Beurling-Malliavin乘子定理的次调和加法
Subharmonic addition to the Beurling-Malliavin multiplier theorem
摘要：
我们证明了一个版本的Beurling-Malliavin乘数定理。这个版本在这里以简化的形式表述。设$u\not\equiv -\infty$和$M\not\equiv -\infty$是复平面$\mathbb C$上的一对次调和函数，具有正部分$u^+:=\sup\{u,0\}$和$M^+$，使得$$ \operatorname{type}[u]:=\limsup_{z\to \infty} \frac{u^+(z)}{|z|}<+\infty, \qquad \operatorname{type}[M]<+\infty, \qquad \int_{-\infty}^{+\infty}\frac{u^+(x)+M^+(x)}{1+x^2}\operatorname{d}x<+\infty. $$如果$\operatorname{type}[u]<a<+\infty$，$0<b<+\infty$和$\operatorname{type}[M]<c<+\infty$，然后有整个函数$h\not\equiv 0$与$\operatorname{type}[\log|h|]<c$和在线性Lebesgue测度$<b$的虚轴$i\mathbb R$中的子集$iY$使得函数$h$在实轴上有界并且$u(z)-M(z)+\log|h(z)|\leq a|\Im z|$在每条直线上有界平行于实轴且不与$iY$相交的直线。
英文摘要：
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 $iY$.
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