Subharmonic addition to the Beurling-Malliavin multiplier theorem

TL;DR

This paper presents a simplified version of the Beurling-Malliavin multiplier theorem, establishing conditions for the existence of entire functions with controlled growth and boundedness properties related to subharmonic functions.

Contribution

It introduces a subharmonic addition to the classical theorem, providing new bounds and conditions involving entire functions and their types in relation to subharmonic functions.

Findings

Existence of entire functions with prescribed growth bounds.

Construction of functions bounded on the real axis with controlled behavior.

Extension of the Beurling-Malliavin theorem to subharmonic functions.

Abstract

We prove a version of the Beurling-Malliavin multiplier theorem. This version is formulated here in a simplified form. Let $u\not\equiv -\infty$ and $M\not\equiv -\infty$ be a pair of subharmonic functions on the complex plane $\mathbb C$ with positive parts $u^+:=\sup\{u,0\}$ and $M^+$ such that $$ \operatorname{type}[u]:=\limsup_{z\to \infty} \frac{u^+(z)}{|z|}<+\infty, \qquad \operatorname{type}[M]<+\infty, \qquad \int_{-\infty}^{+\infty}\frac{u^+(x)+M^+(x)}{1+x^2}\operatorname{d}x<+\infty. $$ If $\operatorname{type}[u]<a<+\infty$, $0<b<+\infty$, and $\operatorname{type}[M]<c<+\infty$, then there are an entire function $h\not\equiv 0$ with $\operatorname{type}[\log|h|]<c$ and a subset $iY$ in the imaginary axis $i\mathbb R$ of linear Lebesgue measure $<b$ such that the function $h$ is bounded on the real axis and $u(z)-M(z)+\log|h(z)|\leq a|\Im z|$ on each straight line parallel to the real axis and not intersecting $iY$.