Analyticity properties and estimates of resolvent kernels near thresholds

Resolvent estimates are derived for the family of ordinary differential operators (formula presented) It is assumed that c(y) = c_± > 0, ρ(y) = p_± for ±y > y_c, and the kernels are studied in neighborhoods of the points {c 2_±p2}, uniformly in compact intervals of p. This family arises in the direct integral decomposition of the acoustic propagator in layered media, -c²(y) ρ(y) ▽_x,y (1/ρ(y) ▽_x,y), x ∈ ℝⁿ, and the Lemma A.2 For 0 ≤ arg z ≤ τ < π, s > 3/2, n ≥ 3, (A.10) ∥M(z)∥_{B(L2,s(Rn+1),L2,-s(Rn+1))} = O(1) as z → 0.