$L^2$-Caffarelli--Kohn--Nirenberg inequalities on metric measure spaces

Motivated by the sharp constants in the $L^2$-Caffarelli--Kohn--Nirenberg (or $L^2$-CKN for short) inequalities on Euclidean spaces, we study, in a unified framework, a sequence of $L^2$-CKN inequalities on metric measure spaces. On a general metric measure space, this sequence implies a reverse volume comparison of G\"unther type. Moreover, on a subclass of spaces admitting the measure contraction property, we show that this sequence of $L^2$-CKN inequalities are valid if and only if the spaces are volume cones. We also provide a stability result for inequalities of this type on volume cones.