Maximal functions related to subelliptic operators invariant under an action of a nilpotent Lie group
Volume 103 / 1992
Studia Mathematica 103 (1992), 239-264
DOI: 10.4064/sm-103-3-239-264
Abstract
On the domain $Ω_a = {(x,b) : x ∈ N, b ∈ ℝ^+, b > a}$, where N is a simply connected nilpotent Lie group and a ≥ 0, certain N-invariant second order subelliptic operators L are considered. Every bounded L-harmonic function F is the Poisson integral $F(x,b) = f ∗ μ̌_a^b(x)$ for an $f ∈ L^∞(N)$. The main theorem of the paper asserts that under some assumptions the maximal functions $M_1f(x) = sup_{b≥a+1} |f∗μ̌_a^b(x)|$, $M_2f(x) = sup_{a
