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 $\{ \sqrt{ t}\nabla(1+t\Delta)^{ -m}\}_{ t>0}$ where $m\geq1$. 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 $p=\min_{ i}n_i$.
Categories: math.AP