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We prove a fractional Hardy-type inequality for vector fields over the halfspace based on a modified fractional semi-norm. A priori, the modifiedsemi-norm is not known to be equivalent to the standard fractional semi-normand in fact gives a smaller norm, in general. As such, the inequality we proveimproves the classical fractional Hardy inequality for vector fields. We willuse the inequality to establish the equivalence of a space of functions (ofinterest) defined over the half space with the classical fractional Sobolevspaces, which amounts to proving a fractional version of the classical Korn'sinequality.

Fractional Korn and Hardy-type inequalities for vector fields in half space