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Zhenzhen Wei

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Preprint Jul 2026

Sharp anisotropic $L^2$-Caffarelli-Kohn-Nirenberg inequalities associated with the Minkowski functional

Let $K\subset \RN$ be a convex body containing the origin in its interior, and let $\hK{\cdot}$ be its Minkowski functional. In this paper, we develop an identity-based framework for sharp anisotropic $L^2$-Caffarelli-Kohn-Nirenberg inequalities associated with the anisotropic radial derivative $$ \mathcal R_K(u)(x)=\frac{x\cdot\nabla u(x)}{\hK{x}}, \quad x\in\RN\setminus\{o\}. $$ A key point of the present work is that $K$ is not assumed to be origin-symmetric. Consequently, the Minkowski functional $\hK{\cdot}$ need not be even, and the usual norm-based anisotropic arguments do not apply directly. The main tools are anisotropic $L^2$-Hardy and $L^2$-Caffarelli-Kohn-Nirenberg identities with explicit nonnegative remainders. These identities yield sharp anisotropic $L^2$-Caffarelli-Kohn-Nirenberg inequalities whose best constants depend on the parameter region of $(a,b)\in\mathbb R^2$. 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 $L^2$ theory; when $K$ is origin-symmetric, they are consistent with the usual norm-based anisotropic framework. In particular, the present results extend the sharp $L^2$-Caffarelli-Kohn-Nirenberg theory to general convex bodies containing the origin in their interiors, for which the Minkowski functional may be non-even.

Zhenzhen Wei · 1 citation

Preprint Aug 2026

Anisotropic Caffarelli-Kohn-Nirenberg inequalities for the Minkowski functional: $L^p$-remainder identities and extremals

We establish exact anisotropic $L^p$-Hardy and Caffarelli-Kohn-Nirenberg identities associated with the Minkowski functional $\|\cdot\|_K$ and the anisotropic radial derivative $\mathcal R_K$, where $1<p<\infty$ and $K\subset\RN$ is a smooth and strictly convex body containing the origin in its interior, not necessarily origin-symmetric. These identities contain explicit nonnegative $R_p$-remainders and yield sharp anisotropic radial $L^p$-Hardy and $L^p$-Caffarelli-Kohn-Nirenberg inequalities. In the noncritical parameter regions, we characterize the full family of extremal functions in the natural weighted completion space, while in the critical case the sharp constant is not attained by any nonzero function in that space. As applications, we establish sharp anisotropic $L^p$-Heisenberg uncertainty principles and obtain explicit anisotropic Gaussian-type optimizers. Finally, letting $K^*$ denote the polar body of $K$, we derive norm-based anisotropic gradient inequalities associated with the origin-symmetric body $K^*\cap(-K^*)$. When $K$ is origin-symmetric, the corresponding gradient constants are sharp, including those in the critical case.

Zhenzhen Wei · 0 citations