On discrete fractional integral operators and mean values of Weyl Sums

TL;DR

This paper establishes new bounds for a discrete fractional integral operator by linking Fourier analysis with mean values of Weyl sums, advancing understanding in harmonic analysis and number theory.

Contribution

It introduces novel $( ext{ell}^p, ext{ell}^q)$ bounds for the operator using circle method techniques and explores explicit interactions with Weyl sum mean values.

Findings

Derived new $( ext{ell}^p, ext{ell}^q)$ bounds for the operator.

Connected Fourier multiplier behavior with mean values of Weyl sums.

Extended results using recent advances in Waring's problem.

Abstract

In this paper we prove new $(\ell^p, \ell^q)$ bounds for a discrete fractional integral operator by applying techniques motivated by the circle method of Hardy and Littlewood to the Fourier multiplier of the operator. From a different perspective, we describe explicit interactions between the Fourier multiplier and mean values of Weyl sums. These mean values express the average behaviour of the number $r_{s,k}(l)$ of representations of a positive integer $l$ as a sum of $s$ positive $k$-th powers. Recent deep results within the context of Waring's problem and Weyl sums enable us to prove a further range of complementary results for the discrete operator under consideration.