On the domain S_a = {(x,e^b): x ∈ N, b ∈ ℝ, b > a} where N is a simply connected nilpotent Lie group, a certain N-left-invariant, second order, degenerate elliptic operator L is considered. N × {e^a} is the Poisson boundary for L-harmonic functions F, i.e. F is the Poisson integral F(xe^b) = ʃ_N f(xy)dμ^b_a(x), for an f in L^∞(N). The main theorem of the paper asserts that the maximal function M^a f(x) = sup{|ʃf(xy)dμ_a^b(y)| : b > a} is of weak type (1,1).
@article{bwmeta1.element.bwnjournal-article-smv101i1p33bwm,
author = {Ewa Damek and Andrzej Hulanicki},
title = {Maximal functions related to subelliptic operators invariant under an action of a solvable Lie group},
journal = {Studia Mathematica},
volume = {100},
year = {1991},
pages = {33-68},
zbl = {0811.43001},
language = {en},
url = {http://dml.mathdoc.fr/item/bwmeta1.element.bwnjournal-article-smv101i1p33bwm}
}
Damek, Ewa; Hulanicki, Andrzej. Maximal functions related to subelliptic operators invariant under an action of a solvable Lie group. Studia Mathematica, Tome 100 (1991) pp. 33-68. http://gdmltest.u-ga.fr/item/bwmeta1.element.bwnjournal-article-smv101i1p33bwm/
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