A maximal oscillatory operator on compact manifolds

Abstract

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