Wave Operator on Compact Manifolds

Abstract

On an n-dimensional (n ≥ 2) compact connected manifold without boundary, we obtain the sharp range of p to ensure the Lp convergence of the wave operator. We are not able to show the optimum result at the end point $$\left| {{1 \over p} - {1 \over 2}} \right| = {1 \over {n - 1}}$$ if n ≠ 3, but an interesting result for n = 3 is given. The main result is an extension of a result on n-dimensional Euclidean space ℝn.