A sharp monomial Caffarelli-Kohn-Nirenberg inequality

We consider a monomial Caffarelli-Kohn-Nirenberg inequality, find the optimal constant and classify the optimizers under an integrated curvature dimension condition. We take advantage of the $\mathrm{\Gamma }$-calculus to exploit geometrical techniques to tackle the problem and regularity results to justify some integration by parts. A symmetry-breaking result is also provided.