$ell^p(mathbb{Z}^d)$-Improving Properties and Sparse Bounds for Discrete Spherical Maximal Averages
$ell^p(mathbb{Z}^d)$-Improving Properties and Sparse Bounds for Discrete Spherical Maximal Averages
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$ell^p(mathbb{Z}^d)$-改进离散球最大平均值的属性和稀疏界限
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发表时间：
2018
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通讯作者：
R. Kesler
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作者：
R. Kesler
我们展示了$ell ^{p}(mathbb{Z}^d)$的范围-改进了离散球面极大平均值在每个维度$dgeq 5$上的性质。用于显示这些改进性质的策略随后被用于建立稀疏边界，这将Magyar， Stein和Wainger的离散极大定理扩展到加权空间。特别地，稀疏界意味着离散球面最大平均值是从$ell^2(w)$到$ell^2(w)$的有界映射，条件是$w^{frac{d}{d-4}+delta}$对于某些$delta> .$属于Muckenhoupt类$A_2$
We exhibit a range of $ell ^{p}(mathbb{Z}^d)$-improving properties for the discrete spherical maximal average in every dimension $dgeq 5$. The strategy used to show these improving properties is then adapted to establish sparse bounds, which extend the discrete maximal theorem of Magyar, Stein, and Wainger to weighted spaces. In particular, the sparse bounds imply that the discrete spherical maximal average is a bounded map from $ell^2(w)$ into $ell^2(w)$ provided $w^{frac{d}{d-4}+delta}$ belongs to the Muckenhoupt class $A_2$ for some $delta>0.$