Restricted weak-type endpoint estimates for k-spherical maximal functions
Restricted weak-type endpoint estimates for k-spherical maximal functions
复制标题
k 球最大函数的受限弱类型端点估计
DOI：
10.1007/s00209-016-1802-y
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发表时间：
2017
影响因子：
0.8
通讯作者：
Kevin A. Hughes
中科院分区：
文献类型：
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作者：
Kevin A. Hughes
本文利用k-球面上格点解集的傅里叶变换的近似公式和Bourgain和Ionescu的方法，将离散k-球面极大函数的$$\ell ^p（\mathbb {Z}^d）$$\ell ^p（Zd）-有界性结果改进为端点处的限制弱型结果.我们引入了一个密度参数，它可以被看作是Minkowski维数的离散形式，用于连续模拟的相关工作中，以利用Wooley和Bourgain-Demeter-Guth关于Vinogradov均值代数的最新进展，通过一个新的单平均近似公式，得到缺项离散k-球面极大函数当$$k \ge 3$$k≥3时的改进界.
In this paper, we use the Approximation Formula for the Fourier transform of the solution set of lattice points on k-spheres and methods of Bourgain and Ionescu to refine the $$\ell ^p(\mathbb {Z}^d)$$ℓp(Zd)-boundedness results for discrete k-spherical maximal functions to a restricted weak-type result at the endpoint. We introduce a density-parameter, which may be viewed as a discrete version of Minkowski dimension used in related works on the continuous analgoue, in order to exploit recent progress of Wooley and Bourgain–Demeter–Guth on the Vinogradov mean value conjectures via a novel Approximation Formula for a single average, and obtain improved bounds for lacunary discrete k-spherical maximal functions when $$k \ge 3$$k≥3.