Subharmonic addition to the Beurling-Malliavin multiplier theorem

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 $$\mathrm{type}[u]:=\underset{z\to \infty }{\mathrm{limsup}}\frac{u^{+}(z)}{|z|}<+\infty ,\mathrm{type}[M]<+\infty ,{\int }_{-\infty }^{+\infty }\frac{u^{+}(x)+M^{+}(x)}{1+x^2}dx<+\infty .$$ If $\mathrm{type}[u]<a<+\infty$, $0<b<+\infty$, and $\mathrm{type}[M]<c<+\infty$, then there are an entire function $h\not\equiv 0$ with $\mathrm{type}[\log |h|]<c$ 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 $u(z)-M(z)+\log |h(z)|\le a|\Im z|$ on each straight line parallel to the real axis and not intersecting $iY$.

12 pages, in Russian