On discrete fractional integral operators and mean values of Weyl Sums

Details On discrete fractional integral operators and mean values of Weyl Sums On discrete fractional integral operators and mean values of Weyl Sums

2010-05-21

arxiv.org/abs/1005.4052v1

Mathematics

Number Theory

Scientific paper

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.

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