On first order interpolation inequalities with weights on the Heisenberg group

Han, Ya; Niu, Peng Cheng; Zhang, Shu Tao

doi:10.1007/s10114-011-8584-2

Cited by 3 publications

(1 citation statement)

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“…In [ZHD15] and [HZ11], weighted Hardy type inequalities were obtained for the generalised Baouendi-Grushin vector fields, which is when γ = 0 gives the standard gradient in R n . We also refer to [HNZ11], [Han15] for weighted Hardy inequalities on the Heisenberg group, to [HZD11] and [ZHD14] on the H-type groups, and a recent paper [Yac17] on Lie groups of polynomial growth as well as to references therein.…”

Section: Introductionmentioning

confidence: 99%

Extended Caffarelli-Kohn-Nirenberg inequalities, and remainders, stability, and superweights for 𝐿^{𝑝}-weighted Hardy inequalities

Ruzhansky

Сураган

Yessirkegenov

2018

Trans. Amer. Math. Soc. Ser. B

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Abstract. In this paper we give an extension of the classical Caffarelli-KohnNirenberg inequalities: we show that for 1 < p, q < ∞, 0 < r < ∞ with , where the constantMoreover, 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 L p -weighted Hardy inequalities on homogeneous groups, which are also new in the Euclidean setting of R n . The critical Hardy inequalities of logarithmic type and uncertainty type principles on homogeneous groups are obtained. Moreover, we investigate another improved version of L p -weighted Hardy inequalities involving a distance and stability estimate. The relation between the critical and the subcritical Hardy inequalities on homogeneous groups is also investigated. We also establish sharp Hardy type inequalities in L p , 1 < p < ∞, with superweights, i.e., with the weights of the formallowing for different choices of α and β. There are two reasons why we call the appearing weights the superweights: the arbitrariness of the choice of any homogeneous quasi-norm and a wide range of parameters.

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Section: Introductionmentioning

confidence: 99%