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.
Article PDF
Similar content being viewed by others
Avoid common mistakes on your manuscript.
References
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.
J. Bourgain and C. Demeter, The proof of the ℓ2decoupling conjecture, Ann. of Math. (2) 182 (2015), 351–389.
A. Carbery, Variants of the Calderón-Zygmund theory for Lp-spaces, Rev. Mat. Iberoamericana 2 (1986), 381–396.
R. R. Coifman and G. Weiss, Analyse Harmonique Non-Commutative sur Certains Espaces Homogènes, Springer, Berlin-New York, 1971.
G. Folland and E. M. Stein, Hardy Spaces on Homogeneous Groups, Princeton University Press, Princeton, NJ, 1982.
A. Greenleaf and A. Seeger, Fourier integral operators with fold singularities, J. Reine Angew. Math. 455 (1994), 35–56.
A. Greenleaf and A. Seeger, On oscillatory integral operators with folding canonical relations, Studia Math. 132 (1999), 125–139.
L. Hörmander, Fourier integral operators. I, Acta Math. 127 (1971), 79–183.
I. Łaba and T. Wolff, A local smoothing estimate in higher dimensions, J. Anal. Math. 88 (2002), 149–171.
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.
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.
A. Nevo and S. Thangavelu, Pointwise ergodic theorems for radial averages on the Heisenberg group, Adv. Math. 127 (1997), 307–334.
D. H. Phong and E. M. Stein, Hilbert integrals, singular integrals, and Radon transforms. I, Acta Math. 157 (1986), 99–157.
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.
M. Pramanik and A. Seeger, Lpregularity of averages over curves and bounds for associated maximal operators, Amer. J. Math. 129 (2007), 61–103.
M. Pramanik and A. Seeger, Optimal Lp-Sobolev regularity of a class of generalized Radon transforms, unpublished manuscript; Lp-Sobolev estimates for a class of integral operators with folding canonical relations, J. Geom.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.
C. D. Sogge and E. M. Stein, Averages over hypersurfaces. Smoothness of generalized Radon transforms, J. Anal. Math. 54 (1990), 165–188.
E. M. Stein, Harmonic Analysis: Real-Variable Methods, Orthogonality, and Oscillatory Integrals, Princeton University Press, Princeton, NJ, 1993.
T. Wolff, Local smoothing type estimates on Lpfor large p, Geom. Funct. Anal. 10 (2000), 1237–1288.
Author information
Authors and Affiliations
Corresponding author
Additional information
Research supported in part by the National Science Foundation.
Rights and permissions
About this article
Cite this article
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
Received:
Revised:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s11854-021-0171-6