The CKN inequality for spinors: symmetry and symmetry breaking

Technical Summary: The CKN Inequality for Spinors: Symmetry and Symmetry Breaking

Problem Statement
This paper investigates the Caffarelli-Kohn-Nirenberg (CKN) inequalities for spinor-valued functions in three dimensions ($d=3$). While the scalar CKN inequality is well-understood regarding the existence of optimizers and their symmetry properties (specifically, whether optimizers are radial or if rotational symmetry is broken), the analogous problem for spinors presents distinct challenges. The authors consider the inequality:
$$\int_{\mathbb{R}^3} \frac{|\sigma \cdot \nabla \psi(x)|^2}{|x|^{2\alpha}} dx \geq C_{\alpha,\beta} \left( \int_{\mathbb{R}^3} \frac{|\psi(x)|^p}{|x|^{\beta p}} dx \right)^{2/p}$$
where $\psi: \mathbb{R}^3 \to \mathbb{C}^2$ is a 2-spinor, $\sigma$ represents the Pauli matrices, and the exponent $p$ is determined by scaling invariance. The central question is to determine the range of parameters $(\alpha, \beta)$ for which the optimal constant $C_{\alpha,\beta}$ is achieved by "symmetric" spinors, and conversely, where "symmetry breaking" occurs (i.e., where the global optimizer is not symmetric).

A key difficulty identified is that, unlike the scalar case where the group $O(3)$ acts on functions, the spinorial inequality is invariant under the group $SU(2)$. However, there are no non-trivial $SU(2)$-invariant spinors. Consequently, the authors must rigorously redefine "symmetry" for spinors. They define a spinor as symmetric if it belongs to an irreducible representation of the lowest dimension ($H_0$ or $H_{-2}$), which corresponds to spinors of the form $f(|x|)\chi_0$ or $f(|x|)(\sigma \cdot \hat{x})\chi_0$, where $\chi_0$ is a constant spinor.

Methodology
The authors employ a multi-faceted approach combining spectral analysis, variational methods, and interpolation inequalities:

1. Logarithmic Coordinates and Spectral Decomposition: The problem is transformed into logarithmic coordinates ($s = \log|x|$) on the cylinder $\mathbb{R} \times S^2$. This converts the scaling invariance of the original inequality into translation invariance. The angular part of the operator is analyzed using the decomposition of spinors into spherical harmonics (eigenfunctions of $\sigma \cdot L$). The effective angular operator is identified as $K_\alpha = (\sigma \cdot L)^2 + (1-2\alpha)\sigma \cdot L$. Unlike the scalar case where the effective operator is independent of the parameter, $K_\alpha$ depends non-monotonically on $\alpha$, creating significant technical obstacles.

2. Existence of Optimizers: Using concentration-compactness arguments adapted from the scalar case (specifically following Catrina and Wang), the authors prove the existence of optimizers for the logarithmic inequality for all $\alpha \notin \Lambda$ (where $\Lambda$ is a set of singular values) and $p \in (2,6)$.

3. Symmetry Proofs (Monotonicity and Interpolation):

   • Monotonicity: A scaling argument is used to establish that if symmetry holds for a specific parameter, it holds for a range of parameters. This relies on analyzing the functional $G_\alpha[\phi]$ and distinguishing cases where $\alpha < 1/2$ and $\alpha > 1/2$.
   • Gagliardo-Nirenberg Inequalities on the Sphere: A novel contribution is the derivation of a Gagliardo-Nirenberg interpolation inequality for spinors on the sphere $S^2$. This inequality bounds the quadratic form associated with $K_\alpha$ by $L^q$ norms. The authors prove that for certain ranges of $q$ (and thus $p$), the inequality is saturated only by constant spinors (type $H_0$). This allows them to prove that global optimizers must be symmetric in specific parameter regions.

4. Symmetry Breaking (Instability Analysis): To identify regions of symmetry breaking, the authors analyze the linear stability of symmetric solutions. They compute the second variation of the energy functional around a symmetric optimizer. By constructing specific test functions (perturbations) in higher angular momentum modes, they demonstrate that the symmetric solution becomes a saddle point (linearly unstable) when the parameters exceed certain thresholds. This instability implies that the global optimizer must be non-symmetric, leading to $C_{\alpha,\beta} < C^*_{\alpha,\beta}$.

Key Results
The paper establishes the following main results:

• Existence: Optimizers exist for all admissible parameters where the inequality is non-trivial ($\alpha \notin \Lambda$).
• Symmetry Regions: The authors identify explicit regions in the $(\alpha, p)$ plane where all optimizers are symmetric. These regions are bounded by two functions, $\alpha_*(p)$ and $\alpha^*(p)$. Within these regions, the optimal constant for spinors coincides with the optimal constant restricted to symmetric spinors.
  • For $\alpha \in (-1/2, 0]$, symmetry holds if $p$ is sufficiently small (specifically $p \leq 2\frac{3+4(\alpha+1)^2}{3+4\alpha^2}$).
  • For $\alpha \in (0, 1/2) \cup (1/2, 1)$, symmetry holds under similar bounds involving rational functions of $\alpha$.
• Symmetry Breaking: The authors prove that symmetry breaking occurs in specific regions where symmetric solutions are linearly unstable. This happens when $p$ is large relative to $\alpha$.
  • For $\alpha \in (-1/2, 0)$, symmetry breaking occurs if $p > \min\{p_1(\alpha), p_2(\alpha)\}$.
  • For $\alpha > 1/2$, symmetry breaking occurs for sufficiently large $p$.
• Structural Differences from the Scalar Case: The paper highlights that the spinorial case is qualitatively different from the scalar CKN inequality. In the scalar case, symmetry breaking typically occurs for large $a$ (or $\alpha$) and large $p$. In the spinorial case, the symmetry region is bounded in $\alpha$ (specifically $-2 \leq \alpha \leq 1$), and the symmetry region for a fixed $\alpha$ consists of two disjoint intervals of $p$ (or $\beta$), separated by a symmetry-breaking region. This "island" structure of symmetry is unique to the spinorial setting.

Significance and Claims
The authors claim that this work provides the first detailed analysis of symmetry and symmetry breaking for CKN-type inequalities involving spinors. They emphasize that:

1. No Symmetrization: Unlike the scalar case, no rearrangement or symmetrization technique is available for spinors. The results rely entirely on spectral methods and stability analysis.
2. Complexity: The problem is significantly more involved than the scalar case due to the non-monotone dependence of the effective angular operator on the parameter $\alpha$ and the complex-linear nature of the spinor space.
3. New Inequalities: The derivation of the Gagliardo-Nirenberg inequality for spinors on the sphere is presented as a result of independent interest.
4. Modesty on Sharpness: The authors explicitly state that their results are not sharp in the sense of covering the entire parameter regime. The exact boundaries between symmetry and symmetry breaking (the precise functions $\alpha_*$ and $\alpha^*$) are not fully determined; the paper provides rigorous upper and lower bounds for these transition curves. Determining the complete phase diagram remains an open problem.

In summary, the paper successfully maps out a rich phase diagram for spinorial CKN inequalities, demonstrating that symmetry breaking is a robust phenomenon in this context, but with a structure more complex and constrained than its scalar counterpart.