Boundedness of maximal functions on non-doubling manifolds with ends

Abstract

Let $M$ be a manifold with ends constructed in $[2]$ and $\Delta$ be the Laplace-Beltrami operator on $M$. In this note, we show the weak type $(1,1)$ and $L^p$ boundedness of the Hardy-Littlewood maximal function and of the maximal function associated with the heat semigroup $\mathcal{M}_{\Delta}f(x) = sup_{t \gt o} | exp(-t \Delta)f(x)|$ on $L^p(M)$ for $1 \lt p \leq \infty$. The significance of these results comes from the fact that $M$ does not satisfies the doubling condition.