Acta Mathematica Sinica, English Series

, Volume 34, Issue 4, pp 681–690 | Cite as

On a Conformally Invariant Integral Equation Involving Poisson Kernel

  • Jin Gang Xiong


We study a prescribing functions problem of a conformally invariant integral equation involving Poisson kernel on the unit ball. This integral equation is not the dual of any standard type of PDE. As in Nirenberg problem, there exists a Kazdan–Warner type obstruction to existence of solutions. We prove existence in the antipodal symmetry functions class.


Poisson kernel conformal invariance blow up analysis 

MR(2010) Subject Classification

45G05 35B33 


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. [1]
    Blumenthal, R. M., Getoor, R. K.: Some theorems on stable processes. Trans. Amer. Math. Soc., 95, 263–273 (1960)MathSciNetCrossRefzbMATHGoogle Scholar
  2. [2]
    Carleman, T.: Zur Theorie der Minimalflächen. Math. Z., 9, 154–160 (1921)MathSciNetCrossRefzbMATHGoogle Scholar
  3. [3]
    Chang, S. Y. Alice, Yang, P.: Prescribing Gaussian curvature on S2. Acta Math., 159, 215–259 (1987)MathSciNetCrossRefzbMATHGoogle Scholar
  4. [4]
    Chang, S. Y. Alice, Yang, P.: Conformal deformation of metrics on S2. J. Differential Geom., 27, 259–296 (1988)MathSciNetCrossRefGoogle Scholar
  5. [5]
    Christ, M., Shao, S.: On the extremizers of an adjoint Fourier restriction inequality. Adv. Math., 230(3), 957–977 (2012)MathSciNetCrossRefzbMATHGoogle Scholar
  6. [6]
    Christ, M., Shao, S.: Existence of extremals for a Fourier restriction inequality. Anal. PDE, 5(2), 261–312 (2012)MathSciNetCrossRefzbMATHGoogle Scholar
  7. [7]
    Dou, J., Guo, Q., Zhu, M.: Subcritical approach to sharp Hardy–Littlewood–Sobolev type inequalities on the upper half space. Adv. Math., 312, 1–45 (2017)MathSciNetCrossRefzbMATHGoogle Scholar
  8. [8]
    Escobar, J. F., Schoen, R.: Conformal metrics with prescribed scalar curvature. Invent. Math., 86, 243–254 (1986)MathSciNetCrossRefzbMATHGoogle Scholar
  9. [9]
    Foschi, D.: Maximizers for the Strichartz inequality. J. Eur. Math. Soc. (JEMS), 9, 739–774 (2007)MathSciNetCrossRefzbMATHGoogle Scholar
  10. [10]
    Frank, R., Lieb, E., Sabin, J.: Maximizers for the Stein–Tomas inequality. Geom. Funct. Anal., 26, 1095–1134 (2016)MathSciNetCrossRefzbMATHGoogle Scholar
  11. [11]
    Hang, F., Wang, X., Yan, X.: Sharp integral inequalities for harmonic functions. Comm. Pure Appl. Math., 61, 54–95 (2008)MathSciNetCrossRefzbMATHGoogle Scholar
  12. [12]
    Hang, F., Wang, X., Yan, X.: An integral equation in conformal geometry. Ann. Inst. H. Poincaré Anal. Non Linéaire, 26, 1–21 (2009)MathSciNetCrossRefzbMATHGoogle Scholar
  13. [13]
    Jin, T., Li, Y. Y., Xiong, J.: On a fractional Nirenberg problem, part II: existence of solutions. Int. Math. Res. Not., 2015(6), 1555–1589Google Scholar
  14. [14]
    Jin, T., Li, Y. Y., Xiong, J.: The Nirenberg problem and its generalizations: A unified approach. Math. Ann., 369(1-2), 109–151 (2017)MathSciNetCrossRefzbMATHGoogle Scholar
  15. [15]
    Jin, T., Xiong, J.: On the isoperimetric quotient over scalar-flat conformal classes. Preprint., arXiv:1709.03644Google Scholar
  16. [16]
    Li, Y. Y., Xiong, J.: Compactness of conformal metrics with constant Q-curvature. I. Preprint., arXiv:1506.00739Google Scholar
  17. [17]
    Stein, E. M.: Singular Integrals and Differentiability Properties of Functions, Princeton Mathematical Series, No. 30 Princeton University Press, Princeton, N. J., 1970Google Scholar
  18. [18]
    Sun, L., Xiong, J.: Classification theorems for solutions of higher order boundary conformally invariant problems, I. J. Funct. Anal., 271, 3727–3764 (2016)MathSciNetCrossRefzbMATHGoogle Scholar
  19. [19]
    Xiong, J.: The critical semilinear elliptic equation with boundary isolated singularities. J. Differential Equations, 263(3), 1907–1930 (2017)MathSciNetCrossRefzbMATHGoogle Scholar

Copyright information

© Institute of Mathematics, Academy of Mathematics and Systems Science, Chinese Academy of Sciences, Chinese Mathematical Society and Springer-Verlag GmbH Germany, part of Springer Nature 2018

Authors and Affiliations

  1. 1.School of Mathematical sciencesBeijing Normal UniversityBeijingP. R. China

Personalised recommendations