Sharp Adams and Hardy-Adams inequalities of any fractional order on hyperbolic spaces of all dimensions
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- by Jungang Li, Guozhen Lu and Qiaohua Yang PDF
- Trans. Amer. Math. Soc. 373 (2020), 3483-3513 Request permission
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
- David R. Adams, A sharp inequality of J. Moser for higher order derivatives, Ann. of Math. (2) 128 (1988), no. 2, 385–398. MR 960950, DOI 10.2307/1971445
- Lars V. Ahlfors, Möbius transformations in several dimensions, Ordway Professorship Lectures in Mathematics, University of Minnesota, School of Mathematics, Minneapolis, Minn., 1981. MR 725161
- Jean-Philippe Anker, $\textbf {L}_p$ Fourier multipliers on Riemannian symmetric spaces of the noncompact type, Ann. of Math. (2) 132 (1990), no. 3, 597–628. MR 1078270, DOI 10.2307/1971430
- J.-P. Anker and L. Ji, Heat kernel and Green function estimates on noncompact symmetric spaces, Geom. Funct. Anal. 9 (1999), no. 6, 1035–1091. MR 1736928, DOI 10.1007/s000390050107
- V. Banica, The nonlinear Schrödinger equation on hyperbolic space, Comm. Partial Differential Equations 32 (2007), no. 10-12, 1643–1677. MR 2372482, DOI 10.1080/03605300600854332
- Valeria Banica, María del Mar González, and Mariel Sáez, Some constructions for the fractional Laplacian on noncompact manifolds, Rev. Mat. Iberoam. 31 (2015), no. 2, 681–712. MR 3375868, DOI 10.4171/RMI/850
- William Beckner, On the Grushin operator and hyperbolic symmetry, Proc. Amer. Math. Soc. 129 (2001), no. 4, 1233–1246. MR 1709740, DOI 10.1090/S0002-9939-00-05630-6
- William Beckner, On Lie groups and hyperbolic symmetry—from Kunze-Stein phenomena to Riesz potentials, Nonlinear Anal. 126 (2015), 394–414. MR 3388886, DOI 10.1016/j.na.2015.06.009
- William Beckner, Sharp Sobolev inequalities on the sphere and the Moser-Trudinger inequality, Ann. of Math. (2) 138 (1993), no. 1, 213–242. MR 1230930, DOI 10.2307/2946638
- William Beckner, Sharp inequalities and geometric manifolds, Proceedings of the conference dedicated to Professor Miguel de Guzmán (El Escorial, 1996), 1997, pp. 825–836. MR 1600195, DOI 10.1007/BF02656488
- Haïm Brezis and Moshe Marcus, Hardy’s inequalities revisited, Ann. Scuola Norm. Sup. Pisa Cl. Sci. (4) 25 (1997), no. 1-2, 217–237 (1998). Dedicated to Ennio De Giorgi. MR 1655516
- E. B. Davies and N. Mandouvalos, Heat kernel bounds on hyperbolic space and Kleinian groups, Proc. London Math. Soc. (3) 57 (1988), no. 1, 182–208. MR 940434, DOI 10.1112/plms/s3-57.1.182
- I. M. Gelfand, S. G. Gindikin, and M. I. Graev, Selected topics in integral geometry, Translations of Mathematical Monographs, vol. 220, American Mathematical Society, Providence, RI, 2003. Translated from the 2000 Russian original by A. Shtern. MR 2000133, DOI 10.1090/mmono/220
- I. S. Gradshteyn and I. M. Ryzhik, Table of integrals, series, and products, 6th ed., Academic Press, Inc., San Diego, CA, 2000. Translated from the Russian; Translation edited and with a preface by Alan Jeffrey and Daniel Zwillinger. MR 1773820
- Alexander Grigor’yan and Masakazu Noguchi, The heat kernel on hyperbolic space, Bull. London Math. Soc. 30 (1998), no. 6, 643–650. MR 1642767, DOI 10.1112/S0024609398004780
- Sigurdur Helgason, Groups and geometric analysis, Pure and Applied Mathematics, vol. 113, Academic Press, Inc., Orlando, FL, 1984. Integral geometry, invariant differential operators, and spherical functions. MR 754767
- Sigurdur Helgason, Geometric analysis on symmetric spaces, 2nd ed., Mathematical Surveys and Monographs, vol. 39, American Mathematical Society, Providence, RI, 2008. MR 2463854, DOI 10.1090/surv/039
- Andreas Juhl, Explicit formulas for GJMS-operators and $Q$-curvatures, Geom. Funct. Anal. 23 (2013), no. 4, 1278–1370. MR 3077914, DOI 10.1007/s00039-013-0232-9
- Loo Keng Hua, Starting with the unit circle, Springer-Verlag, New York-Berlin, 1981. Background to higher analysis; Translated from the Chinese by Kuniko Weltin. MR 665916
- Debabrata Karmakar and Kunnath Sandeep, Adams inequality on the hyperbolic space, J. Funct. Anal. 270 (2016), no. 5, 1792–1817. MR 3452717, DOI 10.1016/j.jfa.2015.11.019
- Hideo Kozono, Tokushi Sato, and Hidemitsu Wadade, Upper bound of the best constant of a Trudinger-Moser inequality and its application to a Gagliardo-Nirenberg inequality, Indiana Univ. Math. J. 55 (2006), no. 6, 1951–1974. MR 2284552, DOI 10.1512/iumj.2006.55.2743
- Nguyen Lam and Guozhen Lu, Sharp Adams type inequalities in Sobolev spaces $W^{m,\frac {n}{m}} (\Bbb R^n)$ for arbitrary integer $m$, J. Differential Equations 253 (2012), no. 4, 1143–1171. MR 2925908, DOI 10.1016/j.jde.2012.04.025
- Nguyen Lam and Guozhen Lu, Sharp Moser-Trudinger inequality on the Heisenberg group at the critical case and applications, Adv. Math. 231 (2012), no. 6, 3259–3287. MR 2980499, DOI 10.1016/j.aim.2012.09.004
- Nguyen Lam and Guozhen Lu, A new approach to sharp Moser-Trudinger and Adams type inequalities: a rearrangement-free argument, J. Differential Equations 255 (2013), no. 3, 298–325. MR 3053467, DOI 10.1016/j.jde.2013.04.005
- X. Liang, G. Lu, X. Wang and Q. Yang, Sharp Hardy-Trudinger-Moser inequalities in any $N$-dimensional hyperbolic spaces, Preprint.
- Jungang Li, Guozhen Lu, and Qiaohua Yang, Fourier analysis and optimal Hardy-Adams inequalities on hyperbolic spaces of any even dimension, Adv. Math. 333 (2018), 350–385. MR 3818080, DOI 10.1016/j.aim.2018.05.035
- Congwen Liu and Lizhong Peng, Generalized Helgason-Fourier transforms associated to variants of the Laplace-Beltrami operators on the unit ball in $\Bbb R^n$, Indiana Univ. Math. J. 58 (2009), no. 3, 1457–1491. MR 2542095, DOI 10.1512/iumj.2009.58.3588
- Genqian Liu, Sharp higher-order Sobolev inequalities in the hyperbolic space $\Bbb {H}^n$, Calc. Var. Partial Differential Equations 47 (2013), no. 3-4, 567–588. MR 3070556, DOI 10.1007/s00526-012-0528-x
- Guozhen Lu and Hanli Tang, Best constants for Moser-Trudinger inequalities on high dimensional hyperbolic spaces, Adv. Nonlinear Stud. 13 (2013), no. 4, 1035–1052. MR 3115151, DOI 10.1515/ans-2013-0415
- Guozhen Lu and Hanli Tang, Sharp Moser-Trudinger inequalities on hyperbolic spaces with exact growth condition, J. Geom. Anal. 26 (2016), no. 2, 837–857. MR 3472818, DOI 10.1007/s12220-015-9573-y
- Guozhen Lu and Qiaohua Yang, A sharp Trudinger-Moser inequality on any bounded and convex planar domain, Calc. Var. Partial Differential Equations 55 (2016), no. 6, Art. 153, 16. MR 3571204, DOI 10.1007/s00526-016-1077-5
- Guozhen Lu and Qiaohua Yang, Sharp Hardy-Adams inequalities for bi-Laplacian on hyperbolic space of dimension four, Adv. Math. 319 (2017), 567–598. MR 3695883, DOI 10.1016/j.aim.2017.08.014
- Guozhen Lu and Qiaohua Yang, Paneitz operators on hyperbolic spaces and high order Hardy-Sobolev-Maz’ya inequalities on half spaces, Amer. J. Math. 141 (2019), no. 6, 1777–1816. MR 4030527, DOI 10.1353/ajm.2019.0047
- G. Lu and G. Yang, Green’s functions of Paneitz and GJMS operators on hyperbolic spaces and sharp Hardy-Sobolev-Maz’ya inequalities on half spaces, arXiv:1903.10365.
- G. Lu and Q. Yang, Sharp Hardy-Sobolev-Maz’ya, Adams and Hardy-Adams inequalities on the Siegel domain and complex hyperbolic spaces, Preprint 2019, arxiv.org.
- Gianni Mancini, Kunnath Sandeep, and Cyril Tintarev, Trudinger-Moser inequality in the hyperbolic space ${\Bbb H}^N$, Adv. Nonlinear Anal. 2 (2013), no. 3, 309–324. MR 3089744, DOI 10.1515/anona-2013-0001
- Vladimir G. Maz’ja, Sobolev spaces, Springer Series in Soviet Mathematics, Springer-Verlag, Berlin, 1985. Translated from the Russian by T. O. Shaposhnikova. MR 817985, DOI 10.1007/978-3-662-09922-3
- J. Moser, A sharp form of an inequality by N. Trudinger, Indiana Univ. Math. J. 20 (1970/71), 1077–1092. MR 301504, DOI 10.1512/iumj.1971.20.20101
- Richard O’Neil, Convolution operators and $L(p,\,q)$ spaces, Duke Math. J. 30 (1963), 129–142. MR 146673
- Mark P. Owen, The Hardy-Rellich inequality for polyharmonic operators, Proc. Roy. Soc. Edinburgh Sect. A 129 (1999), no. 4, 825–839. MR 1718522, DOI 10.1017/S0308210500013160
- S. I. Pohožaev, On the eigenfunctions of the equation $\Delta u+\lambda f(u)=0$, Dokl. Akad. Nauk SSSR 165 (1965), 36–39 (Russian). MR 0192184
- Elias M. Stein, Singular integrals and differentiability properties of functions, Princeton Mathematical Series, No. 30, Princeton University Press, Princeton, N.J., 1970. MR 0290095
- Audrey Terras, Harmonic analysis on symmetric spaces and applications. I, Springer-Verlag, New York, 1985. MR 791406, DOI 10.1007/978-1-4612-5128-6
- Neil S. Trudinger, On imbeddings into Orlicz spaces and some applications, J. Math. Mech. 17 (1967), 473–483. MR 0216286, DOI 10.1512/iumj.1968.17.17028
- Guofang Wang and Dong Ye, A Hardy-Moser-Trudinger inequality, Adv. Math. 230 (2012), no. 1, 294–320. MR 2900545, DOI 10.1016/j.aim.2011.12.001
- V. I. Judovič, Some estimates connected with integral operators and with solutions of elliptic equations, Dokl. Akad. Nauk SSSR 138 (1961), 805–808 (Russian). MR 0140822
- Qiaohua Yang, Dan Su, and Yinying Kong, Sharp Moser-Trudinger inequalities on Riemannian manifolds with negative curvature, Ann. Mat. Pura Appl. (4) 195 (2016), no. 2, 459–471. MR 3476683, DOI 10.1007/s10231-015-0472-4
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
- MR Author ID: 322112
- Email: guozhen.lu@uconn.edu
- Qiaohua Yang
- Affiliation: School of Mathematics and Statistics, Wuhan University, Wuhan, 430072, People’s Republic of China
- MR Author ID: 761680
- Email: qhyang.math@whu.edu.cn
- 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.
- © Copyright 2020 American Mathematical Society
- Journal: Trans. Amer. Math. Soc. 373 (2020), 3483-3513
- MSC (2010): Primary 42B15, 42B35, 42B37, 46E35
- DOI: https://doi.org/10.1090/tran/7986
- MathSciNet review: 4082245