An interpolation result for A₁ weights with applications to fractional Poincaré inequalities

We characterize the real interpolation space between weighted \ (L¹\) and \ (W^1, 1\) spaces on arbitrary domains different from \ (Rⁿ\), when the weights are positive powers of the distance to the boundary multiplied by an \ (A₁\) weight. As an application of this result we obtain weighted fractional Poincaré inequalities with sharp dependence on the fractional parameter \ (s\) (for \ (s\) close to 1) and show that they are equivalent to a weighted Poincaré inequality for the gradient.