On Sharp Anisotropic Hardy Inequalities.

The article focuses on determining the best constants in recently established anisotropic Hardy inequalities with weighted norms, originally studied by Yanyan Li and Xukai Yan. It provides explicit formulas for the sharp constants in these inequalities when the exponent \( p=2 \) or when the weight parameter \(\beta \geq 0\), extending and refining Li–Yan's results. The work also offers an alternative proof and explicit estimates for anisotropic \( L^p \)-Caffarelli–Kohn–Nirenberg interpolation inequalities under specific parameter conditions. The approach relies on integral identities involving weighted divergence operators and positive test functions, enabling the characterization of optimal constants and the demonstration of their sharpness. Additionally, the article discusses recent generalizations by Musina–Nazarov and situates the results within the broader context of anisotropic weighted inequalities in analysis.