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