Interior energy estimates for the linear wave equation

We give a few new interior energy estimates for some linear wave equations inside the light cone for all space dimensions except two. For the free case, using only integration by parts argument essentially, we give more sharp estimate than the previous results by R. Strichratz or L. Vega and N. Visciglia. For the perturbed case, based on the method for the free case, we show the same type of the estimates as the free case under the assumption that the linear potential is of short range. For the proof, we derive a new propagation estimate for the first-order differential operators on space time region. As a Corollary, we give an extension result of R. Strichartz for the perturbed case.