Sharp comparison exponents for Hilbert-transform and UMD constants

At p=2, how sharply can a Banach space’s Hilbert-transform and UMD constants differ? The target tests whether both classical quadratic comparison exponents are optimal, over real and complex spaces, using 2ⁿ-dimensional families.

Why it matters and what counts as progress

Why it matters: It measures the quantitative difference between two equivalent analytic properties, rather than their already known qualitative equivalence.

What counts as progress: Construct spaces attaining either quadratic comparison, or establish a smaller uniform exponent; both directions require separate treatment.

How the problem developed

9 Sept 2026 · Sharpness result [1]
Emiel Lorist and Jan van Neerven Constructed two explicit 2ⁿ-dimensional Banach-space families showing that both quadratic comparisons between the Hilbert-transform and UMD constants are sharp.

AI contribution

The bounds ℏₚ,ₓ ≲ βₚ,ₓ² and βₚ,ₓ ≲ ℏₚ,ₓ² are sharp

Emiel Lorist, Jan van Neerven

Lorist and van Neerven construct two explicit 2^n-dimensional Banach-space families at p=2. One has Hilbert-transform constant of order n and UMD constant of order √n; the other reverses them, making both classical quadratic comparison exponents sharp over real and complex spaces.

AI involvementAI-centered

ImportanceMajor

Technical verificationFormal (Lean 4 (reported comparator and kernel replay) + human)

Human understandingFull