Remote Access Transactions of the American Mathematical Society Series B
Gold Open Access

Transactions of the American Mathematical Society Series B

ISSN 2330-0000

   
 
 

 

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


Authors: Michael Ruzhansky, Durvudkhan Suragan and Nurgissa Yessirkegenov
Journal: Trans. Amer. Math. Soc. Ser. B 5 (2018), 32-62
MSC (2010): Primary 22E30, 43A80
DOI: https://doi.org/10.1090/btran/22
Published electronically: February 14, 2018
Full-text PDF
View in AMS MathViewer New

Abstract | References | Similar Articles | Additional Information

Abstract: In this paper we give an extension of the classical Caffarelli-Kohn-Nirenberg inequalities: we show that for $ 1<p,q<\infty $, $ 0<r<\infty $ with $ p+q\geq r$, $ \delta \in [0,1]\cap \left [\frac {r-q}{r},\frac {p}{r}\right ]$ with $ \frac {\delta r}{p}+\frac {(1-\delta )r}{q}=1$ and $ a$, $ b$, $ c\in \mathbb{R}$ with $ c=\delta (a-1)+b(1-\delta )$, and for all functions $ f\in C_{0}^{\infty }(\mathbb{R}^{n}\backslash \{0\})$ we have

$\displaystyle \Vert\vert x\vert^{c}f\Vert _{L^{r}(\mathbb{R}^{n})} \leq \left \... ...)} \left \Vert\vert x\vert^{b}f\right \Vert^{1-\delta }_{L^{q}(\mathbb{R}^{n})}$    

for $ n\neq p(1-a)$, where the constant $ \left \vert\frac {p}{n-p(1-a)}\right \vert^{\delta }$ is sharp for $ p=q$ with $ a-b=1$ or $ p\neq q$ with $ p(1-a)+bq\neq 0$. In the critical case $ n=p(1-a)$ we have

$\displaystyle \left \Vert\vert x\vert^{c}f\right \Vert _{L^{r}(\mathbb{R}^{n})}... ...} \left \Vert\vert x\vert^{b}f\right \Vert^{1-\delta }_{L^{q}(\mathbb{R}^{n})}.$    

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 $ L^{p}$-weighted Hardy inequalities on homogeneous groups, which are also new in the Euclidean setting of $ \mathbb{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<\infty $, with superweights, i.e., with the weights of the form $ \frac {(a+b\vert x\vert^{\alpha })^{\frac {\beta }{p}}}{\vert x\vert^{m}}$ allowing for different choices of $ \alpha $ and $ \beta $. 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.

References [Enhancements On Off] (What's this?)


Similar Articles

Retrieve articles in Transactions of the American Mathematical Society, Series B with MSC (2010): 22E30, 43A80

Retrieve articles in all journals with MSC (2010): 22E30, 43A80


Additional Information

Michael Ruzhansky
Affiliation: Department of Mathematics, Imperial College London, 180 Queen’s Gate, London SW7 2AZ, United Kingdom
Email: m.ruzhansky@imperial.ac.uk

Durvudkhan Suragan
Affiliation: Institute of Mathematics and Mathematical Modelling, 125 Pushkin Street, 050010 Almaty, Kazakhstan–and–RUDN University, 6 Miklukho-Maklay Street, Moscow 117198, Russia
Email: suragan@math.kz

Nurgissa Yessirkegenov
Affiliation: Institute of Mathematics and Mathematical Modelling, 125 Pushkin Street, 050010 Almaty, Kazakhstan–and–Department of Mathematics, Imperial College London, 180 Queen’s Gate, London SW7 2AZ, United Kingdom
Email: n.yessirkegenov15@imperial.ac.uk

DOI: https://doi.org/10.1090/btran/22
Keywords: Hardy inequality, weighted Hardy inequality, Caffarelli-Kohn-Nirenberg inequality, remainder term, homogeneous Lie group.
Received by editor(s): March 14, 2017
Received by editor(s) in revised form: August 6, 2017
Published electronically: February 14, 2018
Additional Notes: The authors were supported in part by the EPSRC grant EP/K039407/1 and by the Leverhulme Grant RPG-2014-02, as well as by the MESRK grant AP05130981. The second author was also supported by the Ministry of Science of the Russian Federation (the Agreement number No. 02.a03.21.0008). The third author was also supported by the MESRK grant AP05133271. No new data was collected or generated during the course of research.
Article copyright: © Copyright 2018 by the author under Creative Commons Attribution 3.0 License (CC BY 3.0)

American Mathematical Society