A maximal oscillatory operator on compact manifolds

This is a continuation of our previous research about an oscillatory integral operator T, on compact manifolds M. We prove the sharp H^p-L^p, boundedness on the maximal operator T^*, for all 0<p<1. As applications, we first prove the sharp H^p-L^p, boundedness on the maximal operator corresponding to the Riesz means I₊, (|L|) associated with the Schr\"odinger type group e^isL^{/2} and obtain the almost everywhere convergence of I₊, (|L|) f (x, t) f (x) for all f H^p. Also, we are able to obtain the convergence speed of a combination operator from the solutions of the Cauchy problem of fractional Schr\"odinger equations. All results are even new on the n-torus T^n.