The Hilbert transform-UMD dependence problem

Neither exponent can be lowered. Theorem 1.1 constructs explicit $2^{n}$-dimensional spaces $X_n$ and $Y_n$ with $\hbar_{2,X_n}\asymp n$, $\beta_{2,X_n}\asymp\sqrt n$ and $\beta_{2,Y_n}\asymp n$, $\hbar_{2,Y_n}\asymp\sqrt n$, with universal comparison constants, so both quadratic comparisons are sharp at $p=2$; Corollary 1.3 extends the growth rates to every fixed $1<p<\infty$ by extrapolation, with constants depending on $p$. The spaces are built from the summation operators Wenzel had proposed as candidates for exactly this separation. The Lean formalisation covers Theorem 1.1 at $p=2$ over both real and complex scalars; the extrapolation to other $p$ and the exact coefficients of Remark 1.2 are outside it.

Neither exponent can be lowered. Theorem 1.1 constructs explicit $2^{n}$-dimensional spaces $X_n$ and $Y_n$ with $\hbar_{2,X_n}\asymp n$, $\beta_{2,X_n}\asymp\sqrt n$ and $\beta_{2,Y_n}\asymp n$, $\hbar_{2,Y_n}\asymp\sqrt n$, with universal comparison constants, so both quadratic comparisons are sharp at $p=2$; Corollary 1.3 extends the growth rates to every fixed $1<p<\infty$ by extrapolation, with constants depending on $p$. The spaces are built from the summation operators Wenzel had proposed as candidates for exactly this separation. The Lean formalisation covers Theorem 1.1 at $p=2$ over both real and complex scalars; the extrapolation to other $p$ and the exact coefficients of Remark 1.2 are outside it.