A1 weights and d-homogeneous measures on ahlfors d-regular space

Let X be an Ahlfors d-regular space and m a d-regular measure on X. We prove that a measure µ on X is d-homogeneous if and only if µ is mutually absolutely continuous with respect to m and the derivative D m µ(x) is an A 1 weight. Also, we show by an example that every Ahlfors d-regular space carries a measure which is d-homogeneous but not d-regular.