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Discrete analogues in harmonic analysis, I: ℓ2 estimates for singular radon transforms
- American Journal of Mathematics
- Johns Hopkins University Press
- Volume 121, Number 6, December 1999
- pp. 1291-1336
- 10.1353/ajm.1999.0046
- Article
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This paper studies the discrete analogues of singular Radon transforms. We prove the ℓ2 boundedness for those operators that are "quasi-translation-invariant." The approach used is related to the "circle-method" of Hardy and Littlewood, and requires multi-dimensional extensions of Weyl sums and Gauss sums, as well as variants that replace scalar sums by operator sums.