Weighted Gagliardo–Nirenberg Inequalities via Optimal Transport Theory and Applications

We prove Gagliardo–Nirenberg inequalities with three weights—verifying a joint concavity condition—on open convex cones of ℝ𝑛 . If the weights are equal to each other the inequalities become sharp and we compute explicitly the sharp constants. For a certain range of parameters we can characterize the class of extremal functions; in this case, we also show that the sharpness in the main three-weighted Gagliardo–Nirenberg inequality implies that the weights must be equal up to some constant multiplicative factors. Our approach uses optimal mass transport theory and a careful analysis of the joint concavity condition of the weights. As applications we establish sharp weighted 𝑝-log-Sobolev, Faber–Krahn, and isoperimetric inequalities with explicit sharp constants.