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Published online by Cambridge University Press: 11 March 2020
Let 
$M$ be a nondoubling parabolic manifold with ends. First, this paper investigates the boundedness of the maximal function associated with the heat semigroup 
${\mathcal{M}}_{\unicode[STIX]{x1D6E5}}f(x):=\sup _{t>0}|e^{-t\unicode[STIX]{x1D6E5}}f(x)|$ where 
$\unicode[STIX]{x1D6E5}$ is the Laplace–Beltrami operator acting on 
$M$. Then, by combining the subordination formula with the previous result, we obtain the weak type 
$(1,1)$ and 
$L^{p}$ boundedness of the maximal function 
${\mathcal{M}}_{\sqrt{L}}^{k}f(x):=\sup _{t>0}|(t\sqrt{L})^{k}e^{-t\sqrt{L}}f(x)|$ on 
$L^{p}(M)$ for 
$1<p\leq \infty$ where 
$k$ is a nonnegative integer and 
$L$ is a nonnegative self-adjoint operator satisfying a suitable heat kernel upper bound. An interesting thing about the results is the lack of both doubling condition of 
$M$ and the smoothness of the operators’ kernels.
Communicated by C. Meaney
This paper is part of the PhD thesis of H. C. Doan who is supported by Macquarie University scholarship iMQRES.
