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Hardy-Littlewood maximal functions on some solvable Lie groups

Part of: Abstract harmonic analysis Lie groups Harmonic analysis in several variables

Published online by Cambridge University Press:  09 April 2009

G. Gaudry ,
S. Giulini ,
A. Hulanicki  and
A. M. Mantero
G. Gaudry
Affiliation:
School of Mathematical SciencesFlinders University of South AustraliaBedford Park, South Australia 5042, Australia
S. Giulini
Affiliation:
Dipartimento di Matematica “F. Enriques”Università di MilanoMilano, Italy
A. Hulanicki
Affiliation:
Mathematics InstituteWroclaw UniversityWroclaw, Poland
A. M. Mantero
Affiliation:
Istituto di MathematicaUniversità di GenovaGenova, Italy
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Abstract

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Let N be a nilpotent simply connected Lie group, and A a commutative connected d-dimensional Lie group of automorphisms of N which correspond to semisimple endomorphisms of the Lie algebra of N with positive eigenvalues. Form the split extension S = N × A ≅ N × a, a being the Lie algebra of A. We consider a family of “rectangles” Br in S, parameterized by r > 0, such that the measure of Br behaves asymptotically as a fixed power of r. One can construct the Hardy-Littlewood maximal function operator f → Mf relative to left translates of the family {Br}. We prove that M is of weak type (1, 1). This complements a result of J.-O. Strömberg concerning maximal functions defined relative to hyperbolic balls in a symmetric space.

MSC classification

Secondary: 43A80: Analysis on other specific Lie groups 22E30: Analysis on real and complex Lie groups 42B25: Maximal functions, Littlewood-Paley theory
Type
Research Article
Information
Journal of the Australian Mathematical Society , Volume 45 , Issue 1 , August 1988 , pp. 78 - 82
Copyright
Copyright © Australian Mathematical Society 1988

References

[1]Clerc, J.-L. and Stein, E. M., ‘L p-multipliers for non-compact symmetric spaces’, Proc. Nat. Acad. Sci. U.S.A 71 (1974), 3911–3912.CrossRefGoogle Scholar
[2]Strömberg, Jan-Olov, ‘Weak type L 1 estimates for maximal functions on non-compact symmetric spaces’, Ann. of Math. 114 (1981), 115–126.CrossRefGoogle Scholar