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on Bohnenblust--Hille constant on the Boolean cube ($C_{26a}$)

The statement

Degree at most $d$ functions $f:\lbrace \pm 1\rbrace^n\to\mathbb{R}$ have Fourier--Walsh expansion

f(x)=\sum_{S\subseteq [n], |S|\leq d} \widehat f(S) \ x^S, \qquad x^S:=\prod_{i\in S}x_i, \qquad [n]:=\lbrace 1,\dots,n\rbrace.

For $d\in\mathbb{N}$ set $p_d:=\frac{2d}{d+1}$ . The (degree $\le d$ ) Bohnenblust--Hille inequality asks for the smallest constant $C$ such that for every $n$ and every function $f:\lbrace \pm 1\rbrace^n\to\mathbb{R}$ of degree at most $d$ ( $d\leq n$ ),

\left(\sum_{\substack{S\subseteq[n]\\ |S|\le d}} |\widehat f(S)|^{p_d}\right)^{1/p_d} \le C \|f\|_\infty, \qquad \|f\|_{\infty}:=\max_{x\in \lbrace \pm 1\rbrace^n}|f(x)|.

Let $\mathrm{BH}^{\le d}_{\lbrace\pm1\rbrace}$ denote this best constant $C$ (which depends on $d$ ). We define

C_{26a}:=\sup_{d\ge 1}\mathrm{BH}^{\le d}_{\lbrace\pm1\rbrace}.

Equivalently, $C_{26a}$ is the smallest constant for which the above inequality holds simultaneously for all degrees $d$ (with the exponent $p_d=\frac{2d}{d+1}$ depending on $d$ as above).

What gets accepted

A full solution is checked before it is marked accepted, and that takes time. Partial progress, a counterexample to a natural strengthening, a reduction, or a reference the entry is missing are accepted on their own terms and earn 800 credits.

Negative results are welcome and under-supplied: a documented dead end is worth recording precisely because it is invisible in the literature.