Subharmonic functions satisfying the local Levin condition (Q2782545)

Let \(\rho(r)\) be a proximate order, let \(\rho= \lim_{r\to\infty} \rho(r)\) and \(V(r)= r^{\rho(r)}\). Then \(\rho(r)\) is called a formal proximate order of a subharmonic function \(v\) on the upper halfplane \(\mathbb{C}_+\) if \(v(z)\leq V(|z|)\) for all \(z\) in \(\mathbb{C}_+\). In this case \(v\) is said to satisfy the local Levin condition if, for some \(N<\infty\) and \(\delta_1\in (0,\pi/2)\), the set NEWLINE\[NEWLINEA_N= \{r:\sup_{\theta\in [\delta_2,\pi-\delta_1]} v(re^{i\theta})\geq -NV(r)\}NEWLINE\]NEWLINE is unbounded. This paper studies the consequences of this condition in the case where \(0\leq \rho\leq 1\). Given \(q\in (0,1)\) let \(E= \bigcup_{r\in A_N} (q/2,2r/q)\). Also, the complete measure \(\lambda\) of \(v\) is defined for Borel sets \(F\subset\mathbb{C}\) by NEWLINE\[NEWLINE\lambda(F)= 2\pi \int_{F\cap\mathbb{C}_+} \text{Im }\zeta d\mu(\zeta)- \nu(F\cap\mathbb{R}),NEWLINE\]NEWLINE where \(\mu\) is the Riesz measure of \(v\) and \(\nu\) is a boundary measure associaed with \(v\) (in terms of harmonic majorization in half-discs). As a sample result we state Theorem 1: If \(v\) is as above, then there is a constant \(M\), depending only on \(N\), \(\delta_1\), \(q\) and \(\rho(\cdot)\), such that \(|\lambda|(B(0, r))\leq MrV(r)\) and \(\int^\pi_0|v(re^{i\theta})|\sin \theta d\theta\leq MV(r)\) whenever \(r\in E\).NEWLINENEWLINEFor the entire collection see [Zbl 0980.00031].