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 the family of vertical resolvent \(\{\sqrt{t}\nabla (1+t\Delta )^{-m}\}_{t>0}\), 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 \(p=\min _{i}n_i\).