论文标题
Beurling-malliavin乘数定理的次谐音添加
Subharmonic addition to the Beurling-Malliavin multiplier theorem
论文作者
论文摘要
我们证明了Beurling-Malliavin乘数定理的版本。此版本以简化的形式在这里配制。让$ u \ not \ equiv - \ infty $和$ m \ equiv -equiv - \ infty $是复杂平面$ \ mathbb c $带有正零件$ u^+：= \ u^+：= \ sup \ {u，0 \} $和$ m^+$ m^+$ $^+operatoreorneornearnemeameameameameameameameameame {type}的$ u^+： \ infty} \ frac {u^+（z）} {| z |} <+\ infty，\ qquad \ ququad \ operatatorName {type} [m] \ int _ { - \ infty}^{+\ infty} \ frac {u^+（x）+m^+（x）} {1+x^2} \ operatatorName {d} x <+\ infty。 $$如果$ \ operatorName {type} [u] <a <a <+\ infty $，$ 0 <b <b <+\ infty $，以及$ \ operatotorName {type} [m] <c <+\ infty $，那么整个函数$ h \ not f \ e equiv 0 $ equiv 0 $ equiv 0 $ equiv 0 $ equiv 0 $ e y $ \ opereTornAme $ im {线性lebesgue量$ <b $的假想轴$ i \ mathbb r $，以使函数$ h $在真实轴上有界限，$ u（z）-m（z）-m（z）+\ log | h（z）| \ leq | \ leq a | \ im z | $在每个直线上与真实的轴心平行，而不是相交的$ iy $。
We prove a version of the Beurling-Malliavin multiplier theorem. This version is formulated here in a simplified form. Let $u\not\equiv -\infty$ and $M\not\equiv -\infty$ be a pair of subharmonic functions on the complex plane $\mathbb C$ with positive parts $u^+:=\sup\{u,0\}$ and $M^+$ such that $$ \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 $iY$.
