Convergence of Multiplier Operators on Compact Manifolds

We study a family of multiplier operators $T_{m_{\gamma }\left(t\cdot \right)}\left(f\right)$ on compact manifolds $M^n$, which is an analogue of the spherical average $S_t^{\gamma }\left(f\right)$ on $R^n$. We establish the almost everywhere convergence of $T_{m_{\gamma }\left(t\cdot \right)}\left(f\right)$ as $t\to 0$. The result is an extension of a Stein’s theorem on $R^n$. Let ${\tilde{S}}_t^{\gamma }$ be an analogue of $S_t^{\gamma }$on the $n-$torus $T^n$. As a consequence, we obtain that ${lim}_{t\to 0}{\tilde{S}}_t^{\gamma }\left(f\right)\left(x\right)=f\left(x\right)$ almost everywhere if $f\in L^{\frac{n}{n-1+\gamma }}{\left(Lo{g}^{+}L\right)}^{\theta }\left(T^n\right)$ with $\theta >\frac{1-\gamma }{n-1+\gamma }$, $-\frac{n-2}{2}<\gamma <1$ and that there exits an $f\in L^{\frac{n}{n-1+\gamma }}{\left(Lo{g}^{+}L\right)}^{\theta }\left(T^n\right)$ with $\theta <\frac{1-\gamma }{n-1+\gamma }$, $0<\gamma <1$ such that $lim{sup}_{t\to 0}\left({\tilde{S}}_t^{\gamma }\left(f\right)\left(x\right)\right)=\infty$ almost everywhere.