由Birch-Magyar平均给出的缺项极大函数的界
Bounds for Lacunary maximal functions given by Birch--Magyar averages
摘要：
我们得到了关于由多元丢番图方程中的充分非奇异超曲面的扩张所定义的缺项离散极大算子的正、反结果。我们的否定结果表明，这个问题有很大的不同，从缺沿非奇异超曲面定义的离散极大算子。我们的积极成果是相应的全极大函数，最初由Magyar研究的界限上的改进。为了得到正的结果，我们利用第二作者的一种插值技巧，将问题归结为一个主项的极大函数。主要术语采用了第一作者作品中介绍的那些术语，这是马扎尔作品中出现的主要术语的更本地化版本。本文的主要内容是对$\ell^1$附近的主要项给出一个新的界。对于我们的否定结果，我们推广了Zienkiewicz的一个论点。
英文摘要：
We obtain positive and negative results concerning lacunary discrete maximal operators defined by dilations of sufficiently nonsingular hypersurfaces arising from Diophantine equations in many variables. Our negative results show that this problem differs substantially from that of lacunary discrete maximal operators defined along a nonsingular hypersurface. Our positive results are improvements over bounds for the corresponding full maximal functions which were initially studied by Magyar. In order to obtain positive results, we use an interpolation technique of the second author to reduce problem to a maximal function of main terms. The main terms take the shape of those introduced in work of the first author, which is a more localized version of the main terms that appear in work of Magyar. The main ingredient of this paper is a new bound on the main terms near $\ell^1$. For our negative results we generalize an argument of Zienkiewicz.
机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。
