Distribution of the resonances and local energy decay in the transmission problem

We study the resonances associated to the transmission problem for a strictly convex obstacle provided that the speed of propagation of the waves in the interior of the obstacle is strictly greater than the speed in the exterior. We prove that there are no resonances in a region of the form ${\rm Im}\,z\leqslant C_1|z|^{-1},\ |{\rm Re}\,z|\geqslant C_2>0$ . Using this we obtain some uniform estimates on the decay of the local energy.