## $L^p$ boundedness of discrete singular Radon transforms

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- by Alexandru D. Ionescu and Stephen Wainger
- J. Amer. Math. Soc.
**19**(2006), 357-383 - DOI: https://doi.org/10.1090/S0894-0347-05-00508-4
- Published electronically: 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 \begin{equation*} T(f)(x)=\sum _{n\in \mathbb {Z}^{d_1}\setminus \{0\}}f(x-P(n))K(n) \end{equation*} 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.## References

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## Bibliographic Information

**Alexandru D. Ionescu**- Affiliation: Department of Mathematics, University of Wisconsin, 480 Lincoln Drive, Van Vleck Hall, Madison, Wisconsin 53706
- MR Author ID: 660963
- Email: ionescu@math.wisc.edu
**Stephen Wainger**- Affiliation: Department of Mathematics, University of Wisconsin, 480 Lincoln Drive, Van Vleck Hall, Madison, Wisconsin 53706-1313
- MR Author ID: 179960
- Email: wainger@math.wisc.edu
- Received by editor(s): February 27, 2004
- Published electronically: 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 2005
American Mathematical Society

The copyright for this article reverts to public domain 28 years after publication. - Journal: J. Amer. Math. Soc.
**19**(2006), 357-383 - MSC (1991): Primary 11L07, 42B20
- DOI: https://doi.org/10.1090/S0894-0347-05-00508-4
- MathSciNet review: 2188130