The bounds ${\hslash }_{p,X}\lesssim ({\beta }_{p,X}{)}^2$ and ${\beta }_{p,X}\lesssim ({\hslash }_{p,X}{)}^2$ are sharp

Abstract

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 $L^p(\mathbb{R};X)$, and that the UMD constant ${\beta }_{p,X}$ and the Hilbert transform constant ${\hslash }_{p,X}$ 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.

14 pages; the main result has been checked in Lean4