WEAK TYPE (1, 1) ESTIMATES FOR HEAT KERNEL MAXIMAL FUNCTIONS ON LIE-GROUPS

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, .) = p(t) *f solves the heat equation (partial/partial-t - DELTA)u = 0 with initial condition u(0, .) = f(.). We establish weak-type (1, 1) estimates for the maximal operator M (Mf = sup(t > 0) \p(t)*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 M0 (M0f = sup0 < t < 1 \p(t)* f\) and related Hardy-Littlewood operators for all Lie groups.