Israel Journal of Mathematics

, Volume 144, Issue 2, pp 211–219

# An optimal theorem for the spherical maximal operator on the Heisenberg group

• E. K. Narayanan
• S. Thangavelu
Article

## Abstract

Let$$\mathbb{I}^n = \mathbb{C}^n \times \mathbb{R}$$ be the Heisenberg group and μ r be the normalized surface measure on the sphere of radiusr in ℂ n . Let$$Mf = \sup _{r > 0} \left| {f * \mu _r } \right|$$. We prove an optimalL p-boundedness result for the spherical maximal functionMf, namely we prove thatM is bounded onL p(I n ) if and only ifp>2n/2n−1.

## Keywords

Group Theory Maximal Operator Heisenberg Group Surface Measure Optimal Theorem
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

## References

1. [1]
D. Mueller and A. Seeger,Singular spherical maximal operators on a class of two step nilpotent Lie groups, Israel Journal of Mathematics141 (2004), 315–340.
2. [2]
A. Nevo and S. Thangavelu,Pointwise ergodic theorems for radial averages on the Heisenberg group, Advances in Mathematics127 (1997), 307–334.
3. [3]
E. Stein and S. Wainger,Problems in harmonic analysis related to curvature, Bulletin of the American Mathematical Society84 (1978), 1239–1295.
4. [4]
G. Szego,Orthogonal Polynomials, American Mathematical Society Colloquium Publications, Providence, RI, 1967.Google Scholar
5. [5]
S. Thangavelu,Lectures on Hermite and Laguerre Expansions, Mathematical Notes No. 42, Princeton University Press, Princeton, 1993.
6. [6]
S. Thangavelu,Harmonic Analysis on the Heisenberg Group, Progress in Mathematics 159, Birkhäuser, Boston, 1998.
7. [7]
S. Thangavelu,Local ergodic theorems for K-spherical averages on the Heisenberg group, Mathematische Zeitschrift234 (2000), 291–312.