Sharp $L^2$ stability for anisotropic radial Caffarelli-Kohn-Nirenberg inequalities

Abstract

We establish quantitative stability for radial Caffarelli-Kohn-Nirenberg inequalities involving the anisotropic radial derivative and the Minkowski functional of a smooth, strictly convex body in $\mathbb R^N$, $N\ge2$. Here, the convex body need not be origin-symmetric. For the $L^2$-Caffarelli-Kohn-Nirenberg inequality, we establish the sharp stability constant $|b+1-a|$ for $a\ne b+1$, and classify all equality cases in the natural weighted completion spaces, by optimizing over the angular coefficients and a common dilation parameter. We also obtain the sharp stability constants $2$ and $4$ for the scale-invariant and scale-noninvariant $L^2$-Heisenberg inequality, respectively. For $2\le p<\infty$, we also establish weighted stability of the anisotropic radial $L^p$-Caffarelli-Kohn-Nirenberg inequalities when $a\neq b+1$ and $(b+1-a)(N-p(b+1))\geq0$, and the anisotropic radial $L^p$-Heisenberg inequalities. As a byproduct, we establish weighted $L^p$-Poincaré inequalities for Gamma measures, which serve as the main tool for our stability estimates.