Improving estimates for discrete polynomial averages

TL;DR

This paper establishes conditions under which polynomial averaging operators on integers improve $ ext{l}^p$ to $ ext{l}^q$ norms, with sharp results for quadratic polynomials and near-sharp for higher degrees, using advanced harmonic analysis tools.

Contribution

It provides new $ ext{l}^p$-improving inequalities for polynomial averages, leveraging recent breakthroughs like the Vinogradov Mean Value Theorem.

Findings

Sharp inequalities for quadratic polynomials.

Near-sharp inequalities for degree three and higher.

Application of Vinogradov Mean Value Theorem to discrete averages.

Abstract

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{equation*} \|A_N f\|_{\ell^q(\mathbb{Z})} \lesssim_{P,p,q} N^{-d(\frac{1}{p}-\frac{1}{q})} \|f\|_{\ell^p(\mathbb{Z})}, \qquad N \in\mathbb{N}, \end{equation*} where $1\leq p \leq q \leq \infty$. 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.