Subharmonic functions and their Riesz measure (Q2740871)

We write \(SH\) for the class of functions \(u\) that are subharmonic on \(\mathbb{R}^N\) \((N\geq 2)\), harmonic on some neighbourhood of 0 and satisfy \(u(0)=0\). Given positive constants \(\gamma\), \(B\), we write \(u\in SH(\gamma,H)\) if \(u\in SH\) and \(u(x)\leq A+B|x|^\gamma\) for some constant \(A\) and for all \(x\in \mathbb{R}^N\). Several inequalities involving the Riesz measure \(\mu\) of a function \(u\in SH(\gamma,B)\) are proved. For example, in the case \(N\geq 3\) the inequality NEWLINE\[NEWLINE\int_{|\zeta |\leq s}|\zeta |^{2-N} d\mu(\zeta)\leq A+D \Biggl( \int_{|\zeta |\leq s} d\mu(\zeta) \Biggr)^{\gamma/ (\gamma+ N-2)}NEWLINE\]NEWLINE holds for all \(s>0\); here \(D\) is an explicitly determined constant depending on \(\gamma\), \(B\), \(N\). NEWLINENEWLINENEWLINELet \(\mu_1\), \(\mu_2\) be the Riesz measures of functions \(u_1,u_2\in SH(\gamma,B)\). It is shown that if \(0< B'< 2B\), then \(\mu_1+\mu_2\) is not necessarily the Riesz measure of any function in \(SH(\gamma,B')\). NEWLINENEWLINENEWLINELet \(\varphi: [0,+\infty)\to (0,+\infty)\) be a decreasing \(C^1\) function such that \(\varphi(s)\log s\to 0\) (if \(N=2\)) or \(\varphi(s) s^{(N-2)/2}\to 0\) (if \(N\geq 3\)) as \(s\to +\infty\). It is shown that if \(u\in SH\) and \(\int_{\mathbb{R}^N} u^+(x) \varphi'(|x|^2) dx> -\infty\), then NEWLINE\[NEWLINE\int_{\mathbb{R}^N}|x|^{-2} \varphi(1+|x|^2) d\mu(x)< +\infty,NEWLINE\]NEWLINE where \(\mu\) is the Riesz measure of \(u\). Stronger conclusions hold if \(\varphi(s)= o((s\log s)^{-1})\) (if \(N=2\)) or \(\varphi(s)= o(s^{1-N})\) (if \(N\geq 3\)).