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We characterize the real interpolation space between weighted $L^1$ and$W^{1,1}$ spaces on arbitrary bounded domains, when the weights are positivepowers of the distance to the boundary multiplied by an $A_1$ weight. As anapplication of this result we obtain weighted fractional Poincar\'einequalities with sharp dependence on the fractional parameter $s$ (for $s$close to 1) and show that they are equivalent to a weighted Poincar\'einequality for the gradient.

An interpolation result for $A_1$ weights with applications to fractional Poincar\'e inequalities