Linear comparison of the Hilbert transform and UMD constants

Determine whether the Hilbert transform constant and the UMD constant admit uniform linear comparisons over all Banach spaces at a fixed exponent p; specifically, establish whether there are constants depending only on p such that hbar_{p,X} is bounded by beta_{p,X} and beta_{p,X} is bounded by hbar_{p,X}.

Background

For a Banach space X and 1<p<infty, the UMD constant beta_{p,X} measures the boundedness of martingale transforms, while the Hilbert transform constant hbar_{p,X} measures the boundedness of the vector-valued Hilbert transform on Lp(R;X). Classical results establish equivalence of these finiteness properties, but yield only quadratic estimates: hbar_{p,X} <= 2(beta_{p,X})2 and beta_{p,X} <= 2(hbar_{p,X})2.

The paper explains that the possible improvement of these quadratic powers to linear dependence had been unresolved. It then resolves the issue negatively by constructing finite-dimensional spaces X_n and Y_n for which the two constants have respective growth rates n and sqrt(n), showing that neither quadratic exponent can be reduced.

References

The question whether the powers in eq:quadratic-comparison can be reduced, in particular whether

\hbar_{p,X}\lesssim_p\beta_{p,X} \qquad\text{and/or}\qquad \beta_{p,X}\lesssim_p\hbar_{p,X},

has been a longstanding problem; see Burkholder p. 249 and Problem O.6.

eq:quadratic-comparison:

ℏp,X≤2(βp,X)2,βp,X≤2(ℏp,X)2;\hbar_{p,X}\leq 2(\beta_{p,X})^2, \qquad \beta_{p,X}\leq 2 (\hbar_{p,X})^2;

— The bounds $\hbar_{p,X}\lesssim(β_{p,X})^2$ and $β_{p,X}\lesssim(\hbar_{p,X})^2$ are sharp  (2609.10444 - Lorist et al., 9 Sep 2026) in Section 1, Introduction