Mathematical Research Letters

Volume 17 (2010)

Number 4

A note on twisted discrete singular Radon transforms

Pages: 701 – 720

DOI: http://dx.doi.org/10.4310/MRL.2010.v17.n4.a10

Author

Lillian B. Pierce (Institute for Advanced Study, Princeton NJ)

Abstract

In this paper we consider three types of discrete operators stemming from singular Radon transforms. We first extend an $\ell^p$ result for translation invariant discrete singular Radon transforms to a class of twisted operators including an additional oscillatory component, via a simple method of descent argument. Second, we note an $\ell^2$ bound for quasi-translation invariant discrete twisted Radon transforms. Finally, we extend an existing $\ell^2$ bound for a closely related non-translation invariant discrete oscillatory integral operator with singular kernel to an $\ell^p$ bound for all $1<p<\infty$. This requires an intricate induction argument involving layers of decompositions of the operator according to the Diophantine properties of the coefficients of its polynomial phase function.

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