发表机构
Delft University of Technology（代尔夫特理工大学）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
本文构造显式Banach空间族，证明Hilbert变换常数与UMD常数之间的二次互界均为精确，不可改进。
AI 中文摘要
Burkholder 和 Bourgain 在 1980 年代证明了：对于任意 Banach 空间 $X$ 和 $1<p<\infty$，$X$ 的 UMD$_p$ 性质等价于 Hilbert 变换在 $L^p(\R;X)$ 上的有界性，并且 UMD 常数 $\beta_{p,X}$ 与 Hilbert 变换常数 $\hbar_{p,X}$ 满足二次界 \begin{equation*} \hbar_{p,X}\lesssim(\beta_{p,X})^2, \qquad \beta_{p,X}\lesssim(\hbar_{p,X})^2. \end{equation*} 本文给出例子表明这两个界都是精确的。更具体地，我们构造了显式的 $2^n$ 维 Banach 空间，其中 Hilbert 变换常数随 $n$ 增长，而 UMD 常数随 $\sqrt n$ 增长；还构造了另一族空间，其行为相反。
英文摘要
It was proved in the 1980s by Burkholder and Bourgain that, for any Banach space $X$ and $1<p<\infty$, the UMD$_p$ property for $X$ is equivalent to boundedness of the Hilbert transform on $L^p(\R;X)$, and that the UMD constant $β_{p,X}$ and the Hilbert transform constant $\hbar_{p,X}$ are related by the quadratic bounds \begin{equation*} \hbar_{p,X}\lesssim(β_{p,X})^2, \qquad β_{p,X}\lesssim(\hbar_{p,X})^2. \end{equation*} In this paper we present examples showing that both bounds are sharp. More precisely, we construct explicit $2^n$-dimensional Banach spaces for which the Hilbert transform constant grows like $n$ and the UMD constant like $\sqrt n$, and a second family with the reverse behaviour.
Comments14 pages; the main result has been checked in Lean4
