We study the regularity properties of the centered fractional maximal function $M_{\beta}$. More precisely, we prove that the map $f \mapsto |\nabla M_\beta f|$ is bounded and continuous from $W^{1,1}(\mathbb{R}^d)$ to $L^q(\mathbb{R}^d)$ in the endpoint case $q=d/(d-\beta)$ if $f$ is radial function. For $d=1$, the radiality assumption can be removed. This corresponds to the counterparts of known results for the non-centered fractional maximal function. The main new idea consists in relating the centered and non-centered fractional maximal function at the derivative level.

中文翻译：

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中心分数极大函数在径向函数上的正则性

我们研究了中心分数极大函数 $M_{\beta}$ 的正则性。更准确地说，我们证明了映射 $f \mapsto |\nabla M_\beta f|$ 从 $W^{1,1}(\mathbb{R}^d)$ 到 $L^q( \mathbb{R}^d)$ 在端点情况 $q=d/(d-\beta)$ 如果 $f$ 是径向函数。对于 $d=1$，可以去除径向假设。这对应于非中心分数极大函数的已知结果的对应物。主要的新思想在于在导数水平上关联中心和非中心分数极大函数。