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

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.