Explicit counterexamples to Frank-Loss optimizer conjectures for critical Dirac and vector Sobolev inequalities

Abstract

Frank and Loss proposed critical Sobolev inequalities for the Dirac operator and for vector fields. They conjectured optimality of their spinorial family and wrote that it is conceivable that the higher-dimensional Loss-Yau potential is an optimizer. Wang and Zhang recently disproved both optimizer claims using direct comparison and instability arguments. We compute closed-form quotients for two elementary families of explicit competitors with strictly smaller quotients. For every $d\geq2$, the spinorial competitor is a constant spinor multiplied by a scalar Aubin-Talenti optimizer. For every $d\geq3$, the vector competitor is the planar azimuthal field \[ A_d(x)=\frac{d}{(1+\lvert x\rvert^2)^2} (-x_2,x_1,0,\ldots,0). \] It lies in the nonlinear gauge and its quotient is computed in closed form. For every odd $d\geq3$, it gives an explicit proof of nonoptimality of the higher-dimensional Loss-Yau-Dunne-Min potential. In dimension three, this field is a specific member of the competitor family of Wang and Zhang: it is a linear combination of the Loss-Yau potential and its reflected twin. Its axisymmetry gives a negative answer to their question about minimization within the $\mathrm{SO}(2)\times 1$-invariant class.