2026/07/28 by Zhenzhen Wei
#math.AP
Let K⊂ \RN be a convex body containing the origin in its interior, and let \hK⋅ be its Minkowski functional. In this paper, we develop an identity-based framework for sharp anisotropic L2-Caffarelli-Kohn-Nirenberg inequalities associated with the anisotropic radial derivative \mathcal RK(u)(x)=\fracx⋅∇ u(x)\hKx, x∈\RN∖\o\. A key point of the present work is that K is not assumed to be origin-symmetric. Consequently, the Minkowski functional \hK⋅ need not be even, and the usual norm-based anisotropic arguments do not apply directly. The main tools are anisotropic L2-Hardy and L2-Caffarelli-Kohn-Nirenberg identities with explicit nonnegative remainders. These identities yield sharp anisotropic L2-Caffarelli-Kohn-Nirenberg inequalities whose best constants depend on the parameter region of (a,b)∈\mathbb R2. We also study the attainability of the sharp constants in a natural completion space and obtain the corresponding extremal functions. As further consequences, we derive sharp anisotropic Heisenberg-type uncertainty principles and max-type anisotropic gradient inequalities. When K is the Euclidean unit ball, our results recover the classical Euclidean L2 theory; when K is origin-symmetric, they are consistent with the usual norm-based anisotropic framework. In particular, the present results extend the sharp L2-Caffarelli-Kohn-Nirenberg theory to general convex bodies containing the origin in their interiors, for which the Minkowski functional may be non-even.