Quasi-nearly subharmonic functions in locally uniformly homogeneous spaces
The authors define non-negative quasi-nearly subharmonic functions on so-called locally uniformly homogeneous spaces. This function class is rather general and includes quasi-nearly subharmonic (thus also subharmonic, quasi-subharmonic and nearly subharmonic) functions on domains of Euclidean spaces \(\mathbb{R}^n\), \(n\geq2\). In addition, quasi-nearly subharmonic functions with respect to various measures on domains of \(\mathbb{R}^n\), \(n\geq2\), are included. As examples the authors 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 \(\mathbb{C}^n\), \(n\geq1\), and the hyperbolic measure on any domain \(D\subset\mathbb{R}^n\), \(D\neq\mathbb{R}^n\). For \(Y\subset X\) and \(x\in X\), let \[ \delta^Y_K(x):=\inf\big\{d_K(x,y):y\in Y^c\big\}, \] where \(d_K(x,y)\) is a quasi-distance on \(X\) and \(Y^c\) is the complement of \(Y\) taken in \(X\). The main result of the paper is the following theorem. Theorem 2.5. Let \(X\) be a locally uniformly homogeneous space satisfying the following additional condition: if \(B:=B_K(x_0,r_0)\) is a ball in \(X\), \(0\leq r_0\leq\rho_0\), and \(x\in B\), then \[ \delta^B_K(x)=\sup\big\{\varrho>0:B_K(x,\varrho)\subset B\big\}\,. \] Let \(u:X\rightarrow[0,+\infty)\) be \(C\)-quasi-nearly subharmonic. If \(\psi:[0,+\infty)\rightarrow[0,+\infty)\) is permissible, then \(\psi\circ u\) is \(C_1\)-quasi-nearly subharmonic on \(X\), where \(C_1=C_1(C,K,\psi,\varepsilon_0,\rho_0)\).
scientific article; zbMATH DE number 5900667
scientific article