The Hilbert transform-UMD constant dependence problem asks how the operator norm 
of the vector-valued Hilbert
transform on 
compares with the unconditional martingale differences
(UMD) constant 
of a Banach space 
. The latter is the least constant that bounds all sign changes
of the differences of an 
-valued martingale in 
. The classical comparisons (Bourgain 1983, Burkholder 1983)
are
 |  |  |
(1)
|
 |  |  |
(2)
|
Lorist and van Neerven (2026) construct 
-dimensional Banach spaces 
and 
for which 
and 
, while 
and 
. Thus both quadratic exponents are sharp,
already for 
.
The examples were found in conversations with GPT-6 Astra. The authors reviewed the resulting statements and proofs, and the main 
theorem was subsequently checked in Lean 4 with Astra. The
precise numerical coefficients and the extrapolation to 
lie outside the formalization. Independent specialist review
had not been reported as of Sep. 12, 2026.
and 
Are Sharp." 9 Sep 2026.