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arXiv · 2012.06247

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

Spyridon Dendrinos·Kevin Hughes·Marco Vitturi
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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.

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Spyridon Dendrinos, Kevin Hughes, Marco Vitturi. 2020-12-11. Some subcritical estimates for the $\ell^p$-improving problem for discrete curves. https://arxiv.org/abs/2012.06247

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