Vertical Maximal Functions on Manifolds with Ends

We consider the setting of manifolds with ends which are obtained by compact perturbation (gluing) of ends of the form ${\mathbb{R}}^{n_i}\times {\mathcal{M}}_i$. We investigate family of vertical resolvent $\{\sqrt{t}\mathrm{\nabla }(1+t\mathrm{\Delta }{)}^{-m}{\}}_{t>0}$ where $m\ge 1$. We show that the family is uniformly continuous on all $L^p$ for $1\le p\le {\min }_i{n}_i$. Interestingly this is a closed-end condition in the considered setting. We prove that the corresponding Maximal function is bounded in the same range except that it is only weak-type $(1,1)$ for $p=1$. The Fefferman-Stein vector-valued maximal function is again of weak-type $(1,1)$ but bounded if and only if $1<p<{\min }_i{n}_i$, and not at $p={\min }_i{n}_i$.