Multilinear Bohnenblust–Hille constant (real)

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

$T : (ℓ_{\infty}^{n})^{m} \to R$

satisfies the (multilinear) Bohnenblust--Hille inequality

$(j_{1}, \dots, j_{m} = 1 \sum n T (e_{j_{1}}, \dots, e_{j_{m}})^{\frac{2 m}{m + 1}})^{\frac{m + 1}{2 m}} \leq B_{R, m} (n) ∥ T ∥,$

where

$∥ T ∥ := ∥ x^{(1)} ∥_{\infty}, \dots, ∥ x^{(m)} ∥_{\infty} \leq 1 sup T (x^{(1)}, \dots, x^{(m)}) .$

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

$B_{R, m} := n \geq 1 sup B_{R, m} (n) .$

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

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

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

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

The exponent $\frac{2 m}{m + 1}$ 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

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

which would imply the exact value $C_{26 b} = 2$ .

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 $(ℓ_{1}, ℓ_{2})$ -Littlewood inequality. (2016). https://arxiv.org/abs/1604.07595

A better upper or lower bound, with a proof or a construction whose value is re-computed by published code (reproducible), ideally with a certificate a deterministic checker can validate.
A formal proof (Lean) of a known bound, or a precise error in a claimed one.
New references for the tables above (literature claims).

Imported from the crowdsourced repository of optimization constants (Terence Tao and contributors), commit 2c1968cd520b, Apache License 2.0; reformatted for this page. New records should also be reported there.

Current state