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 Rnᵢ× Mᵢ R n i × M i . We investigate the family of vertical resolvent \√t∇ (1+tΔ )⁻ᵐ>0 { t ∇ ( 1 + t Δ ) - m } t > 0 , where m≥ 1 m ≥ 1 . We show that the family is uniformly continuous on all Lᵖ L p for 1≤ ~p~≤ ~min ᵢnᵢ 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 ᵢnᵢ 1 < p < min i n i , and not at p=min ᵢnᵢ p = min i n i .