An extension operator on bounded domains and applications

  • Mathew Gluck
  • Meijun ZhuEmail author


In this paper we study a sharp Hardy–Littlewood–Sobolev type inequality with Riesz potential on bounded smooth domains. We obtain the inequality for a general bounded domain \(\Omega \) and show that if the extension constant for \(\Omega \) is strictly larger than the extension constant for the unit ball \(B_1\) then extremal functions exist. Using suitable test functions we show that this criterion is satisfied by an annular domain whose hole is sufficiently small. The construction of the test functions is not based on any positive mass type theorems, neither on the nonflatness of the boundary. By using a similar choice of test functions with the Poisson-kernel-based extension operator we prove the existence of an abstract domain having zero scalar curvature and strictly larger isoperimetric constant than that of the Euclidean ball.

Mathematics Subject Classification

35Jxx 45Gxx 53-xx 



  1. 1.
    Aubin, T.: Equations différentielles nonlinéaires et Problème de Yamabe concernant la courbure scalaire. J. Math. Pures Appl. 55, 269–296 (1976)MathSciNetzbMATHGoogle Scholar
  2. 2.
    Beckner, W.: Sharp Sobolev inequalities on the sphere and the Moser–Trudinger inequality. Ann. Math. 138, 213–242 (1993)MathSciNetCrossRefGoogle Scholar
  3. 3.
    Dou, J., Zhu, M.: Sharp Hardy–Littlewood–Sobolev inequality on the upper half space. Int. Math. Res. Not. 3, 651687 (2015). CrossRefzbMATHGoogle Scholar
  4. 4.
    Dou, J., Zhu, M.: Reversed Hardy–Littewood–Sobolev inequality. arXiv:1309.1974v3. International Mathematics Research Notices 19, 96969726 (2015).
  5. 5.
    Evans, L.C.: Partial Differential Equations, Graduate Studies in Mathematics, vol. 19. American Mathematical Society, Providence (1998)Google Scholar
  6. 6.
    Frank, R., Lieb, E.: Sharp constants in several inequalities on the Heisenberg group. Ann. Math. 176, 349381 (2012). MathSciNetCrossRefGoogle Scholar
  7. 7.
    Gross, L.: Logarighmic Sobolev inequalities. Am. J. Math. 97, 1061–1083 (1976)CrossRefGoogle Scholar
  8. 8.
    Hang, F., Wang, X., Yan, X.: Sharp integral inequalities for harmonic functions. Commun. Pure Appl. Math. 61(1), 54–95 (2008)MathSciNetCrossRefGoogle Scholar
  9. 9.
    Hang, F., Wang, X., Yan, X.: An integral equation in conformal geometry. Ann. Inst. H. Poincaré Anal. Non Linéaire 26, 1–21 (2009)MathSciNetCrossRefGoogle Scholar
  10. 10.
    Hardy, G.H., Littlewood, J.E.: Some properties of fractional integrals (1). Math. Zeitschr. 27, 565–606 (1928)MathSciNetCrossRefGoogle Scholar
  11. 11.
    Hardy, G.H., Littlewood, J.E.: On certain inequalities connected with the calculus of variations. J. Lond. Math. Soc. 5, 34–39 (1930)CrossRefGoogle Scholar
  12. 12.
    Jin, T., Xiong, J.: On the isoperimetric constant over scalar-flat conformal classes (preprint)Google Scholar
  13. 13.
    Lee, J.M., Parker, T.H.: The Yamabe problem. Bull. Am. Math. Soc. (N.S.) 17(1), 37–91 (1987)MathSciNetCrossRefGoogle Scholar
  14. 14.
    Lee, J.M., Parker, T.H.: the method of moving spheres. J. Eur. Math. Soc. 6, 153–180 (2004)Google Scholar
  15. 15.
    Lieb, E.: Sharp constants in the Hardy–Littlewood–Sobolev and related inequalities. Ann. Math. 118, 349–374 (1983)MathSciNetCrossRefGoogle Scholar
  16. 16.
    Morgan, F., Johnson, D.L.: Some sharp isoperimetric theorems for Riemannian manifolds. Indiana Univ. Math. J. 49(3), 1017–1041 (2000). MR MR1803220 (2002e:53043)MathSciNetCrossRefGoogle Scholar
  17. 17.
    Ngo, Q.A., Nguyen, V.H.: Sharp reversed Hardy–Littlewood–Sobolev inequality on \(\mathbb{R}^n\). Isr. J. Math. (2017). CrossRefzbMATHGoogle Scholar
  18. 18.
    Schoen, R.: Conformal deformation of a Riemannian metric to constant scalar curvature. J. Differ. Geom. 20, 479–495 (1984)MathSciNetCrossRefGoogle Scholar
  19. 19.
    Sobolev, S.L.: On a theorem of functional analysis. Math. Sb. (N.S.) 4, 471–479 (1938). (A. M. S. transl. Ser. 2, 34 (1963), 39-68)Google Scholar
  20. 20.
    Trudinger, N.: Remarks concerning the conformal deformation of Riemannian structures on compact manifolds. Ann. Scuola Norm. Sup. Pisa 22, 265–274 (1968)MathSciNetzbMATHGoogle Scholar

Copyright information

© Springer-Verlag GmbH Germany, part of Springer Nature 2019

Authors and Affiliations

  1. 1.Department of MathematicsThe University of OklahomaNormanUSA

Personalised recommendations