  We prove a fractional Hardy-type inequality for vector fields over the half
space based on a modified fractional semi-norm. A priori, the modified
semi-norm is not known to be equivalent to the standard fractional semi-norm
and in fact gives a smaller norm, in general. As such, the inequality we prove
improves the classical fractional Hardy inequality for vector fields. We will
use the inequality to establish the equivalence of a space of functions (of
interest) defined over the half space with the classical fractional Sobolev
spaces, which amounts to proving a fractional version of the classical Korn's
inequality.
