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We introduce weighted Riesz bounded variation spaces defined on an opensubset of the $n$-dimensional Euclidean space and use them to characterizeweighted Sobolev spaces when the weight belongs to the Muckenhoupt class. As anapplication, using Rubio de Francia's extrapolation theory, a similarcharacterization of the variable exponent Sobolev spaces via variable exponentRiesz bounded variation spaces is obtained.

Sobolev meets Riesz: a characterization of weighted Sobolev spaces via weighted Riesz bounded variation space