Subharmonic functions of order less than one
The author studies subharmonic functions in \(\mathbb R^N\) where \(N\geq 3\) of order at most one. The purpose is to give estimates from below on such functions. The paper is quite technical, with references in the statement of one theorem to many other results in the paper. I will try to explain one of the results. For a subharmonic function \(u\) we put \(p(r)=\int_{B(0,r)}\,d\mu (t)\), where \(\mu\) is the Riesz measure of \(u\). We put, furthermore, \[ J(r)=\int_0^r\frac{p(t)}{t^{N-1}}\;dt. \] Cartan's lemma says that (for given \(\alpha\) and \(R\)) \[ \int_{| \xi| \leq \mathbb R}\frac{d\mu (\xi)}{| x-\xi| ^{N-2}}\leq \frac{p(R)}{\alpha} \] holds for all \(x\in \mathbb R^N\) except for a union of balls \(\Gamma(\alpha, R)\). The author supposes that we are given a function \(\phi(r)\) defined on the positive axis. This function should have some technical properties: first of all it should be positive and increasing to infinity, and if we consider the auxiliary function \[ \psi(r)=\phi'(r)r^{N-1}/(N-2) \] we should also have the properties that \(r/\psi(r)\) tends to infinity and \(\psi(2r)\leq \Lambda \psi(r)\) for \(r\) tending to infinity and some constant \(\Lambda\). Theorem 1 of the paper says the following: let \(u\) be a subharmonic function and suppose that \(\liminf_{r}J(r)/\phi(r)\leq 1\). For any \(\varepsilon>0\) and \(\alpha >0\) there is a sequence \(R_n\) tending to infinity such that \[ u(x)\geq -\psi(x)(\varepsilon+(1+\varepsilon)(1/\alpha -1/R_n^{N-2})) \] for all \(x\in B(0,R_n/2)\) except on \(\Gamma (\alpha, R_n)\).
scientific article; zbMATH DE number 2164749