The CKN inequality for spinors: symmetry and symmetry breaking

TL;DR

The paper develops a spinor version of the Caffarelli-Kohn-Nirenberg inequality in three dimensions and analyzes symmetry of optimizers under SU(2) actions. By reformulating in logarithmic variables and employing spectral methods on the sphere, the authors establish existence of optimizers, derive a spinor Béckner-type interpolation on S^2, and prove symmetry regions via Keller-Lieb-Thirring-type arguments, including extensions by a duality symmetry α ↔ −α−1. They show that symmetry holds in explicit parameter ranges and provide a monotonicity mechanism near α = 0, while also demonstrating linear instability and symmetry breaking in several regimes with explicit threshold curves p(α). The work highlights fundamental differences from the scalar CK N theory, introduces a new GN inequality for spinors on S^2, and sets a framework for understanding symmetry breaking in non-scalar variational problems arising in physics. Overall, the results give qualitative and quantitative insights into when spinor optimizers are symmetric and when symmetry is broken, with implications for Dirac-type operators and related quantum systems.

Abstract

This paper is devoted to Sobolev interpolation inequalities for spinors, with weights of Caffarelli-Kohn-Nirenberg (CKN) type. In view of the corresponding results for scalar functions, a natural question is to determine whether optimal spinors have symmetry properties, or whether spinors with symmetry properties are linearly unstable, in which case we shall say that symmetry breaking occurs. What symmetry means has to be carefully defined and the overall picture turns out to be richer than in the scalar case. So far, no symmetrization technique is available in the spinorial case. We can however determine a range of the parameters for which symmetry holds using a detailed analysis based mostly on spectral methods.

Figures (3)

• Figure 1: Symmetry regions are shown in green and symmetry breaking regions appear in red. The upper left corner represents the parameter domain for \ref{['CKN']} inequalities (scalar functions), while the main figure is the parameter domain for \ref{['SCKN']} inequalities (spinors). In the latter case, the threshold between symmetry and symmetry breaking is determined by the functions $\overline\alpha_\star$ and $\overline\alpha^\star$ which take values in the white areas of the strip $\alpha\le\beta\le\alpha+1$.
• Figure 2: Inequality \ref{['logarithmicSCKN']} for spinors and, for sake of comparison, its counterpart for scalars. The threshold between symmetry and symmetry breaking is determined by the functions $p\mapsto\alpha_\star(p)$ and $p\mapsto\alpha^\star(p)$ which take values in the white areas of the strip $2\le p\le6$. The curves corresponding to the ansatz for symmetry breaking and those which determine the symmetry regions will be made clear in the proofs; also see Figs. \ref{['Fig:SCKNlog-Detail-1']} and \ref{['Fig:SCKNlog-Detail-2']}.
• Figure 3: For any $\alpha\in[-1/2,1)$, the range of symmetry in Theorem \ref{['Thm:symmetry1']} is represented in green. The dotted curve corresponds to the condition $q=\mathsf q(\alpha)$, rewritten in terms of $p$ using \ref{['pqgamma']}. The plain curves correspond to the condition that $$α-1/2$^2=\tfrac{1}{2}\,\mathsf m\,(6-p)/(p-2)$ with either $\mathsf m=1-\alpha$ or $\mathsf m=1+2\,\alpha$ and determine \ref{['SymZone1']}-\ref{['SymZone2']}: symmetry occurs below these curves.

Theorems & Definitions (38)

• Proposition 1
• Definition 2
• Theorem 3
• Lemma 4
• Definition 5
• Theorem 6
• Remark 7
• Proposition 8
• proof
• Proposition 9