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on Multilinear Bohnenblust--Hille constant (real) ($C_{26b}$)

The statement

For integers $m,n\ge 1$ , let $B_{\mathbb R,m}(n)$ be the smallest constant such that every $m$ -linear form

T:(\ell_\infty^n)^m \to \mathbb R

satisfies the (multilinear) Bohnenblust--Hille inequality

\left(\sum_{j_1,\dots,j_m=1}^n \bigl|T(e_{j_1},\dots,e_{j_m})\bigr|^{\frac{2m}{m+1}}\right)^{\frac{m+1}{2m}} \le B_{\mathbb R,m}(n)\ \|T\|,

where

\|T\|:=\sup_{\|x^{(1)}\|_\infty,\dots,\|x^{(m)}\|_\infty \le 1}\bigl|T(x^{(1)},\dots,x^{(m)})\bigr|.

Define the optimal dimension-free (real) Bohnenblust--Hille constant of order $m$ by

B_{\mathbb R,m}:=\sup_{n\ge 1} B_{\mathbb R,m}(n).

Finally, define $C_{26b}:=\sup_{m\ge 1} B_{\mathbb R,m}.$

Equivalently, $C_{26b}<\infty$ if and only if the sequence $\bigl(B_{\mathbb R,m}\bigr)_{m\ge 1}$ is bounded.

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.