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Weighted Gagliardo-Nirenberg inequalities via Optimal Transport Theory and Applications

Zoltán M. Balogh, Sebastiano Don, Alexandru Kristály

TL;DR

The paper develops three-weight Gagliardo–Nirenberg inequalities on open convex cones via an optimal mass transport framework, establishing a central joint concavity condition $(C)$ that ties together three homogeneous weights. It delivers sharp constants and extremal characterizations in the equal-weight case, and proves a rigidity result showing that sharpness with attainment forces the weights to coincide up to a scalar and the transport map to be affine. The authors derive a wide spectrum of sharp weighted inequalities as applications, including sharp weighted Sobolev and GN inequalities, a limit $p$-log-Sobolev inequality as $\gamma\to1$, and sharp weighted Faber–Krahn and isoperimetric inequalities; they also discuss the dual formulation for $\gamma>1$ and the open problem of full equality characterization in that regime. Overall, the work provides a unified transport-based mechanism to obtain and analyze sharp weighted functional inequalities in conic domains with explicit constants and rigidity results.

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.

Weighted Gagliardo-Nirenberg inequalities via Optimal Transport Theory and Applications

TL;DR

The paper develops three-weight Gagliardo–Nirenberg inequalities on open convex cones via an optimal mass transport framework, establishing a central joint concavity condition that ties together three homogeneous weights. It delivers sharp constants and extremal characterizations in the equal-weight case, and proves a rigidity result showing that sharpness with attainment forces the weights to coincide up to a scalar and the transport map to be affine. The authors derive a wide spectrum of sharp weighted inequalities as applications, including sharp weighted Sobolev and GN inequalities, a limit -log-Sobolev inequality as , and sharp weighted Faber–Krahn and isoperimetric inequalities; they also discuss the dual formulation for and the open problem of full equality characterization in that regime. Overall, the work provides a unified transport-based mechanism to obtain and analyze sharp weighted functional inequalities in conic domains with explicit constants and rigidity results.

Abstract

We prove Gagliardo-Nirenberg inequalities with three weights -- verifying a joint concavity condition -- on open convex cones of . 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 -log-Sobolev, Faber-Krahn and isoperimetric inequalities with explicit sharp constants.
Paper Structure (12 sections, 12 theorems, 185 equations)

This paper contains 12 sections, 12 theorems, 185 equations.

Table of Contents

  1. Introduction
  2. Weighted Gagliardo-Nirenberg inequalities: proof of Theorems \ref{['th:CKN2weights']} & \ref{['th:CKN2weights-2']}
  3. Applications
  4. Sharp weighted Sobolev and Gagliardo-Nirenberg inequalities
  5. Weighted Sobolev inequalities with two weights
  6. Weighted Gagliardo-Nirenberg inequalities with one weight
  7. Sharp weighted $p$-log-Sobolev inequality
  8. Sharp weighted Faber-Krahn and isoperimetric inequalities
  9. Sharpness versus equality of the weights: proof of Theorem \ref{['prop:equalitycase-intro']}
  10. Final remarks and examples
  11. Characterization of $p$-concavity
  12. Examples of weights satisfying the main condition

Key Result

Theorem 1.1

$(\gamma<1)$ Let $n\geq 2$, $1<p< \infty$, $1>\gamma>\max\{1-1/n,1/{p'}\}$ and $E\subseteq \mathbb R^n$ be an open convex cone, and $\omega_i\colon E\to (0,+\infty)$ be homogeneous weights with degree $\tau_i>-n$ and of class $\mathcal{C}^1$, $i\in \{1,2,3\}$, such that the triplet $(\omega_1,\omega where If $\frac{1}{1-\gamma}+K =0$ we have that $M=0$ and the above formula still holds by using t

Theorems & Definitions (28)

  • Theorem 1.1
  • Theorem 1.2
  • Theorem 1.3
  • Lemma 2.1
  • Proposition 2.1
  • proof
  • Remark 2.1
  • Remark 2.2
  • Remark 2.3
  • Remark 3.1
  • ...and 18 more