Discrete fractional Radon transforms and quadratic forms

TL;DR

This paper studies discrete fractional Radon transforms over paraboloids defined by quadratic forms, establishing their boundedness on certain ℓ^p to ℓ^q spaces using harmonic analysis and number theory techniques.

Contribution

It introduces a novel analysis of discrete fractional Radon transforms involving quadratic forms, extending their boundedness results through spectral and number theoretic methods.

Findings

Boundedness of discrete fractional Radon transforms on ℓ^p to ℓ^q spaces.

Application of circle method techniques to harmonic analysis problems.

Integration of Fourier analysis and number theory in proving operator bounds.

Abstract

We consider discrete analogues of fractional Radon transforms involving integration over paraboloids defined by positive definite quadratic forms. We prove that such discrete operators extend to bounded operators from $\ell^p$ to $\ell^q$ for a certain family of kernels. The method involves an intricate spectral decomposition according to major and minor arcs, motivated by ideas from the circle method of Hardy and Littlewood. Techniques from harmonic analysis, in particular Fourier transform methods and oscillatory integrals, as well as the number theoretic structure of quadratic forms, exponential sums, and theta functions, play key roles in the proof.