B.~Ya.~Levin function for some sets of segments

O. V. Silvanovich ; N. A. Shirokov

Zapiski Nauchnykh Seminarov POMI, Investigations on linear operators and function theory. Part 51, Tome 527 (2023), pp. 183-203

Résumé

Let $\{I_k\}_{k\in\mathbb{Z}}$, $I_k=[a_k,b_k]$, $b_k$, $a_k\rightarrow {-\infty}$ $(k\rightarrow{-\infty})$, $a_k\rightarrow {+\infty}$ $(k\rightarrow{+\infty})$ be a set of disjoint segments of the real axis $\mathbb{R}$. $J_k=[b_k,a_{k+1}]$, $E=\bigcup\limits_{k\in\mathbb{Z}}J_k.$ We assume that $a_0=-1$, $ b_0=1$, $a_1=2^{n_0}\stackrel{\mathrm{def}}{=}C$, $b_{-1}=-2^{n_0}$, $ |I_k|=2^{-m\alpha}$, $\alpha>0$ in case $I_k\subset [2^m,2^{m+1}]$ or $I_k\subset [-2^{m+1},-2^{m}]$, $m\geq n_0.$ We assume further that there exist $k$ and $l$ such that $a_k=2^n$ and $b_l=-2^n$, for any $n\geq n_0$. The B. Ya. Levin function $f_{E,\sigma}(z)$, $\sigma>0$, is defined to be a function satisfying the following conditions: $f_{E,\sigma}(z)$ is subharmonic on the complex plane $\mathbb{C}$ and harmonic on $\mathbb{C}\setminus E$; $f_{E,\sigma}(z)=0$, $x\in E;\ f_{E,\sigma}(z)>0,\ z\in\mathbb{C}\setminus E$; $\underset{z\rightarrow\infty}{\varlimsup}\dfrac{f_{E,\sigma}(z)}{|z|}=\sigma,\ f_{E,\sigma}(\overline z)=f_{E,\sigma}(z)$; if $g$ is subharmonic on $\mathbb{C}$, $g(x)\leq 0,\ x\in E,$ and $\underset{z\rightarrow\infty}{\varlimsup}\dfrac{g(z)}{|z|}\leq\sigma$, then $$ g(z)\leq f_{E,\sigma}(z),\ z\in \mathbb{C}. $$ The B. Ya. Levin function $f_{E,\sigma}(z)$ exists if $C_1|I_l|\geq|J_k|\geq C|I_l|$, $J_k$, $I_l\subset[2^n,2^{n+1}]$ or $J_k$, $I_l\subset[-2^{n+1},-2^{n}]$, $n\geq n_0.$ We prove that if $C\geq c_0(\alpha)$, then $\max\limits_{x\in I_k}f_{E,\sigma}(x)\leq 6\sigma|I_k| $ and describe the behavior of $f_{E,1}(z)$ in a neighborhood of $J_k$, $k\in\mathbb{Z}$.