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We prove $L^{p}$-Caffarelli-Kohn-Nirenberg type inequalities on homogeneousgroups, which is one of most general subclasses of nilpotent Lie groups, allwith sharp constants. We also discuss some of their consequences. Already inthe abelian case of $\mathbb{R}^{n}$ our results provide new insights in viewof the arbitrariness of the choice of the not necessarily Euclidean quasi-norm.

$L^{p}$-Caffarelli-Kohn-Nirenberg type inequalities on homogeneous groups