Boundedness of maximal functions on nondoubling parabolic manifolds with ends

Let M be a nondoubling parabolic manifold with ends. First, this paper investigates the boundedness of the maximal function associated with the heat semigroup M_Δf(x):=supt_>0|e^−tΔf(x)| where Δ 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 M^k_√Lf(x):=sup_t>0|(t√L)^ke^−t√Lf(x)| on L^p(M) for 1<p≤∞ 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.