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The purpose of the present paper is to establish appropriate cut-offresolvent estimates for the Dirichlet Laplacian on exterior domains. Thegeometrical assumptions on domains are rather general, for example,non-trapping condition is not imposed. The first key assumption guarantees theresult on propagation of singularities, the second is the smallness of theLebesgue measure of the portions of the trapped sets in the fibers of cospherebundle, and the third concerns the upper bound of the sojourn time. As aby-product of these estimates, the local energy decay estimate for solutions tothe initial-boundary value problem for wave equation in the case of odd spacedimensions is obtained.

On wave equation outside trapping obstacles and local energy decay in odd space dimensions