Positivity

, Volume 15, Issue 1, pp 1–10

Quasi-nearly subharmonic functions in locally uniformly homogeneous spaces

Authors

  • Miroslav Pavlović
    • Matematic̆ki Fakultet
    • Department of Physics and MathematicsUniversity of Joensuu
Article

DOI: 10.1007/s11117-009-0037-0

Cite this article as:
Pavlović, M. & Riihentaus, J. Positivity (2011) 15: 1. doi:10.1007/s11117-009-0037-0
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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 Bn of \({{\mathbb{R}}^n}\), the \({{\mathcal{M}}}\)-invariant measure on the unit ball B2n 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 up is quasi-nearly subharmonic for all p > 0.

Keywords

Locally uniformly homogeneous spaceHyperbolic measure\({{\mathcal{M}}}\)-invariant measureQuasihyperbolic measureSubharmonicQuasi-nearly subharmonic

Mathematics Subject Classification (2000)

Primary 31B0531C05Secondary 31C45

Copyright information

© Birkhäuser Verlag Basel/Switzerland 2009