For a polynomial P mapping the integers into the integers, define an averaging operator \(A_{N} f(x):=\frac{1}{N}\sum _{k=1}^N f(x+P(k))\) acting on functions on the integers. We prove sufficient conditions for the \(\ell ^{p}\)-improving inequality$$\begin{aligned} \Vert A_N f\Vert _{\ell ^q(\mathbb {Z})} \lesssim _{P,p,q} N^{-d(\frac{1}{p}-\frac{1}{q})} \Vert f\Vert _{\ell ^p(\mathbb {Z})}, \qquad N \in \mathbb {N}, \end{aligned}$$where \(1\le p \le q \le \infty \). 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.

中文翻译：

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改进离散多项式平均值的估计

对于将整数映射为整数的多项式P，定义平均运算符\（A_ {N} f（x）：= \ frac {1} {N} \ sum _ {k = 1} ^ N f（x + P （k））\）作用于整数上的函数。我们证明了\（\ ell ^ {p} \）改善不等式$$ \ begin {aligned} \ Vert A_N f \ Vert _ {\ ell ^ q（\ mathbb {Z}）} \ lesssim _ { P，p，q} N ^ {-d（\ frac {1} {p}-\ frac {1} {q}）} \ Vert f \ Vert _ {\ ell ^ p（\ mathbb {Z}）} ，\ qquad N \ in \ mathbb {N}，\ end {aligned} $$其中\（1 \ le p \ le q \ le \ infty \）。对于一定范围的二次多项式，建立的不等式是尖锐的，直到（p，  q）。对于三级以上，不等式接近于尖锐。在二次情况下，我们求助于Stein和Wainger研究的离散分数积分。在更高的情况下，我们呼吁使用布尔加因，德米特和古斯最近建立的维诺格拉多夫均值定理。