### Abstract or Additional Information

We show that a class of spaces of vector fields whose semi-norms involve the magnitude of ``directional" difference quotients is in fact equivalent to the class of fractional Sobolev-Slobodeckij spaces. The equivalence can be considered a Korn-type characterization of said Sobolev spaces. For vector fields defined on various classes of domains, we obtain a relevant form of the inequality. As an application, we consider variational problems associated to strongly coupled systems of nonlocal equations motivated by a continuum mechanics model known as peridynamics. We use the fractional Korn-type inequalities to characterize vector fields in associated energy spaces and obtain existence and uniqueness of solutions in fractional Sobolev spaces.