  We characterize the real interpolation space between weighted $L^1$ and
$W^{1,1}$ spaces on arbitrary domains different from $\mathbb{R}^n$, when the
weights are positive powers of the distance to the boundary multiplied by an
$A_1$ weight. As an application of this result we obtain weighted fractional
Poincar\'e inequalities with sharp dependence on the fractional parameter $s$
(for $s$ close to 1) and show that they are equivalent to a weighted Poincar\'e
inequality for the gradient.
