A maximal oscillatory operator on compact manifolds

Abstract

This is a continuation of our previous research about an oscillatory integral operator $T_{\hat{I}\pm ,{\hat{I}}^2}$ on compact manifolds $\mathbb{M}$. We prove the sharp $H^p$-$L^{p,\infty }$ boundedness on the maximal operator $T_{\hat{I}\pm ,{\hat{I}}^2}^{\ast }$ 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 $I_{k,\hat{I}\pm }(|\mathcal{L}|)$ associated with the Schrödinger type group $e^{is{\mathcal{L}}^{\hat{I}\pm /2}}$ and obtain the almost everywhere convergence of $I_{k,\hat{I}\pm }(|\mathcal{L}|)f(x,t)\to f(x)$ 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 Schrödinger equations. All results are even new on the n-torus $T^n$.