Remote Access Transactions of the American Mathematical Society
Green Open Access

Transactions of the American Mathematical Society

ISSN 1088-6850(online) ISSN 0002-9947(print)

 
 

 

Sharp Adams and Hardy-Adams inequalities of any fractional order on hyperbolic spaces of all dimensions


Authors: Jungang Li, Guozhen Lu and Qiaohua Yang
Journal: Trans. Amer. Math. Soc. 373 (2020), 3483-3513
MSC (2010): Primary 42B15, 42B35, 42B37, 46E35
DOI: https://doi.org/10.1090/tran/7986
Published electronically: February 20, 2020
Full-text PDF

Abstract | References | Similar Articles | Additional Information

Abstract: We established in our recent work [Adv. Math. 319 (2017), pp. 567-598] the sharp Hardy-Adams inequalities for the bi-Laplacian-Beltrami operator $ (-\Delta _{\mathbb{H}})^2$ on the hyperbolic space $ \mathbb{B}^4$ in dimension four and for $ \frac {n}{2}$th (integer) power of the Laplacian operator $ (-\Delta _{\mathbb{H}})^{\frac {n}{2}}$ on $ \mathbb{B}^n$ of any even dimension $ n\ge 4$ in [Adv. Math., 2018]. The proofs of these inequalities rely on the special structures of the integer ( $ \frac {n}{2}$th) power and the even dimension $ n$. These are the borderline cases of the sharp higher order Hardy-Sobolev-Maz'ya inequalities by the authors [Amer. J. Math., 2019]. Thus, it remains open if such sharp Hardy-Adams inequalities hold for hyperbolic spaces $ \mathbb{B}^n$ of any odd dimension $ n$ and any fractional power $ \alpha $ of the Laplacian $ (-\Delta _{\mathbb{H}})^{\alpha }$. The purpose of this paper is to settle the remaining cases completely.

One of the main purposes of this paper is to establish sharp Adams type inequalities on Sobolev spaces $ W^{\alpha ,\frac {n}{\alpha }}(\mathbb{B}^{n})$ of any positive fractional order $ \alpha <n$ on the hyperbolic spaces $ \mathbb{B}^{n}$ of any dimension $ n$. The second main purpose of our paper is to establish sharp Hardy-Adams inequalities of any fractional order $ \alpha $ on hyperbolic spaces $ \mathbb{B}^n$ of any dimension $ n$. Such results were only recently obtained by the authors when $ n\ge 4$ and $ n$ is even. Fourier analysis on hyperbolic spaces play an important role in establishing the Adams and Hardy-Adams inequalities of any fractional order derivatives on hyperbolic spaces of any dimension. We also establish the fractional order Sobolev embedding theorems on hyperbolic spaces which are of their own independent interest.


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


Similar Articles

Retrieve articles in Transactions of the American Mathematical Society with MSC (2010): 42B15, 42B35, 42B37, 46E35

Retrieve articles in all journals with MSC (2010): 42B15, 42B35, 42B37, 46E35


Additional Information

Jungang Li
Affiliation: Department of Mathematics, University of Connecticut, Storrs, Connecticut 06290
Email: jungang.2.li@uconn.edu

Guozhen Lu
Affiliation: Department of Mathematics, University of Connecticut, Storrs, Connecticut 06290
Email: guozhen.lu@uconn.edu

Qiaohua Yang
Affiliation: School of Mathematics and Statistics, Wuhan University, Wuhan, 430072, People’s Republic of China
Email: qhyang.math@whu.edu.cn

DOI: https://doi.org/10.1090/tran/7986
Keywords: Fourier multipliers, heat kernels, Hardy inequalities; Adams' inequalities, hyperbolic space, sharp constants
Received by editor(s): September 15, 2018
Received by editor(s) in revised form: September 2, 2019
Published electronically: February 20, 2020
Additional Notes: The first two authors were partly supported by a Simons Collaboration Grant from the Simons Foundation and the third author was partly supported by the National Natural Science Foundation of China (No.11201346). Corresponding Authors: Guozhen Lu and Qiaohua Yang.
Article copyright: © Copyright 2020 American Mathematical Society