发表机构
Centre for Mathematical Sciences, Lund University（隆德大学数学中心）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
本文确定了 Fock 空间 $F^2$ 中 Korenblum 最大值原理的最优半径为 1，并给出等号成立条件，首次精确求解该类型空间的最优半径，且方法推广至加权情形。
AI 中文摘要
Korenblum 最大值原理指出，若在边界附近有 $|f| \le |g|$，则 $\\| f \\| \le \\| g \\|$。自 1991 年以来，Bergman 空间中支配区域的最优大小已被研究，自 2006 年以来 Fock 空间中也进行了研究，但仅获得了界限。我们确定了关于 $e^{-|z|^2}\\,dA(z)$ 平方可积的整函数空间 $F^2$ 的最优大小：若 $f, g\in F^2$ 且对所有 $|z|>1$ 有 $|f(z)| \le |g(z)|$，则 $\\| f \\| \le \\| g \\|$。半径 $1$ 是最优的，并且我们确定了所有等号成立的配对。据我们所知，这是 Bergman 或 Fock 类型空间中首次确定 Korenblum 原理的最优半径。证明结合了以无穷远为中心的 Schwarz-Pick 估计与矩对偶论证，且不需要数值计算。同样的方法给出了加权 Fock 型空间的矩准则。对于每个非增径向权重，最优半径等于由配对 $f\equiv c$，$g(z)=z$ 给出的上界。特别地，Wee 和 Le 对此类加权 Fock 空间的上界在 Hilbert 空间情形下是精确的。我们还证明了对于权重 $|z|^{2\beta}e^{-\alpha |z|^2}$，当 $-1<\beta\le5.44$ 时，最优半径为 $\sqrt{(\beta+1)/\alpha}$。
英文摘要
Korenblum's maximum principle states that if $|f| \le |g|$ near the boundary, then $\| f \| \le \| g \|$. The optimal size of the region of domination has been studied in Bergman spaces since 1991 and in Fock spaces since 2006, but only bounds have been obtained. We determine the optimal size for the Fock space $F^2$ of entire functions that are square integrable with respect to $e^{-|z|^2}\,dA(z)$: if $f, g\in F^2$ and $|f(z) \le |g(z)|$ for all $|z|>1$, then $\| f \| \le \| g \|$. The radius $1$ is optimal and we determine all pairs for which equality holds. To our knowledge, this is the first space of Bergman or Fock type in which the optimal radius in Korenblum's principle has been determined. The proof combines a Schwarz-Pick estimate centred at infinity with a moment-duality argument, and requires no numerical computation. The same method gives a moment criterion for weighted Fock-type spaces. For every non-increasing radial weight, the optimal radius equals the upper bound given by the pairs $f\equiv c$, $g(z)=z$. In particular, the upper bounds of Wee and Le for such weighted Fock spaces are sharp in the Hilbert space case. We also show that the optimal radius is $\sqrt{(β+1)/α}$ for the weight $|z|^{2β}e^{-α|z|^2}$, whenever $-1<β\le5.44$.
Comments16 pages
