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BOUNDEDNESS OF MAXIMAL FUNCTIONS ON NONDOUBLING PARABOLIC MANIFOLDS WITH ENDS
- Part of
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- Journal:
- Journal of the Australian Mathematical Society / Volume 111 / Issue 3 / December 2021
- Published online by Cambridge University Press:
- 11 March 2020, pp. 320-344
- Print publication:
- December 2021
-
- Article
- Export citation
-
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.
