Improving the closed formula for subpolynomial constants in the multilinear Bohnenblust-Hille inequalities
The multilinear Bohnenblust-Hille inequality asserts that, for every positive integer \(m\), there is a \(C_{m}\geq 1\) so that \[ \left( \sum\limits_{i_{1},\ldots,i_{m}=1}^{N}\left| U(e_{i_{^{1}}} ,\ldots,e_{i_{m}})\right| ^{\frac{2m}{m+1}}\right) ^{\frac{m+1}{2m}}\leq C_{m}\left\| U\right\| \] for every \(m\)-linear mapping \(U:\ell_{\infty}^{N}\times\cdots\times \ell_{\infty}^{N}\rightarrow\mathbb{C}\) and every positive integer \(N.\) This inequality was proved by \textit{H. F. Bohnenblust} and \textit{E. Hille} [Ann. Math. (2) 32, 600--622 (1931; Zbl 0001.26901)]. Its version for real scalars is also valid. It is well known that the exponent \(\frac{2m} {m+1}\) is optimal but the optimal values for \(C_{m}\) are still a mystery. The estimates for \(C_{m}\) have been improved over time. For instance, for the case of complex scalars: (1) \(C_{m}\leq m^{\frac{m+1}{2m}}2^{\frac{m-1}{2}}\) [loc. cit.]. (2) \(C_{m}\leq2^{\frac{m-1}{2}}\) (\textit{A. M. Davie} [J. Lond. Math. Soc., II. Ser. 7, 31--40 (1973; Zbl 0264.46055)]). (3) \(C_{m}\leq\left( \frac{2}{\sqrt{\pi}}\right) ^{m-1}\) (\textit{H. Queffélec} [J. Anal. 3, 43--60 (1995; Zbl 0881.11068)]). (4) \(C_{m}<1.41\left( m-1\right) ^{0.304975}-0.04\) (\textit{D. Nuñez-Alarcón} et al. [J. Funct. Anal. 264, No. 2, 429--463 (2013; Zbl 1264.46033)]). Up to now, the best known upper estimates for these constants can be found in [\textit{D. Pellegrino} and \textit{J. B. Seoane-Sepúlveda}, J. Math. Anal. Appl. 386, No. 1, 300--307 (2012; Zbl 1234.26061)] for real scalars and \textit{D. Nuñez-Alarcón, D. Pellegrino} and \textit{J. B. Seoane-Sepúlveda} [J. Funct. Anal. 264, No. 1, 326--336 (2013; Zbl 1264.46032)] for complex scalars. However, these formulas are given by puzzling recursive expressions. In the paper under review, the author improves the closed formulas already known for the multilinear Bohnenblust-Hille inequalities. The result is proved by a clever and deep induction argument.
scientific article; zbMATH DE number 6141614
scientific article