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BOUNDEDNESS OF MAXIMAL FUNCTIONS ON NONDOUBLING PARABOLIC MANIFOLDS WITH ENDS

Part of: Harmonic analysis in several variables

Published online by Cambridge University Press:  11 March 2020

HONG CHUONG DOAN
HONG CHUONG DOAN*
Affiliation:
Department of Economic Mathematics, University of Economics and Law, Vietnam National University, Ho Chi Minh City, Vietnam Department of Mathematics and Statistics, Macquarie University, Sydney, NSW 2109, Australia e-mail: [email protected], [email protected]
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Abstract

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.

Keywords

maximal functionsparabolic manifolds with endsheat kernelsPoisson kernels

MSC classification

Primary: 42B25: Maximal functions, Littlewood-Paley theory
Type
Research Article
Information
Journal of the Australian Mathematical Society , Volume 111 , Issue 3 , December 2021 , pp. 320 - 344
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Copyright
© 2020 Australian Mathematical Publishing Association Inc.

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Footnotes

Communicated by C. Meaney

This paper is part of the PhD thesis of H. C. Doan who is supported by Macquarie University scholarship iMQRES.

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