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We prove  L   p  -Caffarelli–Kohn–Nirenberg type inequalities on homogeneous groups, which is one of most general subclasses of nilpotent Lie groups, all with sharp constants. We also discuss some of their consequences. Already in the Abelian cases of isotropic or anisotropicℝ   n    , our results provide new conclusions in view of the arbitrariness of the choice of the not necessarily Euclidean quasi-norm.