Israel Journal of Mathematics

, Volume 141, Issue 1, pp 315–340 | Cite as

Singular spherical maximal operators on a class of two step nilpotent lie groups



LetH n ≅ℝ2n ⋉ℝ be the Heisenberg group and letμ t be the normalized surface measure for the sphere of radiust in ℝ2n . Consider the maximal function defined byM f=supt>0|f*μ t |. We prove forn≥2 thatM defines an operator bounded onL p (H n ) provided thatp>2n/(2n−1). This improves an earlier result by Nevo and Thangavelu, and the range forL p boundedness is optimal. We also extend the result to a more general class of surfaces and to groups satisfying a nondegeneracy condition; these include the groups of Heisenberg type.


Heisenberg Group Fourier Integral Operator Nondegeneracy Condition Schwartz Kernel Heisenberg Type 
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  1. [1]
    J. Bourgain,Averages in the plane over convex curves and maximal operators, Journal d’Analyse Mathématique47 (1986), 69–85.MATHMathSciNetCrossRefGoogle Scholar
  2. [2]
    M. Cowling,On Littlewood-Paley-Stein theory, Rendiconti del Circolo Matematico di Palermo1 (1981), 21–55.MathSciNetGoogle Scholar
  3. [3]
    S. Cuccagna,L 2 estimates for averaging operators along curves with two-sided k-fold singularities, Duke Mathematical Journal89 (1997), 203–216.MATHCrossRefMathSciNetGoogle Scholar
  4. [4]
    G. Folland and E. M. Stein,Hardy Spaces on Homogeneous Groups, princeton University Press, 1982.Google Scholar
  5. [5]
    A. Greenleaf and A. Seeger,Fourier integral operators with fold singularities, Journal für die reine und angewandte Mathematik455 (1994), 35–56.MATHMathSciNetCrossRefGoogle Scholar
  6. [6]
    A. Greenleaf and A. Seeger,On oscillatory integrals with folding canonical relations, Studia Mathematica132 (1999), 125–139.MATHMathSciNetGoogle Scholar
  7. [7]
    L. Hörmander,The Analysis of Linear Partial Differential Operators, Vol. I, Springer-Verlag, New York, Berlin, 1983.Google Scholar
  8. [8]
    L. Hörmander,Oscillatory integrals and multipliers on FL p, Archiv der Mathematik11 (1973), 1–11.MATHGoogle Scholar
  9. [9]
    A. Kaplan,Fundamental solutions for a class of hypoelliptic PDE generated by composition of quadratic forms, Transactions of the American Mathematical Society258 (1980), 147–153.MATHCrossRefMathSciNetGoogle Scholar
  10. [10]
    G. Métivier,Hypoellipticité analytique sur des groupes nilpotents de rang 2, Duke Mathematical Journal47 (1980), 195–221.MATHCrossRefMathSciNetGoogle Scholar
  11. [11]
    G. Mockenhaupt, A. Seeger and C. D. Sogge,Local smoothing of Fourier integral operators and Carleson-Sjölin estimates, Journal of the American Mathematical Society6 (1993), 65–130.MATHCrossRefMathSciNetGoogle Scholar
  12. [12]
    A. Nevo and S. Thangavelu,Pointwise ergodic theorems for radial averages on the Heisenberg group, Advances in Mathematics127 (1997), 307–334.MATHCrossRefMathSciNetGoogle Scholar
  13. [13]
    D. H. Phong and E. M. Stein,Radon transforms and torsion, International Mathematics Research Notices, 1991, pp. 49–60.Google Scholar
  14. [14]
    F. Ricci and E. M. Stein,Harmonic analysis on nilpotent groups and singular integrals II: Singular kernels supported on submanifolds, Journal of Functional Analysis78 (1988), 56–84.MATHCrossRefMathSciNetGoogle Scholar
  15. [15]
    O. Schmidt,Maximaloperatoren zu Hyperflächen in Gruppen vom homogenen Typ, Diplomarbeit, Universität Kiel, 1998.Google Scholar
  16. [16]
    E. M. Stein,Maximal functions: spherical means, Proceedings of the National Academy of Sciences of the United States of America73 (1976), 2174–2175.MATHCrossRefGoogle Scholar
  17. [17]
    E. M. Stein,Harmonic Analysis: Real Variable Methods, Orthogonality and Oscillatory Integrals, Princeton University Press, 1993.Google Scholar

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© Hebrew University 2004

Authors and Affiliations

  1. 1.Mathematisches SeminarChristian-Albrechts-Universität zu KielKielGermany
  2. 2.Department of MathematicsUniversity of WisconsinMadisonUSA

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