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Towards sharp Bohnenblust-Hille constants. (English) Zbl 1403.46037
The Hardy-Littlewood inequalities for \(m\)-linear forms, conceived by T. Praciano-Pereira [J. Math. Anal. Appl. 81, 561–568 (1981; Zbl 0497.46007)] from a classical result of Hardy and Littlewood for bilinear forms, assert that, for any positive integer \(m\) and \(p\geq2m\), there exist constants \(C_{m,n,p}^{\mathbb{K}}\geq1\) such that \[ C_{m,p}^{\mathbb{K}}:=\sup_{n}C_{m,n,p}^{\mathbb{K}}<\infty \] and \[ \left( \sum_{j_{1},\dots,j_{m}=1}^{n} \left\|T(e_{j_{1}},\dots,e_{j_{m}})\right\|^{\frac{2mp}{mp+p-2m}} \right)^{\frac{mp+p-2m}{2mp}} \leq C_{m,n,p}^{\mathbb{K}} \|T\|, \] for all positive integers \(n\) and all \(m\)-linear forms \(T:\ell_{p}^{n}\times\cdots\times\ell_{p}^{n} \to\mathbb{K}\), where \(\mathbb{K}=\mathbb{R}\) or \(\mathbb{C}\). When \(p=\infty\), the above result is the famous Bohnenblust-Hille inequality. The optimal values of the optimal constants \(C_{m,n,p}^{\mathbb{K}}\) and \(C_{m,p}^{\mathbb{K}}\) are, in general, unknown. Besides the challenging mathematical nature, the problem of finding these optimal constants, especially in the case of real scalars and when \(p=\infty\), has important applications in quantum information theory and also in the investigation of the FEI (Fourier entropy-influence) conjecture. The original estimates for \(C_{m,p}^{\mathbb{K}}\) had exponential growth (\(C_{m,p}^{\mathbb{K}}\leq\left( \sqrt{2}\right) ^{m-1}\)) but, for the case \(p=\infty\), F. Bayart et al. [Adv. Math. 264, 726–746 (2014; Zbl 1331.46037)] have proved that, in fact, the optimal estimates are quite smaller: \[ C_{m,\infty}^{\mathbb{K}} \leq 1.3m^{0.365}. \] When \(p<\infty\), G. Araújo and D. Pellegrino [Bull. Braz. Math. Soc. (N.S.) 48, No. 1, 141–169 (2017; Zbl 06767894)] have shown that the constants \(C_{m,p}^{\mathbb{K}}\) also have a surprisingly small asymptotic growth. In the paper under review, the authors investigate in depth several aspects related to the search of the optimal constants of the Hardy-Littlewood and Bohnenblust-Hille inequalities and provide arguments that suggest that even the best recent estimates for these constants may be far from the optimal ones.
The main technical issue explored in the paper under review is related to the use of interpolation arguments that, according to the authors, lose information and are probably sub-optimal. They develop the interesting notions of entropy and complexity as a qualitative tool to estimate the number of monomials that are needed to create a multilinear form that attains an optimal constant. In particular, the authors conjecture that there is a universal constant for the Bohnenblust-Hille inequalities (universality conjecture). More precisely, they conjecture that the optimal constants of the Bohnenblust-Hille inequality for real scalars are \(2^{1-\frac{1}{m}}\).
In essence, this paper shows that a new line of investigation to the search of optimal constants is in order. A natural step further is the investigation of optimization techniques and extreme points for the unit ball of the space of multilinear forms and, following this vein, a recent technique due to W. M. Cavalcante, D. W. Pellegrino and E. V. Teixeira [“On the geometry of multilinear forms”, Preprint (2016), arXiv:1612.08397] was implemented by F. Vieira Costa Júnior in the software Mathematica [“The optimal multilinear Bohnenblust-Hille constants: a computational solution for the real case”, to appear in Numer. Funct. Anal. Optim., doi:10.1080/01630563.2018.1489414], providing the optimal values of \(C_{m,n,\infty}^{\mathbb{R}}\) for any \(m,n\). This approach may, in the near future, reinforce the universality conjecture of Pellegrino and Teixeira from the paper under review. The paper under review also provides a quite elegant proof of a more general Bohnenblust-Hille inequality and contains sharp results related to other families of inequalities; for instance, the paper presents the optimal constants of all mixed \((\ell_{1},\ell_{2})\)-Littlewood’s inequalities, using a class of interesting multilinear forms that the authors call “strongly non-symmetric”.

MSC:
46G25 (Spaces of) multilinear mappings, polynomials
46B70 Interpolation between normed linear spaces
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