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 Δ on G: that is, u(t, .) = p_t * f solves the heat equation (∂/∂t - Δ)u = 0 with initial condition u(0, .) = f(.). We establish weak-type (1, 1) estimates for the maximal operator M (M f = \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 M_0 (M_0 f = \sup_{0<t<1} |p_t * f|) and related Hardy-Littlewood operators for all Lie groups.
Weak type (1,1) estimates for heat kernel maximal functions on Lie groups
GIULINI, SAVERIO;MAUCERI, GIANCARLO
1991-01-01
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 Δ on G: that is, u(t, .) = p_t * f solves the heat equation (∂/∂t - Δ)u = 0 with initial condition u(0, .) = f(.). We establish weak-type (1, 1) estimates for the maximal operator M (M f = \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 M_0 (M_0 f = \sup_{0I documenti in IRIS sono protetti da copyright e tutti i diritti sono riservati, salvo diversa indicazione.