Some Subcritical Estimates for the $$\ell ^p$$-Improving Problem for Discrete Curves

Abstract

We apply Christ’s method of refinements to the $$\ell ^p$$ ℓ p -improving problem for discrete averages $${\mathcal {A}}_N$$ A N along polynomial curves in $${\mathbb {Z}}^d$$ 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 $$p \rightarrow p'$$ p → p ′ estimates for the averages $${\mathcal {A}}_N$$ A N in the subcritical regime. The dependence on N of the constants here obtained is sharp, except maybe for an $$\epsilon $$ ϵ -loss.

On the Inhomogeneous Vinogradov System

“…Systems of this or similar types have recently arisen in a range of different contexts (for example, [1,4,5]). It follows easily by a standard application from the triangle inequality that J s,k (X; a) J s,k (X) X s+ε (1.5) in the subcritical range.…”