Multilinear Bohnenblust–Hille constant (real)

Description of constant

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.

Bound	Reference	Comments
$\infty$	Trivial	The best known general estimates on $B_{\mathbb R,m}$ for each fixed $m$ are sublinear in $m$; for example $B_{\mathbb R,m} < 1.3\ m^{0.365}$ for $m\ge 14$ [CP2018].

Bound	Reference	Comments
$2$	[DMPSS2014]	Proves the general lower bound $B_{\mathbb R,m}\ge 2^{1-\frac1m}$ for every $m\ge 2$. Taking $\sup_m$ gives $C_{26b}\ge 2$. (For $m=2$ this is sharp: $B_{\mathbb R,2}=\sqrt{2}$, i.e. Littlewood’s $4/3$ inequality.)

The exponent $\frac{2m}{m+1}$ in the multilinear Bohnenblust–Hille inequality is sharp. [CP2018]
Universality Conjecture [PT2016]. The optimal Bohnenblust–Hille constants should be bounded uniformly in $m$; in the real case, they conjecture the sharp values

$B_{\mathbb R,m}=2^{1-\frac1m}$ for all $m$,

which would imply the exact value $C_{26b}=2$.
See the survey [CP2018] for background, further references, and related polynomial/Hardy–Littlewood variants.

[BH1931] Bohnenblust, H. F.; Hille, E. On the absolute convergence of Dirichlet series. Ann. of Math. (2) 32 (1931), no. 3, 600–622.
[CP2018] Cavalcante, Wasthenny V.; Pellegrino, Daniel M. Bohnenblust–Hille inequalities: analytical and computational aspects. An. Acad. Bras. Ci\^enc. 91 (2019), suppl. 1, e20170398. doi:10.1590/0001-3765201720170398. (Epub 2018). Full text: https://www.scielo.br/j/aabc/a/TdCkK3xqRHNHgVx9g9VmSMp/?format=pdf&lang=en
[DMPSS2014] Diniz, D.; Mu~noz-Fern'andez, G. A.; Pellegrino, D.; Seoane-Sep'ulveda, J. B. Lower bounds for the constants in the Bohnenblust–Hille inequality: the case of real scalars. Proc. Amer. Math. Soc. 142 (2014), no. 2, 575–580. https://arxiv.org/abs/1111.3253
[L1930] Littlewood, J. E. On bounded bilinear forms in an infinite number of variables. Quart. J. Math. 1 (1930), 164–174.
[PT2016] Pellegrino, Daniel M.; Teixeira, Eduardo. Sharp Bohnenblust–Hille constants for the mixed $(\ell_1,\ell_2)$-Littlewood inequality. (2016). https://arxiv.org/abs/1604.07595

Prepared with assistance from ChatGPT 5.2 Pro.