Quasi-nearly subharmonic functions in locally uniformly homogeneous spaces

Quasi-nearly subharmonic functions in locally uniformly homogeneous spaces 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. http://www.deepdyve.com/assets/images/DeepDyve-Logo-lg.png Positivity Springer Journals

Quasi-nearly subharmonic functions in locally uniformly homogeneous spaces

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Publisher
SP Birkhäuser Verlag Basel
Copyright
Copyright © 2009 by Birkhäuser Verlag Basel/Switzerland
Subject
Mathematics; Econometrics; Operator Theory; Calculus of Variations and Optimal Control; Optimization; Fourier Analysis; Potential Theory
ISSN
1385-1292
eISSN
1572-9281
D.O.I.
10.1007/s11117-009-0037-0
Publisher site
See Article on Publisher Site

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.

Journal

PositivitySpringer Journals

Published: Nov 25, 2009

References

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