We introduce and study fractional Sobolev spaces of functions taking their values in a Banach space. Our approach is based on Riemann-Liouville derivative. In this regard, paper is a continuation of the paper [D. Idczak, S. Walczak, Fractional Sobolev spaces via Riemann-Liouville derivatives, J. of Function Spaces and Appl. 2013 (2013), Art. ID 128043, 15 pp.], where real-valued functions are investigated. Moreover, we study the relation between the fractional derivative of an abstract function of one variable and partial fractional derivative of a real-valued function of two variables.

我们介绍并研究了在 Banach 空间中取值的函数的分数 Sobolev 空间。我们的方法基于 Riemann-Liouville 导数。在这方面，论文是论文[D. Idczak, S. Walczak, Fractional Sobolev 空间通过 Riemann-Liouville 导数，J. of Function Spaces and Appl。2013 (2013)，艺术。ID 128043, 15 pp.]，其中研究了实值函数。此外，我们研究了一个变量的抽象函数的分数导数与两个变量的实值函数的偏分数导数之间的关系。