On discrete fractional integral operators and mean values of Weyl sums

In this paper, we prove new ℓ p →ℓ 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.