Sparse Bounds for Discrete Maximal Functions associated with Birch-Magyar averages

TL;DR

This paper establishes sparse bounds and $ ext{l}^p$-estimates for discrete maximal functions linked to Birch-Magyar averages over sparse sequences, advancing understanding of their boundedness properties.

Contribution

It introduces a sparse domination principle for these operators and derives $ ext{l}^p$-estimates for all $p>1$, based on scale-free $ ext{l}^p$-improving estimates.

Findings

Established sparse domination for discrete Birch-Magyar averages.

Proved $ ext{l}^p$-estimates for all $p>1$ for the maximal functions.

Demonstrated scale-free $ ext{l}^p$-improving estimates for single-scale averages.

Abstract

In this article, we study discrete maximal function associated with the Birch-Magyar averages over sparse sequences. We establish sparse domination principle for such operators. As a consequence, we obtain $\ell^p$-estimates for such discrete maximal function over sparse sequences for all $p>1$. The proof of sparse bounds is based on scale-free $\ell^p-$improving estimates for the single scale Birch-Magyar averages.