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$ L^p$ boundedness of discrete singular Radon transforms

Author(s): Alexandru D. Ionescu; Stephen Wainger
Journal: J. Amer. Math. Soc. 19 (2006), 357-383.
MSC (1991): Primary 11L07, 42B20
Posted: October 24, 2005
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Abstract: We prove that if $ K:\mathbb{R}^{d_1}\to\mathbb{C}$ is a Calderón-Zygmund kernel and $ P:\mathbb{R}^{d_1}\to\mathbb{R}^{d_2}$ is a polynomial of degree $ A\geq 1$ with real coefficients, then the discrete singular Radon transform operator

$\displaystyle T(f)(x)=\sum_{n\in\mathbb{Z}^{d_1}\setminus\{0\}}f(x-P(n))K(n) $

extends to a bounded operator on $ L^p(\mathbb{R}^{d_2})$, $ 1<p<\infty$. This gives a positive answer to an earlier conjecture of E. M. Stein and S. Wainger.

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Additional Information:

Alexandru D. Ionescu
Affiliation: Department of Mathematics, University of Wisconsin, 480 Lincoln Drive, Van Vleck Hall, Madison, Wisconsin 53706
Email: ionescu@math.wisc.edu

Stephen Wainger
Affiliation: Department of Mathematics, University of Wisconsin, 480 Lincoln Drive, Van Vleck Hall, Madison, Wisconsin 53706-1313
Email: wainger@math.wisc.edu

DOI: 10.1090/S0894-0347-05-00508-4
PII: S 0894-0347(05)00508-4
Keywords: Singular Radon transforms, discrete operators, orthogonality, square functions, exponential sums, the circle method
Received by editor(s): February 27, 2004
Posted: October 24, 2005
Additional Notes: The first author was supported in part by an NSF grant and an Alfred P. Sloan research fellowship
The second author was supported in part by an NSF grant
Copyright of article: Copyright 2005, American Mathematical Society
The copyright for this article reverts to public domain after 28 years from publication.

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