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

Cowling, Michael; Gaudry, Garth; Giulini, Saverio; Mauceri, Giancarlo

doi:10.1090/S0002-9947-1991-0967310-X

Weak type $(1,1)$ estimates for heat kernel maximal functions on Lie groups
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by Michael Cowling, Garth Gaudry, Saverio Giulini and Giancarlo Mauceri PDF

Trans. Amer. Math. Soc. 323 (1991), 637-649 Request permission

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 $\mathcal {M}(\mathcal {M}\;f = {\sup _{t > 0}}|{p_t}{\ast }f|)$ 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 ${\mathcal {M}_0}({\mathcal {M}_0}f = {\sup _{0 < t < 1}}|{p_t}{\ast }f|)$ and related Hardy-Littlewood operators for all Lie groups.

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