Some subcritical estimates for the $\ell^p$-improving problem for discrete curves

TL;DR

This paper establishes sharp subcritical $ ext{ell}^p$-improving estimates for discrete averages along polynomial curves in integer lattices, using Christ's refinement method and diophantine solution estimates.

Contribution

It introduces a novel combination of Christ's refinement method with diophantine estimates to obtain sharp subcritical $ ext{ell}^p$-improving bounds for polynomial curve averages.

Findings

Obtained restricted weak-type $p o p'$ estimates in the subcritical regime.

Constants' dependence on $N$ is sharp, up to an $ ext{epsilon}$-loss.

Applied elementary diophantine estimates to improve understanding of discrete polynomial averages.

Abstract

We apply Christ's method of refinements to the $\ell^p$-improving problem for discrete averages $\mathcal{A}_N$ along polynomial curves in $\mathbb{Z}^d$. Combined with certain elementary estimates for the number of solutions to certain special systems of diophantine equations, we obtain some restricted weak-type $p \to p'$ estimates for the averages $\mathcal{A}_N$ in the subcritical regime. The dependence on $N$ of the constants here obtained is sharp, except maybe for an $\epsilon$-loss.