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

Ewa Damek; Andrzej Hulanicki

Studia Mathematica (1991)

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

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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  3. [DH] E. Damek and A. Hulanicki, Boundaries for left-invariant subelliptic operators on semidirect products of nilpotent and abelian groups, J. Reine Angew. Math. 411 (1990), 1-38. Zbl0699.22012
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  9. [H2] A. Hulanicki, A class of convolution semi-groups of measures on a Lie group, in: Lecture Notes in Math. 828, Springer, 1980, 82-101. 
  10. [HJ] A. Hulanicki and J. Jenkins, Nilpotent groups and summability of eigenfunction expansions of Schrödinger operators, Studia Math. 80 (1984), 235-244. Zbl0564.43007
  11. [K] C. Kenig, oral communication. 
  12. [St] E. M. Stein, Boundary behavior of harmonic functions on symmetric spaces: maximal estimates for Poisson integrals, Invent. Math. 74 (1983), 63-83. 
  13. [S] D. Stroock, Lectures on Stochastic Analysis: Diffusion Theory, Cambridge Univ. Press, 1987. Zbl0605.60057
  14. [SV] D. Stroock and S. R. Varadhan, Multidimensional Diffusion Processes, Springer, 1979. Zbl0426.60069
  15. [T] J. C. Taylor, Skew products, regular conditional probabilities and stochastic differential equations: a remark, preprint. Zbl0763.60031
  16. [Z] J. Zienkiewicz, in preparation. 

Citations in EuDML Documents

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  1. Tadeusz Pytlik, Harmonic functions and Hardy spaces on trees with boundaries
  2. Jacek Zienkiewicz, Maximal estimates for nonsymmetric semigroups
  3. Jarosław Sołowiej, The Fatou theorem for NA groups - a negative result
  4. Ewa Damek, Maximal functions related to subelliptic operators invariant under an action of a nilpotent Lie group
  5. Andrzej Hulanicki, Estimates for the Poisson kernels and a Fatou type theorem applications to analysis on Siegel domains
  6. Ewa Damek, Pointwise estimates for the Poisson kernel on NA groups by the Ancona method
  7. Ewa Damek, Andrzej Hulanicki, Jacek Zienkiewicz, Estimates for the Poisson kernels and their derivatives on rank one NA groups

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