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Multilinear Bohnenblust--Hille constant (real) (C26bC_{26b})

Multilinear Bohnenblust--Hille constant (real): determine C26bC_{26b} or improve its bounds; the repository records 22 (lower) and ∞\infty (upper).

MSC 46G25 · (Spaces of) multilinear mappings, polynomials

Editorial MSC2020 classification. Original article codes and source evidence are recorded in the literature directory.

Licensed source statementAs of Repository table at commit 2c1968c (2026-09-26); open while the recorded lower and upper bounds differSource & evidence ↓
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Problem statement

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

T:(ℓ∞n)m→RT:(\ell_\infty^n)^m \to \mathbb R

satisfies the (multilinear) Bohnenblust--Hille inequality

(∑j1,…,jm=1n∣T(ej1,…,ejm)∣2mm+1)m+12m≤BR,m(n) ∥T∥,\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)∥∞,…,∥x(m)∥∞≤1∣T(x(1),…,x(m))∣.\|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 mm by

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

Finally, define C26b:=sup⁡m≥1BR,m.C_{26b}:=\sup_{m\ge 1} B_{\mathbb R,m}.

Equivalently, C26b<∞C_{26b}<\infty if and only if the sequence (BR,m)m≥1\bigl(B_{\mathbb R,m}\bigr)_{m\ge 1} is bounded.

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Background

  • The exponent 2mm+1\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 mm; in the real case, they conjecture the sharp values

BR,m=21−1mB_{\mathbb R,m}=2^{1-\frac1m} for all mm,

which would imply the exact value C26b=2C_{26b}=2.

  • See the survey [CP2018] for background, further references, and related polynomial/Hardy--Littlewood variants.

References (from the repository page)

  • [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 (ℓ1,ℓ2)(\ell_1,\ell_2)-Littlewood inequality. (2016). https://arxiv.org/abs/1604.07595

Progress & source assessment

Repository table (commit 2c1968c, 26 September 2026): best known lower bound 22; best known upper bound ∞\infty. Values marked with an asterisk have minimal verification.

Known upper bounds

Bound Reference Comments
∞\infty Trivial The best known general estimates on BR,mB_{\mathbb R,m} for each fixed mm are sublinear in mm; for example BR,m<1.3 m0.365B_{\mathbb R,m} < 1.3\ m^{0.365} for m≥14m\ge 14 [CP2018].

Known lower bounds

Bound Reference Comments
22 [DMPSS2014] Proves the general lower bound BR,m≥21−1mB_{\mathbb R,m}\ge 2^{1-\frac1m} for every m≥2m\ge 2. Taking sup⁡m\sup_m gives C26b≥2C_{26b}\ge 2. (For m=2m=2 this is sharp: BR,2=2B_{\mathbb R,2}=\sqrt{2}, i.e. Littlewood's 4/34/3 inequality.)
References
  1. [1]Davis, Damek; Ivanisvili, Paata; Tao, Terence (maintainers). Optimization Constants in Mathematics, github.com/teorth/optimizationproblems (commit 2c1968c, 26 September 2026). — Multilinear Bohnenblust--Hille constant (real) ($C_{26b}$) linkApache-2.0 (repository license); constant pages reproduced with attribution; repository references retained.

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