Let \(f\in L^p({\mathbb {R}}^d)\), \(d\ge 3\), and let \(A_t f(x)\) be the average of f over the sphere with radius t centered at x. For a subset E of [1, 2] we prove close to sharp \(L^p\rightarrow L^q\) estimates for the maximal function \(\sup _{t\in E} |A_t f|\). A new feature is the dependence of the results on both the upper Minkowski dimension of E and the Assouad dimension of E. The result can be applied to prove sparse domination bounds for a related global spherical maximal function.

中文翻译：

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$$ \ mathbf {L ^ p \ rightarrow L ^ q} $$ Lp→球面最大值算子的Lq界

令\（f \ in L ^ p（{\ mathbb {R}} ^ d）\），\（d \ ge 3 \）和\（A_t f（x）\）是f在半径为t的球体以x为中心。对于[1，2]的子集E，我们证明接近最大函数\（\ sup _ {t \ in E} | A_t f | \）的锐利\（L ^ p \ rightarrow L ^ q \）估计。一个新的特点是上的两个上部计盒维数的结果的依赖性Ë和的Assouad尺寸ë。该结果可用于证明相关全局球形最大函数的稀疏控制界。