In this paper, the authors present some new characterizations of the Musielak–Orlicz–Sobolev spaces with even smoothness order via ball averages and their derivatives on the radius. Consequently, as special examples of the Musielak–Orlicz–Sobolev spaces studied in this paper, the corresponding characterizations for some weighted Sobolev spaces, Orlicz–Sobolev spaces and variable Sobolev spaces are also obtained. Since these characterizations depend only on ball averages and their derivatives on the radius, they provide some possible ways to introduce the corresponding function spaces on any metric measure space.

中文翻译：

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通过球均值及其导数刻画偶数Musielak-Orlicz-Sobolev空间

在本文中，作者通过球平均值及其在半径上的导数，对Musielak-Orlicz-Sobolev空间的某些新特征进行了描述，它们具有均匀的平滑度顺序。因此，作为本文研究的Musielak-Orlicz-Sobolev空间的特殊示例，还获得了一些加权Sobolev空间，Orlicz-Sobolev空间和可变Sobolev空间的相应特征。由于这些特征仅取决于平均球及其半径的导数，因此它们提供了一些可能的方法来在任何度量度量空间上引入相应的函数空间。