In this paper we generalize the characterization for Sobolev spaces with constant exponent via averages on balls to the variable exponent setting. This method can be used to define Sobolev spaces with variable exponent on metric spaces.
"Characterizations of Sobolev spaces with variable exponent via averages on balls." Tohoku Math. J. (2) 72 (4) 569 - 579, 2020. https://doi.org/10.2748/tmj.20191018