A note on twisted discrete singular Radon transforms

TL;DR

This paper extends bounds for discrete singular Radon transforms to twisted and oscillatory variants, employing descent and induction methods to establish $ ext{l}^p$ and $ ext{l}^2$ bounds across different operator classes.

Contribution

It introduces new $ ext{l}^p$ and $ ext{l}^2$ bounds for twisted and oscillatory discrete Radon transforms, expanding the theoretical understanding of these operators.

Findings

Extended $ ext{l}^p$ bounds to twisted operators with oscillatory components.

Established $ ext{l}^2$ bounds for quasi-translation invariant transforms.

Derived $ ext{l}^p$ bounds for non-translation invariant oscillatory operators.

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.