A maximal oscillatory operator on compact manifolds

This is a continuation of our previous research about an oscillatory integral operator Tα, β on compact manifolds M. We prove the sharp Hᵖ-Lp,∞ boundedness on the maximal operator T*α, β for all $0<p<1$. As applications, we first prove the sharp Hᵖ-Lp,∞ boundedness on the maximal operator corresponding to the Riesz means Ik,α(|L|) associated with the Schr\"odinger type group e^isLα/2 and obtain the almost everywhere convergence of Ik,α(|L|)f(x,t)→ f(x) for all f∈ Hᵖ. 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 Tⁿ.