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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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##### References:
 [1] Adams, R.; Fournier, J. J., Sobolev Spaces, (2003), Elsevier · Zbl 1098.46001 [2] Albuquerque, N.; Bayart, F.; Pellegrino, D.; Seoane-Sepúlveda, J., Sharp generalizations of the multilinear Bohnenblust-Hille inequality, J. Funct. Anal., 266, 3726-3740, (2014) · Zbl 1319.46035 [3] Araujo, G.; Pellegrino, D., On the constants of the Bohnenblust-Hille and Hardy-Littlewood inequalities, Bull. Braz. Math. Soc. New Series, 78, 141-169, (2017) · Zbl 06767894 [4] Araujo, G.; Pellegrino, D.; Diniz, D.; da Silva e Silva, P., On the upper bounds for the constants of the Hardy-Littlewood inequality, J. Funct. Anal., 267, 1878-1888, (2014) · Zbl 1298.26066 [5] Bayart, F.; Pellegrino, D.; Seoane-Sepulveda, J., The Bohr radius of the $$n$$-dimensional polydisk is equivalent to $$\sqrt{\frac{\log n}{n}}$$, Adv. Math., 264, 726-746, (2014) · Zbl 1331.46037 [6] Benedek, A.; Panzone, R., The space $$L^p$$, with mixed norm, Duke Math. J., 28, 301-324, (1961) · Zbl 0107.08902 [7] Boas, H. P., The football player and the infinite series, Notices Amer. Math. Soc., 44, 1430-1435, (1997) · Zbl 0909.30001 [8] Bohnenblust, H. F.; Hille, E., On the absolute convergence of Dirichlet series, Ann. of Math., 32, 600-622, (1931) · JFM 57.0266.05 [9] Botelho, G.; Pellegrino, D., When every multilinear mapping is multiple summing, Math. Nachr., 282, 10, 1414-1422, (2009) · Zbl 1191.46041 [10] J. Campos, W. Cavalcante, V. Fávaro, D. Nuñez-Alarcón, D. Pellegrino and D. M. Serrano-Rodríguez, Polynomial and multilinear Hardy-Littlewood inequalities: Analytical and numerical approaches, preprint (2015); arXiv:1503.00618. · Zbl 1405.46028 [11] W. Cavalcante and D. Pellegrino, Geometry of the closed unit ball of the space of bilinear forms on $$\ell_p$$, preprint (2016); arXiv:1603.01535 [math.FA]. [12] De, A.; Diakonikolas, I.; Servedio, R. A., A robust Khintchine inequality, and algorithms for computing optimal constants in Fourier analysis and high-dimensional geometry, SIAM J. Discrete Math., 30, 2, 1058-1094, (2016) · Zbl 1339.68195 [13] Defant, A.; Frerick, L.; Ortega-Cerdà, J.; Ounaïes, M.; Seip, K., The Bohnenblust-Hille inequality for homogeneous polynomials is hypercontractive, Ann. of Math. (2), 174, 1, 485-497, (2011) · Zbl 1235.32001 [14] Diestel, J.; Jarchow, H.; Tonge, A., Absolutely Summing Operators, 43, (2005), Cambridge University Press · Zbl 0855.47016 [15] Dimant, V.; Sevilla-Peris, P., Summation of coefficients of polynomials on $$\ell_p$$ spaces, Publ. Mat., 60, 2, 289-310, (2016) · Zbl 1378.46032 [16] Haagerup, U., The best constants in the Khintchine inequality, Studia Math., 70, 231-283, (1981) · Zbl 0501.46015 [17] Hardy, G.; Littlewood, J. E., Bilinear forms bounded in space $$[p, q]$$, Quart. J. Math., 5, 241-254, (1934) · JFM 60.0335.01 [18] König, H., On the best constants in the Khintchine inequality for Steinhaus variables, Israel Math. J., 203, 23-57, (2014) · Zbl 1314.46017 [19] Littlewood, J. E., On bounded bilinear forms in an infinite number of variables, Quart. J. (Oxford Ser.), 1, 164-174, (1930) · JFM 56.0335.01 [20] Maligranda, L.; Sabourova, N., On clarkson’s inequality in the real case, Math. Nachr., 280, 1363-1375, (2007) · Zbl 1134.26008 [21] Montanaro, A., Some applications of hypercontractive inequalities in quantum information theory, J. Math. Phys., 53, 12, 15, (2012) · Zbl 1278.81045 [22] Pellegrino, D., The optimal constants of the mixed $$(\ell_1, \ell_2)$$-Littlewood inequality, J. Number Theory, 160, 11-18, (2016) · Zbl 1431.46024 [23] D. Pellegrino and D. M. Serrano-Rodríguez, On the mixed $$(\ell_1, \ell_2)$$-Littlewood inequality for real scalars and applications, preprint (2015); arXiv:1510.00909v1 [math.FA]. [24] Praciano-Pereira, T., On bounded multilinear forms on a class of $$\ell_p$$ spaces, J. Math. Anal. Appl., 81, 561-568, (1981) · Zbl 0497.46007 [25] Saksman, E.; Seip, K., Recent Progress on Operator Theory and Approximation in Spaces of Analytic Functions, Contemp. Math., 679, Some open questions in analysis for Dirichlet series, 179-191, (2016), American Mathematical Society · Zbl 1377.30048 [26] Szarek, J., On the best constants in the Khinchin inequality, Studia Math., 58, 2, 197-208, (1976) · Zbl 0424.42014
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