# American Institute of Mathematical Sciences

March  2015, 35(3): 935-942. doi: 10.3934/dcds.2015.35.935

## Existence of the maximizing pair for the discrete Hardy-Littlewood-Sobolev inequality

 1 Department of Mathematics, INS and MOE-LSC, Shanghai Jiao Tong University, Shanghai, China, China, China

Received  February 2014 Revised  April 2014 Published  October 2014

In this paper, we study the best constant of the following discrete Hardy-Littlewood-Sobolev inequality, $$\sum_{i,j,i\neq j}\frac{f_{i}g_{j}}{\mid i-j\mid^{n-\alpha}}\leq C_{r,s,\alpha} |f|_{l^r} |g|_{l^s},$$where $i,j\in \mathbb Z^n$, $r,s>1$, $0 < \alpha < n$, and $\frac {1} {r} + \frac {1} {s} + \frac {n-\alpha}{n} \geq 2$. Indeed, we prove that the best constant is attainable in the supercritical case $\frac {1}{r} + \frac {1} {s} + \frac {n-\alpha}{n} > 2$.
Citation: Genggeng Huang, Congming Li, Ximing Yin. Existence of the maximizing pair for the discrete Hardy-Littlewood-Sobolev inequality. Discrete & Continuous Dynamical Systems - A, 2015, 35 (3) : 935-942. doi: 10.3934/dcds.2015.35.935
##### References:
 [1] H. Brézis and E. Lieb, A relation between pointwise convergence of functions and convergence of functionals,, Proceedings of the American Mathematical Society, 88 (1983), 486.  doi: 10.1090/S0002-9939-1983-0699419-3.  Google Scholar [2] X. Chen and X. Zhen, Optimal summation interval and nonexistence of positive solutions to a discrete sytem,, Acta Math. Sci. Ser. B Engl. Ed., 34 (2014), 1720.  doi: 10.1016/S0252-9602(14)60117-X.  Google Scholar [3] W. Chen, C. Jin, C. Li and J. Lim, Weighted Hardy-Littlewood-Sobolev inequalities and systems of integral equations,, Discrete Contin. Dyn. Syst. 2005, (2005), 164.   Google Scholar [4] W. Chen and C. Li, An integral system and the Lane-Emden conjecture,, Discrete Contin. Dyn. Syst., 24 (2009), 1167.  doi: 10.3934/dcds.2009.24.1167.  Google Scholar [5] W. Chen and C. Li, Radial symmetry of solutions for some integral systems of Wolff type,, Discrete Contin. Dyn. Syst., 30 (2011), 1083.  doi: 10.3934/dcds.2011.30.1083.  Google Scholar [6] W. Chen and C. Li, Regularity of solutions for a system of integral equations,, Commun. Pure Appl. Anal., 4 (2005), 1.  doi: 10.3934/cpaa.2005.4.1.  Google Scholar [7] W. Chen and C. Li, The best constant in a weighted Hardy-Littlewood-Sobolev inequality,, Proc. Amer. Math. Soci., 136 (2008), 955.  doi: 10.1090/S0002-9939-07-09232-5.  Google Scholar [8] W. Chen, C. Li and B. Ou, Classification of solutions for a system of integral equations,, Communications in Partial Difference Equations, 30 (2005), 59.  doi: 10.1081/PDE-200044445.  Google Scholar [9] W. Chen, C. Li and B. Ou, Qualitative properties of solutions for an integral equation,, Discrete Contin. Dyn. Syst., 12 (2005), 347.  doi: 10.3934/dcds.2005.12.347.  Google Scholar [10] W. Chen, C. Li and B. Ou, Classification of solutions for an integral equation,, Communications on pure and applied mathematics, 59 (2006), 330.  doi: 10.1002/cpa.20116.  Google Scholar [11] Z. Cheng and C. Li, An extended discrete Hardy-Littlewood-Sobolev inequality,, Discrete Contin. Dyn. Syst., 34 (2014), 1951.  doi: 10.3934/dcds.2014.34.1951.  Google Scholar [12] G. Hardy, J. Littlewood and J. Pólya, Inequalities,, $2^{nd}$ edition, (1952).   Google Scholar [13] C. Li and J. Villavert, An extention of the Hardy-Littlewood-Pólya inequality,, Acta Math. Scientia, 31 (2011), 2285.  doi: 10.1016/S0252-9602(11)60400-1.  Google Scholar [14] E. Lieb, Sharp constants in the Hardy-Littlewood-Sobolev and related inequalities,, Annals of Math., 118 (1983), 349.  doi: 10.2307/2007032.  Google Scholar

show all references

##### References:
 [1] H. Brézis and E. Lieb, A relation between pointwise convergence of functions and convergence of functionals,, Proceedings of the American Mathematical Society, 88 (1983), 486.  doi: 10.1090/S0002-9939-1983-0699419-3.  Google Scholar [2] X. Chen and X. Zhen, Optimal summation interval and nonexistence of positive solutions to a discrete sytem,, Acta Math. Sci. Ser. B Engl. Ed., 34 (2014), 1720.  doi: 10.1016/S0252-9602(14)60117-X.  Google Scholar [3] W. Chen, C. Jin, C. Li and J. Lim, Weighted Hardy-Littlewood-Sobolev inequalities and systems of integral equations,, Discrete Contin. Dyn. Syst. 2005, (2005), 164.   Google Scholar [4] W. Chen and C. Li, An integral system and the Lane-Emden conjecture,, Discrete Contin. Dyn. Syst., 24 (2009), 1167.  doi: 10.3934/dcds.2009.24.1167.  Google Scholar [5] W. Chen and C. Li, Radial symmetry of solutions for some integral systems of Wolff type,, Discrete Contin. Dyn. Syst., 30 (2011), 1083.  doi: 10.3934/dcds.2011.30.1083.  Google Scholar [6] W. Chen and C. Li, Regularity of solutions for a system of integral equations,, Commun. Pure Appl. Anal., 4 (2005), 1.  doi: 10.3934/cpaa.2005.4.1.  Google Scholar [7] W. Chen and C. Li, The best constant in a weighted Hardy-Littlewood-Sobolev inequality,, Proc. Amer. Math. Soci., 136 (2008), 955.  doi: 10.1090/S0002-9939-07-09232-5.  Google Scholar [8] W. Chen, C. Li and B. Ou, Classification of solutions for a system of integral equations,, Communications in Partial Difference Equations, 30 (2005), 59.  doi: 10.1081/PDE-200044445.  Google Scholar [9] W. Chen, C. Li and B. Ou, Qualitative properties of solutions for an integral equation,, Discrete Contin. Dyn. Syst., 12 (2005), 347.  doi: 10.3934/dcds.2005.12.347.  Google Scholar [10] W. Chen, C. Li and B. Ou, Classification of solutions for an integral equation,, Communications on pure and applied mathematics, 59 (2006), 330.  doi: 10.1002/cpa.20116.  Google Scholar [11] Z. Cheng and C. Li, An extended discrete Hardy-Littlewood-Sobolev inequality,, Discrete Contin. Dyn. Syst., 34 (2014), 1951.  doi: 10.3934/dcds.2014.34.1951.  Google Scholar [12] G. Hardy, J. Littlewood and J. Pólya, Inequalities,, $2^{nd}$ edition, (1952).   Google Scholar [13] C. Li and J. Villavert, An extention of the Hardy-Littlewood-Pólya inequality,, Acta Math. Scientia, 31 (2011), 2285.  doi: 10.1016/S0252-9602(11)60400-1.  Google Scholar [14] E. Lieb, Sharp constants in the Hardy-Littlewood-Sobolev and related inequalities,, Annals of Math., 118 (1983), 349.  doi: 10.2307/2007032.  Google Scholar
 [1] Gongbao Li, Tao Yang. Improved Sobolev inequalities involving weighted Morrey norms and the existence of nontrivial solutions to doubly critical elliptic systems involving fractional Laplacian and Hardy terms. Discrete & Continuous Dynamical Systems - S, 2020  doi: 10.3934/dcdss.2020469 [2] Yifan Chen, Thomas Y. Hou. Function approximation via the subsampled Poincaré inequality. Discrete & Continuous Dynamical Systems - A, 2021, 41 (1) : 169-199. doi: 10.3934/dcds.2020296 [3] Peter Poláčik, Pavol Quittner. Entire and ancient solutions of a supercritical semilinear heat equation. Discrete & Continuous Dynamical Systems - A, 2021, 41 (1) : 413-438. doi: 10.3934/dcds.2020136 [4] Marco Ghimenti, Anna Maria Micheletti. Compactness results for linearly perturbed Yamabe problem on manifolds with boundary. Discrete & Continuous Dynamical Systems - S, 2020  doi: 10.3934/dcdss.2020453 [5] Denis Bonheure, Silvia Cingolani, Simone Secchi. Concentration phenomena for the Schrödinger-Poisson system in $\mathbb{R}^2$. Discrete & Continuous Dynamical Systems - S, 2020  doi: 10.3934/dcdss.2020447 [6] Noah Stevenson, Ian Tice. A truncated real interpolation method and characterizations of screened Sobolev spaces. Communications on Pure & Applied Analysis, 2020, 19 (12) : 5509-5566. doi: 10.3934/cpaa.2020250 [7] Sumit Arora, Manil T. Mohan, Jaydev Dabas. Approximate controllability of a Sobolev type impulsive functional evolution system in Banach spaces. Mathematical Control & Related Fields, 2020  doi: 10.3934/mcrf.2020049 [8] Anna Canale, Francesco Pappalardo, Ciro Tarantino. Weighted multipolar Hardy inequalities and evolution problems with Kolmogorov operators perturbed by singular potentials. Communications on Pure & Applied Analysis, , () : -. doi: 10.3934/cpaa.2020274 [9] Justin Holmer, Chang Liu. Blow-up for the 1D nonlinear Schrödinger equation with point nonlinearity II: Supercritical blow-up profiles. Communications on Pure & Applied Analysis, , () : -. doi: 10.3934/cpaa.2020264 [10] Lars Grüne, Matthias A. Müller, Christopher M. Kellett, Steven R. Weller. Strict dissipativity for discrete time discounted optimal control problems. Mathematical Control & Related Fields, 2020  doi: 10.3934/mcrf.2020046 [11] Cuicui Li, Lin Zhou, Zhidong Teng, Buyu Wen. The threshold dynamics of a discrete-time echinococcosis transmission model. Discrete & Continuous Dynamical Systems - B, 2020  doi: 10.3934/dcdsb.2020339 [12] Christian Beck, Lukas Gonon, Martin Hutzenthaler, Arnulf Jentzen. On existence and uniqueness properties for solutions of stochastic fixed point equations. Discrete & Continuous Dynamical Systems - B, 2020  doi: 10.3934/dcdsb.2020320 [13] Leilei Wei, Yinnian He. A fully discrete local discontinuous Galerkin method with the generalized numerical flux to solve the tempered fractional reaction-diffusion equation. Discrete & Continuous Dynamical Systems - B, 2020  doi: 10.3934/dcdsb.2020319 [14] Yuri Fedorov, Božidar Jovanović. Continuous and discrete Neumann systems on Stiefel varieties as matrix generalizations of the Jacobi–Mumford systems. Discrete & Continuous Dynamical Systems - A, 2020  doi: 10.3934/dcds.2020375 [15] Haixiang Yao, Ping Chen, Miao Zhang, Xun Li. Dynamic discrete-time portfolio selection for defined contribution pension funds with inflation risk. Journal of Industrial & Management Optimization, 2020  doi: 10.3934/jimo.2020166 [16] Christopher S. Goodrich, Benjamin Lyons, Mihaela T. Velcsov. Analytical and numerical monotonicity results for discrete fractional sequential differences with negative lower bound. Communications on Pure & Applied Analysis, , () : -. doi: 10.3934/cpaa.2020269 [17] Yichen Zhang, Meiqiang Feng. A coupled $p$-Laplacian elliptic system: Existence, uniqueness and asymptotic behavior. Electronic Research Archive, 2020, 28 (4) : 1419-1438. doi: 10.3934/era.2020075 [18] Ahmad Z. Fino, Wenhui Chen. A global existence result for two-dimensional semilinear strongly damped wave equation with mixed nonlinearity in an exterior domain. Communications on Pure & Applied Analysis, 2020, 19 (12) : 5387-5411. doi: 10.3934/cpaa.2020243 [19] Zedong Yang, Guotao Wang, Ravi P. Agarwal, Haiyong Xu. Existence and nonexistence of entire positive radial solutions for a class of Schrödinger elliptic systems involving a nonlinear operator. Discrete & Continuous Dynamical Systems - S, 2020  doi: 10.3934/dcdss.2020436 [20] Zhilei Liang, Jiangyu Shuai. Existence of strong solution for the Cauchy problem of fully compressible Navier-Stokes equations in two dimensions. Discrete & Continuous Dynamical Systems - B, 2020  doi: 10.3934/dcdsb.2020348

2019 Impact Factor: 1.338