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Classes of quasi-nearly subharmonic functions. (English) Zbl 1158.31002

It is known that if \(u\geq 0\) is subharmonic on a domain \(\Omega\subset\mathbb{R}^n\), \(n\geq 2\) and \(p>0\), then there is a constant \(C=C(n,p)\geq\frac{1}{m(B(0,1))}\) such that \[ u(x)^p\leq\frac{c}{r^n}\int_{B(x,r)}u^p\,dm, \] for any ball \(B(x,r)\subset\Omega\). In an attempt to generalize this mean value inequality, the authors define a new class of functions \(QNS\): a nonnegative, locally integrable function \(u\) on \(\Omega\) is called quasi-nearly subharmonic if there is a constant \(K=K(n,u,\Omega)>0\) such that \[ u(x)\leq\frac{K}{r^n}\int_{B(x,r)}u\,dm \] for any \(B(x,r)\subset\Omega\). The authors show that the \(QNS\) class is large. For example, if \(h\) is a real valued and harmonic on \(\Omega\), then the function \(|\nabla h|^p\) is \(QNS\) for every \(p>0\). The authors prove some of the basic properties of these functions and give a characterization of the \(QNS\) functions with the aid of the quasihyperbolic metric.

MSC:

31C05 Harmonic, subharmonic, superharmonic functions on other spaces
31B25 Boundary behavior of harmonic functions in higher dimensions
31B05 Harmonic, subharmonic, superharmonic functions in higher dimensions
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