Transactions of the American Mathematical Society

			Journals Home Search Author Info Subscribe Tech Support My Subscriptions Help
ISSN 1088-6850(e) ISSN 0002-9947(p)

Strongly singular convolution operators on the Heisenberg group

Author(s): Neil Lyall
Journal: Trans. Amer. Math. Soc. 359 (2007), 4467-4488.
MSC (2000): Primary 42B20, 43A80
Posted: April 16, 2007
MathSciNet review: 2309194
Retrieve article in: 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 $\mathbf{H}^n$ ; these are convolution operators on $\mathbf{H}^n$ 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.

References:

1.: R. Askey and S. Wainger, Mean convergence of expansions in Laguerre and Hermite series, American J. Math. 87 (1965), 695-708. MR 0182834 (32:316)
2.: A. Erdélyi, Asymptotic forms for Laguerre polynomials, J. Indian Math. Soc. 24 (1960), 235-250. MR 0123751 (23:A1073)
3.: C. Fefferman, Inequalities for strongly singular convolution operators, Acta Math. 124 (1970), 9-36. MR 0257819 (41:2468)
4.: C. Fefferman and E. M. Stein, spaces of several variables, Acta Math. 129 (1972), 137-193. MR 0447953 (56:6263)
5.: C. L. Frenzen and R. Wong, Uniform asymptotic expansions of Laguerre polynomials, SIAM J. Math. Anal. 19 (1988), 1232-1248. MR 957682 (89k:33012)
6.: D. Geller, Fourier analysis on the Heisenberg group, Proc. Nat. Acad. Sci. U.S.A. 74 (1977), 1328-1331. MR 0486312 (58:6069)
7.: I. I. Hirschman, Multiplier Transforms I, Duke Math. J. 26 (1956), 222-242. MR 0104973 (21:3721)
8.: N. Lyall, A class of strongly singular Radon transforms on the Heisenberg group, to appear in Proc. of Edin. Math. Soc.
9.: E. M. Stein, Harmonic analysis: Real-variable methods, orthogonality, and oscillatory integrals, Princeton Univ. Press, Princeton, 1993. MR 1232192 (95c:42002)
10.: E. M. Stein and G. Weiss, Introduction to Fourier analysis on Euclidean spaces, Princeton Univ. Press, Princeton, 1971. MR 0304972 (46:4102)
11.: S. Thangavelu, Harmonic analysis on the Heisenberg group, Birkhäuser, Boston, 1998. MR 1633042 (99h:43001)
12.: S. Wainger, Special trigonometric series in dimensions, Memoirs of the AMS 59, American Math. Soc., 1965. MR 0182838 (32:320)
13.: T. H. Wolff, Lectures on harmonic analysis, University Lecture Series 29, American Math. Soc., 2003. MR 2003254 (2004e:42002)

Similar Articles:

Retrieve articles in Transactions of the American Mathematical Society with MSC (2000): 42B20, 43A80

Retrieve articles in all Journals with MSC (2000): 42B20, 43A80

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: 10.1090/S0002-9947-07-04187-6
PII: S 0002-9947(07)04187-6
Received by editor(s): November 12, 2004
Received by editor(s) in revised form: October 10, 2005
Posted: April 16, 2007
Copyright of article: Copyright 2007, American Mathematical Society
The copyright for this article reverts to public domain after 28 years from publication.

	Publications Meetings The Profession Membership Programs Math Samplings Policy & Advocacy In the News About the AMS
\|