Boundedness of maximal functions on nondoubling parabolic manifolds with ends

Abstract

Let be a nondoubling parabolic manifold with ends. First, this paper investigates the boundedness of the maximal function associated with the heat semigroup 0 e-t\unicode[STIX]x1D6E5f(x)]]> where is the Laplace-Beltrami operator acting on. Then, by combining the subordination formula with the previous result, we obtain the weak type and boundedness of the maximal function 0|(t&sqrt;L)^k-t&sqrt;Lf(x)] on for <![CDATA[$1 where is a nonnegative integer and 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 and the smoothness of the operators' kernels.