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Proceedings of the American Mathematical Society

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$ L^p$-nondegenerate Radon-like operators with vanishing rotational curvature


Author: Philip T. Gressman
Journal: Proc. Amer. Math. Soc. 143 (2015), 1595-1604
MSC (2010): Primary 28A75, 42B20, 42C99
DOI: https://doi.org/10.1090/S0002-9939-2014-12407-5
Published electronically: November 24, 2014
MathSciNet review: 3314072
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Abstract: We consider the $ L^p \rightarrow L^q$ mapping properties of a model family of Radon-like operators integrating functions over $ n$-dimensional submanifolds of $ \mathbb{R}^{2n}$. It is shown that nonvanishing rotational curvature is never generic when $ n \geq 2$ and is, in fact, impossible for all but finitely many values of $ n$. Nevertheless, operators satisfying the same $ L^p \rightarrow L^q$ estimates as the ``nondegenerate'' case (modulo the endpoint) are dense in the model family for all $ n$.


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Additional Information

Philip T. Gressman
Affiliation: Department of Mathematics, University of Pennsylvania, David Rittenhouse Laboratory, 209 South 33rd Street, Philadelphia, Pennsylvania 19104
Email: gressman@math.upenn.edu

DOI: https://doi.org/10.1090/S0002-9939-2014-12407-5
Received by editor(s): August 6, 2013
Published electronically: November 24, 2014
Additional Notes: The author was partially supported by NSF grant DMS-1101393 and an Alfred P. Sloan Foundation Fellowship.
Communicated by: Alexander Iosevich
Article copyright: © Copyright 2014 American Mathematical Society

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