Boundedness of maximal functions on non-doubling manifolds with ends

Abstract

Let $M$ be a manifold with ends constructed in \cite{GS} and $Δ$ 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 $\M_Δf(x)=\sup_{t> 0} |\exp (-tΔ)f(x)| $ on $L^p(M)$ for $1 < p \le \infty$. The significance of these results comes from the fact that $M$ does not satisfies the doubling condition.