It is known that there are several approaches to define a Sobolev class for Banach valued functions. We compare the usual definition via weak derivatives with the Reshetnyak–Sobolev space and with the Newtonian space; in particular, we provide sufficient conditions when all three agree. Also, we revise the difference quotient criterion and the property of Lipschitz mapping to preserve Sobolev space when it is acting as a superposition operator.

中文翻译：

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基于 Banach 函数空间的向量值 Sobolev 空间

众所周知，有多种方法可以为 Banach 值函数定义 Sobolev 类。我们将通过弱导数的通常定义与 Reshetnyak-Sobolev 空间和牛顿空间进行比较；特别是，当三者都同意时，我们提供充分条件。此外，我们修改了差商准则和 Lipschitz 映射的性质，以在充当叠加算子时保留 Sobolev 空间。