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A sharp monomial Caffarelli-Kohn-Nirenberg inequality

Francesco Pagliarin

TL;DR

This work establishes a sharp monomial Caffarelli–Kohn–Nirenberg inequality by embedding the problem into a weighted geometric framework and exploiting $\Gamma$-calculus on carefully constructed warped-product spaces. By proving an integrated curvature–dimension condition and leveraging $n$-conformal invariance, the authors derive the exact optimal constant $C_{opt}$ and classify optimizers (radial in the modest parameter regime) with the explicit form $f(x)=\left(\frac{1}{s+t|x|^{2\alpha}}\right)^{\frac{n-2}{2}}$. A key feature is translating the inequality to a finite-measure warped space $S$, enabling variational methods, Schauder-type regularity for degenerate elliptic equations, and a precise computation of $Z$ that yields $C_{opt}=\frac{4}{\alpha^2 n(n-2) Z^{2/n}}$. The results also reveal symmetry-breaking phenomena in the corresponding whole-space inequality and connect the optimality to integrated curvature bounds rather than pointwise concavity assumptions. The work blends geometric analysis, degenerate elliptic theory, and optimal transport-inspired insights to deliver sharp constants and a detailed optimizer structure.

Abstract

We consider a monomial Caffarelli-Kohn-Nirenberg inequality, find the optimal constant and classify the optimizers under an integrated curvature dimension condition. We take advantage of the $Γ$-calculus to exploit geometrical techniques to tackle the problem and regularity results to justify some integration by parts. A symmetry-breaking result is also provided.

A sharp monomial Caffarelli-Kohn-Nirenberg inequality

TL;DR

This work establishes a sharp monomial Caffarelli–Kohn–Nirenberg inequality by embedding the problem into a weighted geometric framework and exploiting -calculus on carefully constructed warped-product spaces. By proving an integrated curvature–dimension condition and leveraging -conformal invariance, the authors derive the exact optimal constant and classify optimizers (radial in the modest parameter regime) with the explicit form . A key feature is translating the inequality to a finite-measure warped space , enabling variational methods, Schauder-type regularity for degenerate elliptic equations, and a precise computation of that yields . The results also reveal symmetry-breaking phenomena in the corresponding whole-space inequality and connect the optimality to integrated curvature bounds rather than pointwise concavity assumptions. The work blends geometric analysis, degenerate elliptic theory, and optimal transport-inspired insights to deliver sharp constants and a detailed optimizer structure.

Abstract

We consider a monomial Caffarelli-Kohn-Nirenberg inequality, find the optimal constant and classify the optimizers under an integrated curvature dimension condition. We take advantage of the -calculus to exploit geometrical techniques to tackle the problem and regularity results to justify some integration by parts. A symmetry-breaking result is also provided.
Paper Structure (17 sections, 13 theorems, 199 equations, 1 figure)

This paper contains 17 sections, 13 theorems, 199 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Geometric, functional setting and curvature
  3. Integration by parts
  4. n-conformal invariance
  5. Warped products
  6. Functional setting and approximation results
  7. Properties of $\mathbb{S}^{d-1}_*$ and integrated curvature dimension condition
  8. Proof of Theorem \ref{['thm1.1']}
  9. Computation of $C_{opt}$
  10. Appendix
  11. Lower bound on the solution
  12. Notable properties of $P_t$
  13. The operator $T_\lambda$ and the conclusion
  14. Positivity preserving
  15. Curvature dimension condition on $\mathbb{S}^{d-1}_*$
  16. ...and 2 more sections

Key Result

Theorem 1.1

Under hypotheses eqhpab, $n>4$, $A_d=0$ and the optimal constant $C_{opt}$ in eqmonckn is equal to where Moreover, if $\alpha^2\leq \frac{D-1}{n-1}<1$, then all the optimizers are radial and they have the following form

Figures (1)

  • Figure :

Theorems & Definitions (35)

  • Theorem 1.1
  • Remark 1.2
  • Corollary 1.3
  • proof
  • Definition 2.1
  • Example 2.2
  • Definition 2.3
  • Remark 2.4
  • Proposition 2.5
  • proof
  • ...and 25 more