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Transactions of the American Mathematical Society

ISSN 1088-6850(online) ISSN 0002-9947(print)

 

 

Weak type $ (1,1)$ estimates for heat kernel maximal functions on Lie groups


Authors: Michael Cowling, Garth Gaudry, Saverio Giulini and Giancarlo Mauceri
Journal: Trans. Amer. Math. Soc. 323 (1991), 637-649
MSC: Primary 43A80; Secondary 22E30, 42B25, 58G11
MathSciNet review: 967310
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Abstract: For a Lie group $ G$ with left-invariant Haar measure and associated Lebesgue spaces $ {L^p}(G)$, we consider the heat kernels $ {\{ {p_t}\} _{t > 0}}$ arising from a right-invariant Laplacian $ \Delta $ on $ G$: that is, $ u(t, \cdot ) = {p_t}{\ast}f$ solves the heat equation $ (\partial /\partial t - \Delta )u = 0$ with initial condition $ u(0, \cdot ) = f( \cdot )$. We establish weak-type $ (1,1)$ estimates for the maximal operator $ \mathcal{M}(\mathcal{M}\;f = {\sup _{t > 0}}\vert{p_t}{\ast}f\vert)$ and for related Hardy-Littlewood maximal operators in a variety of contexts, namely for groups of polynomial growth and for a number of classes of Iwasawa $ AN$ groups. We also study the "local" maximal operator $ {\mathcal{M}_0}({\mathcal{M}_0}f = {\sup _{0 < t < 1}}\vert{p_t}{\ast}f\vert)$ and related Hardy-Littlewood operators for all Lie groups.


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DOI: https://doi.org/10.1090/S0002-9947-1991-0967310-X
Article copyright: © Copyright 1991 American Mathematical Society