Potential Analysis

, Volume 38, Issue 4, pp 1103–1122 | Cite as

Blaschke, Privaloff, Reade and Saks Theorems for Diffusion Equations on Lie Groups



We prove some asymptotic characterizations for the subsolutions to a class of diffusion equations on homogeneous Lie groups. These results are the diffusion counterpart of the classical Blaschke, Privaloff, Reade and Saks Theorems for harmonic functions.


Diffusion equations Ultraparabolic equations Subsolutions Mean-value operators Homogeneous Lie groups 

Mathematics Subject Classifications (2010)

Primary 35K70 31B05; Secondary 35R03 35H10 


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. 1.
    Beckenbach, E.F., Radó, T.: Subharmonic functions and surfaces of negative curvature. Trans. Am. Math. Soc. 35, 662–674 (1933)CrossRefGoogle Scholar
  2. 2.
    Blaschke, W.: Ein Mittelwersatz und eine kennzeichnende Eigenschaft des logarithmischen Potentials. Leipz. Ber. 68, 3–8 (1916)Google Scholar
  3. 3.
    Bonfiglioli, A., Lanconelli, E.: Subharmonic functions in sub-Riemannian settings. J. Eur. Math. Soc. (2012, to appear)Google Scholar
  4. 4.
    Bonfiglioli, A., Lanconelli, E., Uguzzoni, F.: Stratified Lie Groups and Potential Theory for Their Sub-Laplacians. Springer, Berlin (2007)MATHGoogle Scholar
  5. 5.
    Cinti, C.: Sub-solutions and mean value operators for ultraparabolic equations on Lie groups. Math. Scand. 101, 83–103 (2007)MathSciNetMATHGoogle Scholar
  6. 6.
    Cinti, C., Lanconelli, E.: Riesz and Poisson–Jensen representation formulas for a class of ultraparabolic operators on Lie groups. Potential Anal. 30, 179–200 (2009)MathSciNetMATHCrossRefGoogle Scholar
  7. 7.
    Freitas, P., Matos, J.P.: On the characterization of harmonic and subharmonic functions via mean-value properties. Potential Anal. 32, 189–200 (2010)MathSciNetMATHCrossRefGoogle Scholar
  8. 8.
    Garofalo, N., Lanconelli, E.: Asymptotic behavior of fundamental solutions and potential theory of parabolic operators with variable coefficients. Math. Ann. 283, 211–239 (1989)MathSciNetMATHCrossRefGoogle Scholar
  9. 9.
    Hörmander, L.: Notions of Convexity. Birkhäuser, Boston (2007)MATHGoogle Scholar
  10. 10.
    Kogoj, A.E., Lanconelli, E.: An invariant Harnack inequality for a class of hypoelliptic ultraparabolic equations. Mediterr. J. Math. 1, 51–80 (2004)MathSciNetMATHCrossRefGoogle Scholar
  11. 11.
    Kogoj, A.E., Lanconelli, E.: Link of groups and homogeneous Hörmander operators. Proc. Am. Math. Soc. 135, 2019–2030 (2007)MathSciNetMATHCrossRefGoogle Scholar
  12. 12.
    Kozakiewicz, W.: Un théorème sur les opérateurs et son application à la théorie des Laplaciens généralisés. C. R. Soc. sc. Varsovie 26, 18–24 (1933)Google Scholar
  13. 13.
    Lanconelli, E., Pascucci, A.: Superparabolic functions related to second order hypoellitic operators. Potential Anal. 11, 303–323 (1999)MathSciNetMATHCrossRefGoogle Scholar
  14. 14.
    Lanconelli, E., Polidoro, S.: On a class of hypoelliptic evolution operators. Rend. Sem. Mat. Univ. Pol. Torino 52(1), 29–63 (Partial Differ. Equ.) (1994)MathSciNetMATHGoogle Scholar
  15. 15.
    Negrini, P., Scornazzani, V.: Superharmonic functions and regularity of boundary points for a class of elliptic-parabolic partial differential operators. Bollettino UMI An. Funz. Appl. Serie VI, Vol. III-C 1, 85–106 (1984)MathSciNetGoogle Scholar
  16. 16.
    Pini, B.: Maggioranti e minoranti delle soluzioni delle equazioni paraboliche. Ann. Math. Pures Appl. 37, 249–264 (1954)MathSciNetMATHCrossRefGoogle Scholar
  17. 17.
    Pini, B.: Su un integrale analogo al potenziale logaritmico. Boll. Unione Mat. Ital. 9(3), 244–250 (1954)MathSciNetMATHGoogle Scholar
  18. 18.
    Potts, D.H.: A note on the operators of Blaschke and Privaloff. Bull. Am. Math. Soc. 54, 782–787 (1948)MathSciNetMATHCrossRefGoogle Scholar
  19. 19.
    Privaloff, I.: Sur les fonctions harmoniques. Rec. Math. Moscou (Mat. Sbornik) 32, 464–471 (1925)MATHGoogle Scholar
  20. 20.
    Privaloff, I.: On a theorem of S. Saks. Rec. Math. Moscou (Mat. Sbornik) 9, 457–460 (1941)MathSciNetGoogle Scholar
  21. 21.
    Reade, M.: Some remarks on subharmonic functions. Duke Math. J. 10, 531–536 (1943)MathSciNetMATHCrossRefGoogle Scholar
  22. 22.
    Rudin, W.: Real and Complex Analysis. McGraw-Hill, New York (1987)MATHGoogle Scholar
  23. 23.
    Saks, S.: On the operators of Blaschke and Privaloff for subharmonic functions. Rec. Math. Moscou (Mat. Sbornik) 9, 451–456 (1941)MathSciNetGoogle Scholar
  24. 24.
    Stein, E.M.: Harmonic Analysis: Real Variable Methods, Orthogonality, and Oscillatory Integrals. Princeton University Press, Princeton (1995)Google Scholar
  25. 25.
    Watson, N.A.: A theory of subtemperatures in several variables. Proc. Lond. Math. Soc. 26, 385–417 (1973)MATHCrossRefGoogle Scholar
  26. 26.
    Watson, N.A.: Nevanlinna’s first fundamental theorem for supertemperatures. Math. Scand. 73, 49–64 (1993)MathSciNetMATHGoogle Scholar
  27. 27.
    Watson, N.A.: Volume mean values of subtemperatures. Colloq. Math. 86, 253–258 (2000)MathSciNetMATHGoogle Scholar
  28. 28.
    Watson, N.A.: Elementary proofs of some basic subtemperatures theorems. Colloq. Math. 94, 111–140 (2002)MathSciNetMATHCrossRefGoogle Scholar
  29. 29.
    Watson, N.A.: A generalized Nevanlinna theorem for supertemperatures. Ann. Acad. Sci. Fenn. Math. 28, 35–54 (2003)MathSciNetMATHGoogle Scholar

Copyright information

© Springer Science+Business Media B.V. 2012

Authors and Affiliations

  1. 1.Dipartimento di MatematicaUniversità di BolognaBolognaItaly

Personalised recommendations