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arXiv · 2205.09051

Weighted Gagliardo-Nirenberg inequalities via Optimal Transport Theory and Applications

Zoltán M. Balogh·Sebastiano Don·Alexandru Kristály
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Abstract

We prove Gagliardo-Nirenberg inequalities with three weights -- verifying a joint concavity condition -- on open convex cones of $\mathbb R^n$. 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.

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Zoltán M. Balogh, Sebastiano Don, Alexandru Kristály. 2022-05-18. Weighted Gagliardo-Nirenberg inequalities via Optimal Transport Theory and Applications. https://arxiv.org/abs/2205.09051

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