On Discrete Fractional Integral Operators and Mean Values of Weyl Sums

In this paper we prove new ` → ` 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 rs,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.