Israel Journal of Mathematics

, Volume 220, Issue 1, pp 189–223 | Cite as

Sharp reversed Hardy–Littlewood–Sobolev inequality on R n

  • Quốc Anh NgôEmail author
  • Van Hoang Nguyen


This is the first in our series of papers that concerns Hardy–Littlewood–Sobolev (HLS) type inequalities. In this paper, the main objective is to establish the following sharp reversed HLS inequality in the whole space R n, \(\int {_{{R^n}}} \int {_{{R^n}}f\left( x \right)} {\left| {x - y} \right|^\lambda }g\left( y \right)dxdy \geqslant {\ell _{n,p,r}}{\left\| f \right\|_{{L^p}\left( {{R^n}} \right)}}{\left\| g \right\|_{{L^r}\left( {{R^n}} \right)}}\), for any non-negative functions fL p(R n), gL r(R n), and p, r ∈ (0, 1), λ > 0 such that 1/p+1/r −λ/n = 2. We will also explore some estimates for ℓn,p,r and the existence of optimal functions for the above inequality, which will shed light on some existing results in literature.


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  1. [Ale58]
    A. D. Aleksandrov, Uniqueness theorems for surfaces in the large. V, Vestnik Leningrad University. Mathematics 13 (1958), 5–8.MathSciNetGoogle Scholar
  2. [Aub76]
    T. Aubin, Problèmes isopérimétriques et espaces de Sobolev, Journal of Differential Geometry 11 (1976), 573–598.CrossRefzbMATHMathSciNetGoogle Scholar
  3. [Bec93]
    W. Beckner, Sharp Sobolev inequalities on the sphere and the Moser–Trudinger inequality, Annals of Mathematics 138 (1993), 213–242.CrossRefzbMATHMathSciNetGoogle Scholar
  4. [BL76]
    H. J. Brascamp and E. H. Lieb, Best constants in Young’s inequality, its converse, and its generalization to more than three functions, Advances in Mathematics 20 (1976), 151–173.CrossRefzbMATHMathSciNetGoogle Scholar
  5. [Bur09]
    A. Burchard, A short course on rearrangement inequalities, June 2009. available at: Scholar
  6. [CGS89]
    L. Caffarelli, B. Gidas and J. Spruck, Asymptotic symmetry and local behavior of semilinear elliptic equations with critical Sobolev growth, Communications on Pure and Applied Mathematics 42 (1989), 271–297.CrossRefzbMATHMathSciNetGoogle Scholar
  7. [CCL10]
    E. Carlen, J. A. Carrillo and M. Loss, Hardy–Littlewood–Sobolev inequalities via fast diffusion flows, Proceedings of the National Academy of Sciences of the United States of America 107 (2010), 19696–19701.CrossRefzbMATHMathSciNetGoogle Scholar
  8. [CL92]
    E. Carlen and M. Loss, Competing symmetries, the logarithmic HLS inequality and Onofri’s inequality on S n, Geometric and Functional Analysis 2 (1992), 90–104.CrossRefzbMATHMathSciNetGoogle Scholar
  9. [CL91]
    W. Chen and C. Li, Classification of solutions of some nonlinear elliptic equations, Duke Mathematical Journal 63 (1991), 615–622.CrossRefzbMATHMathSciNetGoogle Scholar
  10. [CLO05]
    W. Chen, C. Li and B. Ou, Classification of solutions for a system of integral equations, Communications in Partial Differential Equations 30 (2005), 59–65.CrossRefzbMATHMathSciNetGoogle Scholar
  11. [CLO06]
    W. Chen, C. Li and B. Ou, Classification of solutions for an integral equation, Communications on Pure and Applied Mathematics 59 (2006), 330–343.CrossRefzbMATHMathSciNetGoogle Scholar
  12. [DZ13]
    J. Dou and M. Zhu, Sharp Hardy–Littlewood–Sobolev inequality on the upper half space, International Mathematics Research Notices 3 (2015), 651–687.zbMATHMathSciNetCrossRefGoogle Scholar
  13. [DZ14]
    J. Dou and M. Zhu, Reversed Hardy–Littlewood–Sobolev inequality, International Mathematics Research Notices 19 (2015), 9696–9726.CrossRefzbMATHGoogle Scholar
  14. [FL10]
    R. L. Frank and E. H. Lieb, Inversion positivity and the sharp Hardy–Littlewood–Sobolev inequality, Calculus of Variations and Partial Differential Equations 39 (2010), 85–99.CrossRefzbMATHMathSciNetGoogle Scholar
  15. [FL12a]
    R. L. Frank and E. H. Lieb, Sharp constant in several inequalities on Heisenberg group, Annals of Mathematics 176 (2012), 349–381.CrossRefzbMATHMathSciNetGoogle Scholar
  16. [FL12]
    R. L. Frank and E. H. Lieb, A new, rearrangement-free proof of the sharp Hardy–Littlewood–Sobolev inequality, in Spectral Theory, Function Spaces and Inequalities, Operator Theory: Advances and Applications, Vol. 219, Birkhäuser/Springer, Basel, 2012, pp. 55–67.CrossRefzbMATHMathSciNetGoogle Scholar
  17. [GNN79]
    B. Gidas, W. M. Ni and L. Nirenberg, Symmetry and related properties via the maximum principle, Communications in Mathematical Physics 68 (1979), 209–243.CrossRefzbMATHMathSciNetGoogle Scholar
  18. [Gro75]
    L. Gross, Logarithmic Sobolev inequality, American Journal of Mathematics 97 (1976), 1061–1083.CrossRefzbMATHGoogle Scholar
  19. [HZ15]
    Y. Han and M. Zhu, Hardy–Littlewood–Sobolev inequalities on compact Riemannian manifolds and applications, Journal of Differential Equations 260 (2016), 1–25.CrossRefzbMATHMathSciNetGoogle Scholar
  20. [HL28]
    G. H. Hardy and J. E. Littlewood, Some properties of fractional integrals. I, Mathematische Zeitschrift 27 (1928), 565–606.CrossRefzbMATHMathSciNetGoogle Scholar
  21. [HL30]
    G. H. Hardy and J. E. Littlewood, Notes on the theory of series (XII): On certain inequalities connected with the calculus of variations, Journal fo the London Mathematical Society 5 (1930), 34–39.CrossRefzbMATHMathSciNetGoogle Scholar
  22. [HY13]
    Y. Hua and X. Yu, Necessary conditions for existence results of some integral system, Abstract and Applied Analysis (2013), Art. ID 504282, 5 pp.Google Scholar
  23. [Lei15]
    Y. Lei, On the integral systems with negative exponents, Discrete and Continuous Dynamical Systetms 35 (2015), 1039–1057.CrossRefzbMATHMathSciNetGoogle Scholar
  24. [Li04]
    Y. Li, Remark on some conformally invariant integral equations: the method of moving spheres, Journal of the European Mathematical Society 6 (2004), 153–180.CrossRefzbMATHMathSciNetGoogle Scholar
  25. [LZ95]
    Y. Li and M. Zhu, Uniqueness theorems through the method of moving spheres, Duke Mathematical Journal 80 (1995), 383–417.CrossRefzbMATHMathSciNetGoogle Scholar
  26. [Lieb83]
    E. Lieb, Sharp constants in the Hardy–Littlewood–Sobolev and related inequalities, Annals of Mathematics 118 (1983), 349–374.CrossRefzbMATHMathSciNetGoogle Scholar
  27. [LL01]
    E. H. Lieb and M. Loss, Analysis, Graduate Studies in Mathematics, Vol. 14, American Mathematical Society, Providence, RI, 2001.CrossRefzbMATHGoogle Scholar
  28. [Ngo15]
    Q. A. Ngô, Classification of solutions for a system of integral equations with negative exponents via the method of moving spheres, 2015, 15 p., personal notes, available at:\_1515.pdf.Google Scholar
  29. [NN15]
    Q. A. Ngô and V. H. Nguyen, Sharp reversed Hardy–Littlewood–Sobolev inequality on the half space R + n, International Mathematics Research Notices doi: 10.1093/imrn/rnw108.Google Scholar
  30. [Ros71]
    G. Rosen, Minimum value for c in the Sobolev inequality ||ϕ 3|| ≤ c’||▽ϕ3||, SIAM Journal on Applied Mathematics 21 (1971), 30–32.CrossRefMathSciNetGoogle Scholar
  31. [Ser71]
    J. Serrin, A symmetry problem in potential theory, Archive for Rational Mechanics and Analysis 43 (1971), 304–318.CrossRefzbMATHMathSciNetGoogle Scholar
  32. [Sob38]
    S. L. Sobolev, On a theorem of functional analysis, Matematicheskiĭ Sbornik 4 (1938), 471–479; English translaton in American Mathematical Society Translation Series 2 34 (1963), 39–68.Google Scholar
  33. [SW58]
    E. M. Stein and G. Weiss, Fractional integrals in n-dimensional Euclidean space, Journal of Mathematics and Mechanics 7 (1958), 503–514.zbMATHGoogle Scholar
  34. [Tal76]
    G. Talenti, Best constant in Sobolev inequality, Annali di Matematica Pura ed Applicata 110 (1976), 353–372.CrossRefzbMATHMathSciNetGoogle Scholar
  35. [Xu05]
    X. Xu, Exact solutions of nonlinear conformally invariant integral equations in R3, Advances in Mathematics 194 (2005), 485–503.CrossRefzbMATHMathSciNetGoogle Scholar
  36. [Xu07]
    X. Xu, Uniqueness theorem for integral equations and its application, Journal of Functional Analysis 247 (2007), 95–109.CrossRefzbMATHMathSciNetGoogle Scholar
  37. [Zhu14]
    M. Zhu, Prescribing integral curvature equation, Differential and Integral Equations 29 (2016), 889–904.zbMATHMathSciNetGoogle Scholar

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© Hebrew University of Jerusalem 2017

Authors and Affiliations

  1. 1.Department of MathematicsCollege of Science Viêt Nam National UniversityHà NôiViet Nam
  2. 2.School of Mathematical SciencesTel Aviv UniversityTel AvivIsrael

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