Estimates for the wave kernel near focal points on compact manifolds

Abstract

This article deals with analogue statements of the so-called basic Strichartz inequality for certain values of the time variable t on a smooth compact manifold; that is we prove L^(q′) → L^q bounds for the modified half-wave operator e^(itP) P⁻^((n+1)(1/2− 1/q)) where P=√−Δ+c² for a set of times t which depends on the global behavior of the geodesic flow. Then we give estimates for the blow-up of the bounds as approaching the limit points of this set. In doing this we use facts from differential geometry and the calculus of variations.