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

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 $p\to p^{\prime }$ 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.

19 pages, 3 figures