The bounds $\hbar_{p,X}\lesssim(β_{p,X})^2$ and $β_{p,X}\lesssim(\hbar_{p,X})^2$ are sharp

Abstract: It was proved in the 1980s by Burkholder and Bourgain that, for any Banach space $X$ and $1&lt;p&lt;\infty$, the UMD$<em>p$ property for $X$ is equivalent to boundedness of the Hilbert transform on $L<sup>p(\R;X)$, and that the UMD constant $β</em>{p,X}$ 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.