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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
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Abstract: We consider the $ L^2$ 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.

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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.

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