Extended Caffarelli-Kohn-Nirenberg inequalities, and remainders, stability, and superweights for Lᵖ-weighted Hardy inequalities

الأصل · EN

In this paper we give an extension of the classical Caffarelli-Kohn-Nirenberg inequalities: we show that for 1<p,q<∞, 0<r<∞ with p+q≥ r, δ∈[0,1]∩[r-q/r,p/r] with δr/p+(1-δ)r/q=1 and a, b, c with c=δ(a-1)+b(1-δ), and for all functions f∈ C₀∞(Rⁿ{0}) we have |x|ᶜfₗʳ₍ᵣⁿ₎ ≤ |p/n-p(1-a)|δ |x|ᵃ∇ fδₗᵖ₍ᵣⁿ₎ |x|ᵇf¹⁻δLq(Rⁿ) for n≠ p(1-a), where the constant |p/n-p(1-a)|δ is sharp for p=q with a-b=1 or p≠ q with p(1-a)+bq≠0. In the critical case n=p(1-a) we have |x|ᶜfₗʳ₍ᵣⁿ₎ ≤ pδ |x|ᵃ|x|∇ fδₗᵖ₍ᵣⁿ₎ |x|ᵇf¹⁻δLq(Rⁿ). Moreover, we also obtain anisotropic versions of these inequalities which can be conveniently formulated in the language of Folland and Stein's homogeneous groups. Consequently, we obtain remainder estimates for Lᵖ-weighted Hardy inequalities on homogeneous groups, which are also new in the Euclidean setting of Rⁿ. The critical Hardy inequalities of logarithmic type and uncertainty type principles on homogeneous groups are obtained. Moreover, we investigate another improved version of Lᵖ-weighted Hardy inequalities involving a distance and stability estimates. We also establish sharp Hardy type inequalities in Lᵖ, 1<p<∞, with superweights, i.e. with the weights of the form (a+b|x|α)βp|x|ᵐ allowing for different choices of α and β.