engleski

Improving estimates for discrete polynomial averages

For a polynomial P mapping the integers into the integers, define an averaging operator A_N acting on functions f on the integers. We prove sufficient conditions for the l^p-improving inequality. 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.

Improving estimate ; Polynomial ; Discrete average ; Discrete fractional integral

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