AI 中文总结
研究与闵可夫斯基泛函相关的尖锐各向异性\(L^2\)-卡法雷利 - 科恩 - 尼伦伯格不等式，通过基于恒等式的框架及相关工具得出不等式及常数，研究其可达性并推导相关原理和不等式，扩展了该理论到一般凸体。
AI 中文摘要
设\(K\subset \mathbb{R}^N\)是一个在其内部包含原点的凸体，\(\hK{\cdot}\)是其闵可夫斯基泛函。本文针对与各向异性径向导数\(\mathcal R_K(u)(x)=\frac{x\cdot\nabla u(x)}{\hK{x}}\)相关的尖锐各向异性\(L^2\)-卡法雷利 - 科恩 - 尼伦伯格不等式，建立了一个基于恒等式的框架。关键在于不假定\(K\)关于原点对称，所以闵可夫斯基泛函不一定是偶函数，常规基于范数的各向异性论证不直接适用。主要工具是带有明确非负余项的各向异性\(L^2\)-哈代和\(L^2\)-卡法雷利 - 科恩 - 尼伦伯格恒等式，由此得出尖锐不等式，其最佳常数依赖于\((a,b)\in\mathbb{R}^2\)的参数区域。还研究了尖锐常数在自然完备空间中的可达性并得到相应极值函数，进而推导出尖锐各向异性海森堡型不确定性原理和最大型各向异性梯度不等式。当\(K\)是欧几里得单位球时恢复经典欧几里得\(L^2\)理论，当\(K\)关于原点对称时与常规基于范数的各向异性框架一致，特别地，将尖锐\(L^2\)-卡法雷利 - 科恩 - 尼伦伯格理论扩展到内部包含原点的一般凸体，其闵可夫斯基泛函可能非偶。
英文摘要
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.
