, Volume 15, Issue 1, pp 1-10

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.