Vertical maximal functions on manifolds with ends

Abstract

AbstractWe consider the setting of manifolds with ends which are obtained by compact perturbation (gluing) of ends of the form $${\mathbb {R}}^{n_i}\times {\mathcal {M}}_i$$ R n i × M i . We investigate the family of vertical resolvent $$\{\sqrt{t}\nabla (1+t\Delta )^{-m}\}_{t>0}$$ { t ∇ ( 1 + t Δ ) - m } t > 0 , where $$m\ge 1$$ m ≥ 1 . We show that the family is uniformly continuous on all $$L^p$$ L p for $$1\le ~p~\le ~\min _{i}n_i$$ 1 ≤ p ≤ 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 $$p=1$$ p = 1 . The Fefferman-Stein vector-valued maximal function is again of weak-type (1, 1) but bounded if and only if $$1<p<\min _{i}n_i$$ 1 < p < min i n i , and not at $$p=\min _{i}n_i$$ p = min i n i .