Sharp L² stability for anisotropic radial Caffarelli-Kohn-Nirenberg inequalities

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 RN, N≥2. Here, the convex body need not be origin-symmetric. For the L²-Caffarelli-Kohn-Nirenberg inequality, we establish the sharp stability constant $|b+1-a|$ for a≠ 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 L²-Heisenberg inequality, respectively. For 2≤ p<∞, we also establish weighted stability of the anisotropic radial Lᵖ-Caffarelli-Kohn-Nirenberg inequalities when a≠ b+1 and (b+1-a)(N-p(b+1))≥0, and the anisotropic radial Lᵖ-Heisenberg inequalities. As a byproduct, we establish weighted Lᵖ-Poincaré inequalities for Gamma measures, which serve as the main tool for our stability estimates.