Extended Caffarelli-Kohn-Nirenberg inequalities, and remainders, stability, and superweights for Lp-weighted Hardy inequalities
File(s)
Published version
Author(s)
Ruzhansky, M
Suragan, D
Yessirkegenov, N
Type
Journal Article
Abstract
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−qr, pr] with δrp+(1−δ) rq=1and a, b, c∈
R with c=δ(a−1) +b(1−δ), and for all functions f∈C∞0(Rn\{0})we have
‖|x|cf‖Lr(Rn)≤∣∣∣∣pn−p(1−a)∣∣∣∣δ‖|x|a∇f‖δLp(Rn)∥∥∥|x|bf∥∥∥1−δLq(Rn) for
n=p(1−a), where the constant ∣∣∣pn−p(1−a I∣∣δ is sharp for p=q with a−b=1or p=q with p(1−a)+bq= 0. In the critical case n=p(1−a) we have
‖|x|f‖Lr(Rn)≤pδ‖|x| a log |x|∇f‖δLp(Rn)∥∥∥|x|bf∥∥∥1−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 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 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. 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.
Nirenberg inequalities: we show that for 1
<p,q<∞,0<r<∞with p+q≥r, δ∈[0,1]∩[r−qr, pr] with δrp+(1−δ) rq=1and a, b, c∈
R with c=δ(a−1) +b(1−δ), and for all functions f∈C∞0(Rn\{0})we have
‖|x|cf‖Lr(Rn)≤∣∣∣∣pn−p(1−a)∣∣∣∣δ‖|x|a∇f‖δLp(Rn)∥∥∥|x|bf∥∥∥1−δLq(Rn) for
n=p(1−a), where the constant ∣∣∣pn−p(1−a I∣∣δ is sharp for p=q with a−b=1or p=q with p(1−a)+bq= 0. In the critical case n=p(1−a) we have
‖|x|f‖Lr(Rn)≤pδ‖|x| a log |x|∇f‖δLp(Rn)∥∥∥|x|bf∥∥∥1−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 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 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. 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.
Date Issued
2018-02-14
Date Acceptance
2017-09-14
Citation
Transactions of the American Mathematical Society, 2018, 5, pp.32-62
ISSN
0002-9947
Publisher
American Mathematical Society
Start Page
32
End Page
62
Journal / Book Title
Transactions of the American Mathematical Society
Volume
5
Copyright Statement
© Copyright 2018 by the author under Creative Commons Attribution 3.0 License (CC BY 3.0) (https://creativecommons.org/licenses/by/3.0/)
Sponsor
Engineering & Physical Science Research Council (EPSRC)
The Leverhulme Trust
Grant Number
EP/K039407/1
RPG-2014-002
Subjects
0101 Pure Mathematics
General Mathematics
Publication Status
Published