We exhibit a range of ℓ^P(ℤ^D)-improving properties for the discrete spherical maximal average in every dimension d ≥ 5. These improving properties are then used 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 in every dimension d ≥ 5 the discrete spherical maximal average is a bounded map from ℓ²(w) into ℓ²(w) provided \({w^{{d \over {d - 4}}}}\) belongs to the Muckenhoupt class A₂.

中文翻译：

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ℓP（ℤD）-改进球面离散平均平均值的性质和稀疏边界

我们表现出一定范围的ℓ ^P（ℤ ^d -improving用于在离散球形最大平均性质）每个维度d ≥5。这些改进性能，然后用于建立稀疏的边界，这延长了离散的最大的定理匈牙利，Stein和Wainger到加权​​空间。特别地，该稀疏界限意味着，在每一个维度d ≥5离散球形最大平均距离的有界地图ℓ ²（瓦特）插入ℓ ²（瓦特）提供\（{瓦特^ {{d \在{d - 4 }}}} \）属于Muckenhoupt类A ₂。