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Spherical means on the Heisenberg group: Stability of a maximal function estimate

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Abstract

Consider the surface measure μ on a sphere in a nonvertical hyperplane on the Heisenberg group ℍn, n ≥ 2, and the convolution f * μ. Form the associated maximal function Mf = supt>0 ∣f * μt∣ generated by the automorphic dilations. We use decoupling inequalities due to Wolff and Bourgain—Demeter to prove Lp-boundedness of M in an optimal range.

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References

  1. D. Beltran, S. Guo, J. Hickman and A. Seeger, The circular maximal operator on Heisenberg radial functions, Ann. Sc. Norm. Super. Pisa Cl. Sci. (5), to appear.

  2. J. Bourgain and C. Demeter, The proof of the ℓ2decoupling conjecture, Ann. of Math. (2) 182 (2015), 351–389.

    Article  MathSciNet  Google Scholar 

  3. A. Carbery, Variants of the Calderón-Zygmund theory for Lp-spaces, Rev. Mat. Iberoamericana 2 (1986), 381–396.

    Article  MathSciNet  Google Scholar 

  4. R. R. Coifman and G. Weiss, Analyse Harmonique Non-Commutative sur Certains Espaces Homogènes, Springer, Berlin-New York, 1971.

    Book  Google Scholar 

  5. G. Folland and E. M. Stein, Hardy Spaces on Homogeneous Groups, Princeton University Press, Princeton, NJ, 1982.

    MATH  Google Scholar 

  6. A. Greenleaf and A. Seeger, Fourier integral operators with fold singularities, J. Reine Angew. Math. 455 (1994), 35–56.

    MathSciNet  MATH  Google Scholar 

  7. A. Greenleaf and A. Seeger, On oscillatory integral operators with folding canonical relations, Studia Math. 132 (1999), 125–139.

    MathSciNet  MATH  Google Scholar 

  8. L. Hörmander, Fourier integral operators. I, Acta Math. 127 (1971), 79–183.

    Article  MathSciNet  Google Scholar 

  9. I. Łaba and T. Wolff, A local smoothing estimate in higher dimensions, J. Anal. Math. 88 (2002), 149–171.

    Article  MathSciNet  Google Scholar 

  10. D. Müller and A. Seeger, Singular spherical maximal operators on a class of two step nilpotent Lie groups, Israel J. Math. 141 (2004), 315–340.

    Article  MathSciNet  Google Scholar 

  11. E. K. Narayanan and S. Thangavelu, An optimal theorem for the spherical maximal operator on the Heisenberg group, Israel J. Math. 144 (2004), 211–219.

    Article  MathSciNet  Google Scholar 

  12. A. Nevo and S. Thangavelu, Pointwise ergodic theorems for radial averages on the Heisenberg group, Adv. Math. 127 (1997), 307–334.

    Article  MathSciNet  Google Scholar 

  13. D. H. Phong and E. M. Stein, Hilbert integrals, singular integrals, and Radon transforms. I, Acta Math. 157 (1986), 99–157.

    Article  MathSciNet  Google Scholar 

  14. M. Pramanik, K. M. Rogers and A. Seeger, A Calderón-Zygmund estimate with applications to generalized Radon transforms and Fourier integral operators, Studia Math. 202 (2011), 1–15.

    Article  MathSciNet  Google Scholar 

  15. M. Pramanik and A. Seeger, Lpregularity of averages over curves and bounds for associated maximal operators, Amer. J. Math. 129 (2007), 61–103.

    Article  MathSciNet  Google Scholar 

  16. M. Pramanik and A. Seeger, Optimal Lp-Sobolev regularity of a class of generalized Radon transforms, unpublished manuscript; Lp-Sobolev estimates for a class of integral operators with folding canonical relations, J. Geom. Anal. 31 (2021), 6725–6765.

    Article  MathSciNet  Google Scholar 

  17. C. D. Sogge and E. M. Stein, Averages over hypersurfaces. Smoothness of generalized Radon transforms, J. Anal. Math. 54 (1990), 165–188.

    Article  MathSciNet  Google Scholar 

  18. E. M. Stein, Harmonic Analysis: Real-Variable Methods, Orthogonality, and Oscillatory Integrals, Princeton University Press, Princeton, NJ, 1993.

    MATH  Google Scholar 

  19. T. Wolff, Local smoothing type estimates on Lpfor large p, Geom. Funct. Anal. 10 (2000), 1237–1288.

    Article  MathSciNet  Google Scholar 

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Authors and Affiliations

  1. Department of Mathematics, Purdue University, 4150 N. University St., West Lafayette, IN, 47907, USA

    Theresa C. Anderson

  2. Department of Mathematics, UCLA, Box 951555, Los Angeles, CA, 90095-1555, USA

    Laura Cladek

  3. Department of Mathematics, University of British Columbia, 1984 Mathematics Road, Vancouver, BC, Canada, V6T 1Z2

    Malabika Pramanik

  4. Department of Mathematics, University of Wisconsin, 480 Lincoln Drive, Madison, WI, 53706, USA

    Andreas Seeger

Authors
  1. Theresa C. Anderson
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  2. Laura Cladek
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  3. Malabika Pramanik
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  4. Andreas Seeger
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Corresponding author

Correspondence to Andreas Seeger.

Additional information

Research supported in part by the National Science Foundation.

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Anderson, T.C., Cladek, L., Pramanik, M. et al. Spherical means on the Heisenberg group: Stability of a maximal function estimate. JAMA 145, 1–28 (2021). https://doi.org/10.1007/s11854-021-0171-6

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  • DOI: https://doi.org/10.1007/s11854-021-0171-6

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