An interpolation result for \(A_1\) weights with applications to fractional Poincaré inequalities

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 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.