Integrals of subharmonic functions and their differences with weight over small sets on a ray

TL;DR

This paper extends classical theorems of Nevanlinna by estimating integrals of subharmonic functions over small sets on a ray, providing uniform bounds that generalize previous lemmas for small arcs and intervals.

Contribution

It develops a new theorem that generalizes Nevanlinna's classical results and extends Edrei-Fuchs Lemma to small sets on a ray with uniform estimates.

Findings

Provides bounds for integrals of subharmonic functions over small sets.

Generalizes classical lemmas to new settings with uniform constants.

Connects integral estimates with the characteristic function of subharmonic functions.

Abstract

Let $E$ be a measurable subset in a segment $[0,r]$ in the positive part of the real axis in the complex plane, and $U=u-v$ be the difference of subharmonic functions $u\not\equiv -\infty$ and $v\not\equiv-\infty$ on the complex plane. An integral of the maximum on circles centered at zero of $U^+:=\sup\{0,U\} $ or $|u|$ over $E$ with a function-multiplier $g\in L^p(E)$ in the integrand is estimated, respectively, in terms of the characteristic function $T_U$ of $U$ or the maximum of $u$ on circles centered at zero, and also in terms of the linear Lebesgue measure of $E$ and the $ L^p$-norm of $g$. Our main theorem develops the proof of one of the classical theorems of Rolf Nevanlinna in the case $E=[0,R]$, given in the classical monograph by Anatolii A. Gol'dberg and Iosif V. Ostrovskii, and also generalizes analogs of the Edrei-Fuchs Lemma on small arcs for small intervals from the works of A.F. Grishin, M.L. Sodin, T.I. Malyutina. Our estimates are uniform in the sense that the constants in these estimates do not depend on $U$ or $u$, provided that $U$ has an integral normalization near zero or $u(0)\geq 0$, respectively.