Geodesic
Sobolev spaces of vector-valued functions
Zapiski Nauchnykh Seminarov POMI, Investigations on linear operators and function theory. Part 19, Tome 190 (1991), pp. 5-14

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The generalizations of Sobolev spaces in the following two directions are considered in this work. First of all, instead of $L^p$-norm the norm of a rearrangement-invariant space is used. Secondly, the spaces of functions with values in some Banach space are investigated. For this spaces we decide the problem of their connection with the spaces of Bessel potentials. 
@article{ZNSL_1991_190_a0,
     author = {A. V. Bukhvalov},
     title = {Sobolev spaces of vector-valued functions},
     journal = {Zapiski Nauchnykh Seminarov POMI},
     pages = {5--14},
     publisher = {mathdoc},
     volume = {190},
     year = {1991},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/ZNSL_1991_190_a0/}
}
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A. V. Bukhvalov. Sobolev spaces of vector-valued functions. Zapiski Nauchnykh Seminarov POMI, Investigations on linear operators and function theory. Part 19, Tome 190 (1991), pp. 5-14. http://geodesic.mathdoc.fr/item/ZNSL_1991_190_a0/