Bloch and gap subharmonic functions. (Q595883)

Let \({\mathbb B}_\alpha\) be the class of all positive subharmonic functions \(u\) in the open unit ball \(B_N\) of the space \({\mathbb R}^N\) such that \(G_\alpha (u) = \sup_{x\in B_N} (1-\| x\| ^2)^\alpha u(x) < +\infty.\) The analogous class \({\mathbb A}_\alpha\) of holomorphic in the disc functions \(f\) is defined by the condition \(\sup_{| z| <1}(1-| z| )^\alpha | f(z)| < +\infty.\) The author studies the class \({\mathbb B}_\alpha.\) There are many interesting papers concerning the class \({\mathbb A}_\alpha\), but the author does not include them in the list of references. The author uses the function \( \phi_a(x) =\frac{a(\| a\| ^2-(x,a)) - \sqrt{1-\| a\| ^2}(\| a\| ^2 x-(x,a)a)}{\| a\| ^2(1-(x,a))}\) instead of \(\frac{a-z}{1-z\bar{a}}\), and this allows him to investigate the class \({\mathbb B}_\alpha\) in the spaces of any dimension \(N.\) Among the results of the paper we note the following. Given \(\alpha >0\), \(R\in (0,1)\), \(R_a =R\frac{1-\| a\| ^2}{1+R\| a\| }\), a positive subharmonic function \(u\) belongs to \({\mathbb B}_\alpha\) if and only if \[ M_{\alpha, R}(u) =\sup _{a\in B_N}\frac{1}{[\text{Vol}\, B(a,R_a)]^{1- \frac{\alpha}{N}}}\int _{B(a, R_a)}u(x)\,dx <\infty. \] Moreover, \[ \left(\frac{1}{1+R}\right)^\alpha G_\alpha (u) \leq \left(\frac{1}{RV_N^{1/N}}\right)^\alpha M_{\alpha, R}(u) \leq\left(\frac{1+R}{1-R}\right)^\alpha G_\alpha (u), \] where \(V_N\) is the volume of the unit ball in \({\mathbb R}^N.\) It is known that the class \({\mathbb B}_\alpha\) does not have a description in terms of Riesz's measure now. The availability of the unbounded factor \((\frac{1+R}{1-R})^\alpha \) in the above inequality underlines the severity of the problem. The author investigates subharmonic functions of the kind \(u(x) = \sum_{k=0}^\infty c_k\| x\| ^{2^k}\), \(c_k \geq 0,\) as well.