## Reverse Stein–Weiss inequalities and existence of their extremal functions

HTML articles powered by AMS MathViewer

- by Lu Chen, Zhao Liu, Guozhen Lu and Chunxia Tao PDF
- Trans. Amer. Math. Soc.
**370**(2018), 8429-8450 Request permission

## Abstract:

In this paper, we establish the following reverse Stein–Weiss inequality, namely the reversed weighted Hardy–Littlewood–Sobolev inequality, in $\mathbb {R}^n$: \begin{equation*} \int _{\mathbb {R}^n}\int _{\mathbb {R}^n}|x|^\alpha |x-y|^\lambda f(x)g(y)|y|^\beta dxdy\geq C_{n,\alpha ,\beta ,p,q’}\|f\|_{L^{q’}}\|g\|_{L^p} \end{equation*} for any nonnegative functions $f\in L^{q’}(\mathbb {R}^n)$, $g\in L^p(\mathbb {R}^n)$, and $p,\ q’\in (0,1)$, $\alpha$, $\beta$, $\lambda >0$ such that $\frac {1}{p}+\frac {1}{q’}-\frac {\alpha +\beta +\lambda }{n}=2$. We derive the existence of extremal functions for the above inequality. Moreover, some asymptotic behaviors are obtained for the corresponding Euler–Lagrange system. For an analogous weighted system, we prove necessary conditions of existence for any positive solutions by using the Pohozaev identity. Finally, we also obtain the corresponding Stein–Weiss and reverse Stein–Weiss inequalities on the $n$-dimensional sphere $\mathbb {S}^n$ by using the stereographic projections. Our proof of the reverse Stein–Weiss inequalities relies on techniques in harmonic analysis and differs from those used in the proof of the reverse (non-weighted) Hardy–Littlewood–Sobolev inequalities.## References

- William Beckner,
*Functionals for multilinear fractional embedding*, Acta Math. Sin. (Engl. Ser.)**31**(2015), no. 1, 1–28. MR**3285943**, DOI 10.1007/s10114-015-4321-6 - 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,
*Weighted inequalities and Stein-Weiss potentials*, Forum Math.**20**(2008), no. 4, 587–606. MR**2431497**, DOI 10.1515/FORUM.2008.030 - William Beckner,
*Pitt’s inequality with sharp convolution estimates*, Proc. Amer. Math. Soc.**136**(2008), no. 5, 1871–1885. MR**2373619**, DOI 10.1090/S0002-9939-07-09216-7 - 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 - Herm Jan Brascamp and Elliott H. Lieb,
*Best constants in Young’s inequality, its converse, and its generalization to more than three functions*, Advances in Math.**20**(1976), no. 2, 151–173. MR**412366**, DOI 10.1016/0001-8708(76)90184-5 - Eric A. Carlen, José A. Carrillo, and Michael Loss,
*Hardy-Littlewood-Sobolev inequalities via fast diffusion flows*, Proc. Natl. Acad. Sci. USA**107**(2010), no. 46, 19696–19701. MR**2745814**, DOI 10.1073/pnas.1008323107 - E. Carlen and M. Loss,
*Competing symmetries, the logarithmic HLS inequality and Onofri’s inequality on $S^n$*, Geom. Funct. Anal.**2**(1992), no. 1, 90–104. MR**1143664**, DOI 10.1007/BF01895706 - Emanuel Carneiro,
*A sharp inequality for the Strichartz norm*, Int. Math. Res. Not. IMRN**16**(2009), 3127–3145. MR**2533799**, DOI 10.1093/imrn/rnp045 - L. Chen, G. Lu, and C. Tao,
*Hardy-Littlewood-Sobolev inequality with fractional Possion kernel and its appliaction in PDEs*, preprint. - L. Chen, G. Lu and C. Tao,
*Reverse Stein-Weiss inequalities on the upper half space and the existence of their extremals*, Preprint. - L. Chen, Z. Liu, G. Lu and C. Tao,
*Stein-Weiss inequalities with the fractional Poisson kernel*, Preprint. - Lu Chen, Zhao Liu, and Guozhen Lu,
*Symmetry and regularity of solutions to the weighted Hardy-Sobolev type system*, Adv. Nonlinear Stud.**16**(2016), no. 1, 1–13. MR**3456742**, DOI 10.1515/ans-2015-5005 - Wenxiong Chen and Congming Li,
*The best constant in a weighted Hardy-Littlewood-Sobolev inequality*, Proc. Amer. Math. Soc.**136**(2008), no. 3, 955–962. MR**2361869**, DOI 10.1090/S0002-9939-07-09232-5 - Wenxiong Chen, Chao Jin, Congming Li, and Jisun Lim,
*Weighted Hardy-Littlewood-Sobolev inequalities and systems of integral equations*, Discrete Contin. Dyn. Syst.**suppl.**(2005), 164–172. MR**2192671** - Wei Dai, Zhao Liu, and Guozhen Lu,
*Hardy-Sobolev type integral systems with Dirichlet boundary conditions in a half space*, Commun. Pure Appl. Anal.**16**(2017), no. 4, 1253–1264. MR**3637912**, DOI 10.3934/cpaa.2017061 - Wei Dai, Zhao Liu, and Guozhen Lu,
*Liouville type theorems for PDE and IE systems involving fractional Laplacian on a half space*, Potential Anal.**46**(2017), no. 3, 569–588. MR**3630408**, DOI 10.1007/s11118-016-9594-6 - Pablo L. De Nápoli, Irene Drelichman, and Ricardo G. Durán,
*On weighted inequalities for fractional integrals of radial functions*, Illinois J. Math.**55**(2011), no. 2, 575–587 (2012). MR**3020697** - Jingbo Dou and Meijun Zhu,
*Sharp Hardy-Littlewood-Sobolev inequality on the upper half space*, Int. Math. Res. Not. IMRN**3**(2015), 651–687. MR**3340332**, DOI 10.1093/imrn/rnt213 - Jingbo Dou and Meijun Zhu,
*Reversed Hardy-Littewood-Sobolev inequality*, Int. Math. Res. Not. IMRN**19**(2015), 9696–9726. MR**3431607**, DOI 10.1093/imrn/rnu241 - Rupert L. Frank and Elliott H. Lieb,
*Sharp constants in several inequalities on the Heisenberg group*, Ann. of Math. (2)**176**(2012), no. 1, 349–381. MR**2925386**, DOI 10.4007/annals.2012.176.1.6 - Rupert L. Frank and Elliott H. Lieb,
*Inversion positivity and the sharp Hardy-Littlewood-Sobolev inequality*, Calc. Var. Partial Differential Equations**39**(2010), no. 1-2, 85–99. MR**2659680**, DOI 10.1007/s00526-009-0302-x - Rupert L. Frank and Elliott H. Lieb,
*A new, rearrangement-free proof of the sharp Hardy-Littlewood-Sobolev inequality*, Spectral theory, function spaces and inequalities, Oper. Theory Adv. Appl., vol. 219, Birkhäuser/Springer Basel AG, Basel, 2012, pp. 55–67. MR**2848628**, DOI 10.1007/978-3-0348-0263-5_{4} - Loukas Grafakos,
*Classical and modern Fourier analysis*, Pearson Education, Inc., Upper Saddle River, NJ, 2004. MR**2449250** - Xiaolong Han,
*Existence of maximizers for Hardy-Littlewood-Sobolev inequalities on the Heisenberg group*, Indiana Univ. Math. J.**62**(2013), no. 3, 737–751. MR**3164842**, DOI 10.1512/iumj.2013.62.4976 - Xiaolong Han, Guozhen Lu, and Jiuyi Zhu,
*Hardy-Littlewood-Sobolev and Stein-Weiss inequalities and integral systems on the Heisenberg group*, Nonlinear Anal.**75**(2012), no. 11, 4296–4314. MR**2921990**, DOI 10.1016/j.na.2012.03.017 - Yazhou Han and Meijun Zhu,
*Hardy-Littlewood-Sobolev inequalities on compact Riemannian manifolds and applications*, J. Differential Equations**260**(2016), no. 1, 1–25. MR**3411662**, DOI 10.1016/j.jde.2015.06.032 - Fengbo Hang, Xiaodong Wang, and Xiaodong Yan,
*Sharp integral inequalities for harmonic functions*, Comm. Pure Appl. Math.**61**(2008), no. 1, 54–95. MR**2361304**, DOI 10.1002/cpa.20193 - G. H. Hardy and J. E. Littlewood,
*Some properties of fractional integrals. I*, Math. Z.**27**(1928), no. 1, 565–606. MR**1544927**, DOI 10.1007/BF01171116 - Elliott H. Lieb,
*Sharp constants in the Hardy-Littlewood-Sobolev and related inequalities*, Ann. of Math. (2)**118**(1983), no. 2, 349–374. MR**717827**, DOI 10.2307/2007032 - Elliott H. Lieb and Michael Loss,
*Analysis*, 2nd ed., Graduate Studies in Mathematics, vol. 14, American Mathematical Society, Providence, RI, 2001. MR**1817225**, DOI 10.1090/gsm/014 - Guozhen Lu and Jiuyi Zhu,
*Symmetry and regularity of extremals of an integral equation related to the Hardy-Sobolev inequality*, Calc. Var. Partial Differential Equations**42**(2011), no. 3-4, 563–577. MR**2846267**, DOI 10.1007/s00526-011-0398-7 - Quốc Anh Ngô and Van Hoang Nguyen,
*Sharp reversed Hardy-Littlewood-Sobolev inequality on $\textbf {R}^n$*, Israel J. Math.**220**(2017), no. 1, 189–223. MR**3666824**, DOI 10.1007/s11856-017-1515-x - S. L. Sobolev,
*On a theorem in functional analysis (in Russian)*, Mat. Sb,**4**(1938), 471-497. - E. M. Stein and Guido Weiss,
*Fractional integrals on $n$-dimensional Euclidean space*, J. Math. Mech.**7**(1958), 503–514. MR**0098285**, DOI 10.1512/iumj.1958.7.57030 - Elias M. Stein,
*Singular integrals and differentiability properties of functions*, Princeton Mathematical Series, No. 30, Princeton University Press, Princeton, N.J., 1970. MR**0290095**

## Additional Information

**Lu Chen**- Affiliation: School of Mathematical Sciences, Beijing Normal University, Beijing 100875, People’s Republic of China
- Address at time of publication: School of Mathematics and Statistics, Beijing Institute of Technology, Beijing 100081, People’s Republic of China
- MR Author ID: 1147340
- Email: luchen2015@mail.bnu.edu.cn
**Zhao Liu**- Affiliation: School of Mathematics and Computer Science, Jiangxi Science and Technology Normal University, Nanchang 330038, People’s Republic of China
- Email: liuzhao2008tj@sina.com
**Guozhen Lu**- Affiliation: Department of Mathematics, University of Connecticut, Storrs, Connecticut 06269
- MR Author ID: 322112
- Email: guozhen.lu@uconn.edu
**Chunxia Tao**- Affiliation: School of Mathematical Sciences, Beijing Normal University, Beijing 100875, People’s Republic of China
- Email: taochunxia@mail.bnu.edu.cn
- Received by editor(s): November 11, 2016
- Received by editor(s) in revised form: March 25, 2017
- Published electronically: August 21, 2018
- Additional Notes: The first two authors and the fourth author were partly supported by a grant from the NNSF of China (No.11371056).

The third author was partly supported by a US NSF grant and a Simons Fellowship from the Simons Foundation.

The third and fourth authors are the corresponding authors. - © Copyright 2018 American Mathematical Society
- Journal: Trans. Amer. Math. Soc.
**370**(2018), 8429-8450 - MSC (2010): Primary 42B99, 35B40
- DOI: https://doi.org/10.1090/tran/7273
- MathSciNet review: 3864382