Improving Estimates for Discrete Polynomial Averages

For a polynomial P mapping the integers into the integers, define an averaging operator ANf(x):=1N∑k=1Nf(x+P(k)) acting on functions on the integers. We prove sufficient conditions for the ℓp-improving inequality ‖ANf‖ℓq(Z)≲P,p,qN-d(1p-1q)‖f‖ℓp(Z),N∈N,where 1 ≤ p≤ q≤ ∞. For a range of quadratic polynomials, the inequalities established are sharp, up to the boundary of the allowed pairs of (p, q). For degree three and higher, the inequalities are close to being sharp. In the quadratic case, we appeal to discrete fractional integrals as studied by Stein and Wainger. In the higher degree case, we appeal to the Vinogradov Mean Value Theorem, recently established by Bourgain, Demeter, and Guth