# Maximal functions related to subelliptic operators invariant under an action of a solvable Lie group

Studia Mathematica (1991)

• Volume: 101, Issue: 1, page 33-68
• ISSN: 0039-3223

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## Abstract

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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).

## How to cite

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Damek, Ewa, and Hulanicki, Andrzej. "Maximal functions related to subelliptic operators invariant under an action of a solvable Lie group." Studia Mathematica 101.1 (1991): 33-68. <http://eudml.org/doc/215892>.

@article{Damek1991,
abstract = {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).},
author = {Damek, Ewa, Hulanicki, Andrzej},
journal = {Studia Mathematica},
keywords = {simply connected nilpotent Lie group; degenerate elliptic operator; Poisson boundary; -harmonic functions; maximal function; weak type },
language = {eng},
number = {1},
pages = {33-68},
title = {Maximal functions related to subelliptic operators invariant under an action of a solvable Lie group},
url = {http://eudml.org/doc/215892},
volume = {101},
year = {1991},
}

TY - JOUR
AU - Damek, Ewa
AU - Hulanicki, Andrzej
TI - Maximal functions related to subelliptic operators invariant under an action of a solvable Lie group
JO - Studia Mathematica
PY - 1991
VL - 101
IS - 1
SP - 33
EP - 68
AB - 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).
LA - eng
KW - simply connected nilpotent Lie group; degenerate elliptic operator; Poisson boundary; -harmonic functions; maximal function; weak type
UR - http://eudml.org/doc/215892
ER -

## References

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16. [Z] J. Zienkiewicz, in preparation.

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