,y)), x is an element of R-n, and the results im ply "low energy" estimates for this operator, as well as the validity of th e "limiting absorption principle".