Strongly singular convolution operators on the Heisenberg group
Author:
Neil Lyall
Journal:
Trans. Amer. Math. Soc. 359 (2007), 4467-4488
MSC (2000):
Primary 42B20, 43A80
DOI:
https://doi.org/10.1090/S0002-9947-07-04187-6
Published electronically:
April 16, 2007
MathSciNet review:
2309194
Full-text PDF
Abstract | References | Similar Articles | Additional Information
Abstract: We consider the 
mapping properties of a model class of strongly singular integral operators on the Heisenberg group 
; these are convolution operators on 
whose kernels are too singular at the origin to be of Calderón-Zygmund type. This strong singularity is compensated for by introducing a suitably large oscillation.
Our results are obtained by utilizing the group Fourier transform and uniform asymptotic forms for Laguerre functions due to Erdélyi.
- 1. Richard Askey and Stephen Wainger, Mean convergence of expansions in Laguerre and Hermite series, Amer. J. Math. 87 (1965), 695–708. MR 0182834, https://doi.org/10.2307/2373069
- 2. A. Erdélyi, Asymptotic forms for Laguerre polynomials, J. Indian Math. Soc. (N.S.) 24 (1960), 235–250 (1961). MR 0123751
- 3. Charles Fefferman, Inequalities for strongly singular convolution operators, Acta Math. 124 (1970), 9–36. MR 0257819, https://doi.org/10.1007/BF02394567
- 4. C. Fefferman and E. M. Stein, 𝐻^{𝑝} spaces of several variables, Acta Math. 129 (1972), no. 3-4, 137–193. MR 0447953, https://doi.org/10.1007/BF02392215
- 5. C. L. Frenzen and R. Wong, Uniform asymptotic expansions of Laguerre polynomials, SIAM J. Math. Anal. 19 (1988), no. 5, 1232–1248. MR 957682, https://doi.org/10.1137/0519087
- 6. Daryl Geller, Fourier analysis on the Heisenberg group, Proc. Nat. Acad. Sci. U.S.A. 74 (1977), no. 4, 1328–1331. MR 0486312
- 7. I. I. Hirschman Jr., On multiplier transformations, Duke Math. J 26 (1959), 221–242. MR 0104973
- 8. N. Lyall, A class of strongly singular Radon transforms on the Heisenberg group, to appear in Proc. of Edin. Math. Soc.
- 9. Elias M. Stein, Harmonic analysis: real-variable methods, orthogonality, and oscillatory integrals, Princeton Mathematical Series, vol. 43, Princeton University Press, Princeton, NJ, 1993. With the assistance of Timothy S. Murphy; Monographs in Harmonic Analysis, III. MR 1232192
- 10. Elias M. Stein and Guido Weiss, Introduction to Fourier analysis on Euclidean spaces, Princeton University Press, Princeton, N.J., 1971. Princeton Mathematical Series, No. 32. MR 0304972
- 11. Sundaram Thangavelu, Harmonic analysis on the Heisenberg group, Progress in Mathematics, vol. 159, Birkhäuser Boston, Inc., Boston, MA, 1998. MR 1633042
- 12. Stephen Wainger, Special trigonometric series in 𝑘-dimensions, Mem. Amer. Math. Soc. No. 59 (1965), 102. MR 0182838
- 13. Thomas H. Wolff, Lectures on harmonic analysis, University Lecture Series, vol. 29, American Mathematical Society, Providence, RI, 2003. With a foreword by Charles Fefferman and preface by Izabella Łaba; Edited by Łaba and Carol Shubin. MR 2003254
spaces of several variables, Acta Math. 129 (1972), 137-193. 
dimensions, Memoirs of the AMS 59, American Math. Soc., 1965. Retrieve articles in Transactions of the American Mathematical Society with MSC (2000): 42B20, 43A80
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Additional Information
Neil Lyall
Affiliation:
Department of Mathematics, University of Wisconsin, Madison, Wisconsin 53706
Address at time of publication:
Department of Mathematics, The University of Georgia, Boyd GSRC, Athens, Georgia 30602
Email:
lyall@math.wisc.edu, lyall@math.uga.edu
DOI:
https://doi.org/10.1090/S0002-9947-07-04187-6
Received by editor(s):
November 12, 2004
Received by editor(s) in revised form:
October 10, 2005
Published electronically:
April 16, 2007
Article copyright:
© Copyright 2007
American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.
