On Sharp Anisotropic Hardy Inequalities

Recently, Yanyan Li and Xukai Yan showed in [8, 9] the following interesting Hardy inequalities with anisotropic weights: Let n ≥ 2, p ≥ 1, pα > 1 − n, p(α + β) > −n, then there exists C > 0 such that ^͈͈|x|^β|x|^α+¹∇u^͈͈L_p(Rn) ≥ C^͈͈|x|^β|x|^αu^͈͈L_p(Rn), ∀ u ∈ C¹_c (Rⁿ). Here x = (x1, . . . , xn−1, 0) for x = (x_i) ∈ Rⁿ. In this note, we will determine the best constant for the above estimate when p = 2 or β ≥ 0. Moreover, as refinement for very special case of Li–Yan’s result in [9], we provide explicit estimate for the anisotropic L^p-Caffarelli–Kohn–Nirenberg inequality.