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Sharp generalizations of the multilinear Bohnenblust-Hille inequality. (English) Zbl 1319.46035
Let $$\mathbb{K}=\mathbb{R}$$ or $$\mathbb{C}$$. If $$A$$ is a continuous $$m$$-linear form on $$c_{0}\times\cdots\times c_{0}$$, for $$m \geq 2$$, then there is a constant $$C_{m}$$ (depending only on $$m$$) such that $\left( \sum_{i_{1},\dots,i_{m}=1}^{\infty} \left| A(e_{i_{1}},\dots,e_{i_{m}})\right| ^{\frac{2m}{m+1}}\right) ^{\frac{m+1}{2m}}\leq C_{m}\left\| A\right\| .$ This is the Bohnenblust-Hille inequality and, when $$m=2$$, we have the well-known Littlewood $$4/3$$ inequality. During a few decades, some generalizations of the Bohnenblust-Hille inequality have been presented including, as quoted by authors, those that appear in the papers of A. Defant and P. Sevilla-Peris [J. Funct. Anal. 256, No. 5, 1642–1664 (2009; Zbl 1171.46034)] and T. Praciano-Pereira [J. Math. Anal. Appl. 81, 561–568 (1981; Zbl 0497.46007)].
In the main theorem of this interesting paper, the authors show that the Bohnenblust-Hille inequality is a particular case of a more general and sharp family of inequalities and that the above mentioned generalizations can be adapted to their approach, that is, can be recovered with appropriate choices of parameters and objects involved. The paper also attracts attention by the use of economical and/or simple arguments in many proofs.

##### MSC:
 46G25 (Spaces of) multilinear mappings, polynomials 47L22 Ideals of polynomials and of multilinear mappings in operator theory 47B10 Linear operators belonging to operator ideals (nuclear, $$p$$-summing, in the Schatten-von Neumann classes, etc.)
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