关于Minkowski泛函的各向异性Caffarelli-Kohn-Nirenberg不等式：$L^p$余项恒等式与极值函数
Anisotropic Caffarelli-Kohn-Nirenberg inequalities for the Minkowski functional: $L^p$-remainder identities and extremals
浏览论文内容
中文总结 AI 辅助
该研究建立了与Minkowski泛函相关的各向异性$L^p$-Hardy与Caffarelli-Kohn-Nirenberg恒等式，刻画了极值函数族，推导了不确定性原理与梯度不等式，丰富了各向异性泛函不等式理论。
中文摘要 AI 辅助
我们建立了与Minkowski泛函$\\|\cdot\\|_K$和各向异性径向导数$\mathcal{R}_K$相关的精确各向异性$L^p$-Hardy恒等式与Caffarelli-Kohn-Nirenberg恒等式，其中$1<p<\infty$，$K\subset\mathbb{R}^N$是内部包含原点的光滑严格凸体，不一定关于原点对称。这些恒等式包含显式非负$R_p$余项，并可推导出精确的各向异性径向$L^p$-Hardy不等式与$L^p$-Caffarelli-Kohn-Nirenberg不等式。在非临界参数区域，我们刻画了自然加权完备空间中极值函数的完整族；而在临界情形下，该空间中不存在任何非零函数能达到精确常数。作为应用，我们建立了精确的各向异性$L^p$-Heisenberg不确定性原理，并得到了显式的各向异性高斯型最优函数。最后，记$K^*$为$K$的极体，我们推导了与关于原点对称的体$K^*\cap(-K^*)$相关的、基于范数的各向异性梯度不等式。当$K$关于原点对称时，对应的梯度常数是精确的，包括临界情形下的常数。
英文摘要
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.
