  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
Schr\"odinger type group $e^{is\mathcal{L}^{\alpha/2}}$ and obtain the almost
everywhere convergence of $I_{k,\alpha}(|\mathcal{L}|)f(x,t)\to f(x)$ 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
Schr\"odinger equations. All results are even new on the n-torus $T^{n}$.
