Positivity

, Volume 15, Issue 1, pp 1–10 | Cite as

Quasi-nearly subharmonic functions in locally uniformly homogeneous spaces

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Abstract

We define nonnegative quasi-nearly subharmonic functions on so called locally uniformly homogeneous spaces. We point out that this function class is rather general. It includes quasi-nearly subharmonic (thus also subharmonic, quasisubharmonic and nearly subharmonic) functions on domains of Euclidean spaces \({{\mathbb{R}}^n}\), n ≥ 2. In addition, quasi-nearly subharmonic functions with respect to various measures on domains of \({{\mathbb{R}}^n}\), n ≥ 2, are included. As examples we list the cases of the hyperbolic measure on the unit ball B n of \({{\mathbb{R}}^n}\), the \({{\mathcal{M}}}\)-invariant measure on the unit ball B 2n of \({{\mathbb{C}}^n}\), n ≥ 1, and the quasihyperbolic measure on any domain \({D\subset {\mathbb{R}}^n}\), \({D\ne {\mathbb{R}}^n}\). Moreover, we show that if u is a quasi-nearly subharmonic function on a locally uniformly homogeneous space and the space satisfies a mild additional condition, then also u p is quasi-nearly subharmonic for all p > 0.

Keywords

Locally uniformly homogeneous space Hyperbolic measure \({{\mathcal{M}}}\)-invariant measure Quasihyperbolic measure Subharmonic Quasi-nearly subharmonic 

Mathematics Subject Classification (2000)

Primary 31B05 31C05 Secondary 31C45 

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References

  1. 1.
    Coifman R.R., Weiss G.: Analyse harmonique non-commutative sur certains espaces homogènes. Lecture Notes in Mathematics, vol 242. Springer, Berlin (1971)Google Scholar
  2. 2.
    Coifman R.R., Weiss G.: Extensions of Hardy spaces and their use in analysis. Bull. Am. Math. Soc. 83(4), 569–645 (1977)MATHCrossRefMathSciNetGoogle Scholar
  3. 3.
    Djordjević O., Pavlović M.: \({{\mathcal{L}}^p}\)-integrability of the maximal function of a polyharmonic function. J. Math. Anal. Appl. 336, 411–417 (2007)MATHCrossRefMathSciNetGoogle Scholar
  4. 4.
    Domar Y.: On the existence of a largest subharmonic minorant of a given function. Ark. Mat. 3(39), 429–440 (1957)MathSciNetGoogle Scholar
  5. 5.
    Fefferman C., Stein E.M.: Hp spaces of several variables. Acta Math. 129, 137–192 (1972)MATHCrossRefMathSciNetGoogle Scholar
  6. 6.
    Kojić V.: Quasi-nearly subharmonic functions and conformal mappings. Filomat. 21(2), 243–249 (2007)MATHCrossRefMathSciNetGoogle Scholar
  7. 7.
    Kuran Ü.: Subharmonic behavior of | h |p, (p > 0, h harmonic). J. Lond. Math. Soc. (2) 8, 529–538 (1974)MATHCrossRefMathSciNetGoogle Scholar
  8. 8.
    Pavlović M.: Mean values of harmonic congugates in the unit disc. Complex Variables 10, 53–65 (1988)MATHGoogle Scholar
  9. 9.
    Pavlović M.: Inequalities for the gradient of eigenfunctions of the invariant Laplacian on the unit ball. Indag. Math. (N.S.) 2, 89–98 (1991)MATHCrossRefMathSciNetGoogle Scholar
  10. 10.
    Pavlović M.: On subharmonic behavior and oscillation of functions on balls in \({{\mathbb{R}}^n}\). Publ. Inst. Math. (Beograd) 55(69), 18–22 (1994)MathSciNetGoogle Scholar
  11. 11.
    Pavlović M., Riihentaus J.: Classes of quasi-nearly subharmonic functions. Potential Anal. 29, 89–104 (2008)MATHCrossRefMathSciNetGoogle Scholar
  12. 12.
    Riihentaus J.: On a theorem of Avanissian–Arsove. Expo. Math. 7, 69–72 (1989)MATHMathSciNetGoogle Scholar
  13. 13.
    Riihentaus, J.: Subharmonic functions: non-tangential and tangential boundary behavior. In: Mustonen, V., Rákosnik, J. (eds.) Function Spaces, Differential Operators and Nonlinear Analysis (FSDONA’99), Proceedings of the Syöte Conference 1999, pp. 229–238. Math. Inst., Czech Acad. Science, Praha, (2000)Google Scholar
  14. 14.
    Riihentaus J.: A generalized mean value inequality for subharmonic functions. Expo. Math. 19, 187–190 (2001)MATHMathSciNetGoogle Scholar
  15. 15.
    Riihentaus, J.: On the weighted boundary behavior of \({{\mathcal{M}}}\)-subharmonic functions. In: International Workshop on Potential Theory and free Boundary Flows, September 25–30, 2005, Kiev, Ukraine. Transactions of the Institute of Mathematics of the National Academy of Ukraine, vol. 3, No. 4, pp. 92–108 (2006)Google Scholar
  16. 16.
    Stoll, M.: Invariant Potential Theory in the Unit Ball of \({{\mathbb{C}}^n}\). London Mathematical Society Lecture Notes Series, Cambridge (1994)Google Scholar
  17. 17.
    Stoll M.: Boundary limits and non-integrability of \({{\mathcal{M}}}\) -subharmonic functions in the unit ball of \({{\mathbb{C}}^n}\) (n ≥ 1). Trans. Am. Math. Soc. 349(9), 3773–3785 (1997)MATHCrossRefMathSciNetGoogle Scholar
  18. 18.
    Torchinsky A.: Real-Variable Methods in Harmonic Analysis. Academic Press, London (1986)MATHGoogle Scholar
  19. 19.
    Vuorinen M.: On the Harnack constant and the boundary behavior of Harnack functions. Ann. Acad. Sci. Fenn., Ser. A I, Math. 7, 259–277 (1982)MATHMathSciNetGoogle Scholar
  20. 20.
    Vuorinen, M.: Conformal Geometry and Quasiregular Mappings. In: Lecture Notes in Mathematics, vol. 1319. Springer, Berlin (1988)Google Scholar

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© Birkhäuser Verlag Basel/Switzerland 2009

Authors and Affiliations

  1. 1.Matematic̆ki FakultetBelgradeSerbia
  2. 2.Department of Physics and MathematicsUniversity of JoensuuJoensuuFinland

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