The asymptotic growth of the constants in the Bohnenblust-Hille inequality is optimal (Q442185)

In 1931 Bohnenblust and Hille showed that for every positive integer \(n\) there is \(C_n>0\) such that for every positive integer \(N\) and for every \(n\)-linear mapping \(U: {\mathbb C}^N\times\cdots\times {\mathbb C}^N\to {\mathbb C}\), NEWLINE\[NEWLINE\left(\sum_{i_1,\ldots,i_n=1}^N|U(e_{i_1},e_{i_2},\ldots, e_{i_m})|^{{2n\over n+1}} \right)^{{n+1\over2n}}\leq\sup_{|z_j|<1}|U(z_1,\ldots,z_n)|NEWLINE\]NEWLINE where \((e_i)_{i=1}^N\) denotes the canonical basis for \({\mathbb C}^N\). Since 1931 a number of papers have appeared which have improved Bohnenblust and Hille's original estimates for the sequence of constants \((C_n)_n\). In [\textit{D. Pellegrino} and \textit{J. B. Seoane-Sepúlveda}, J. Math. Anal. Appl. 386, No. 1, 300--307 (2012; Zbl 1234.26061)] the last two authors of the paper under review obtained estimates for \(C_n\) in terms of the constant for \(C_{[n/2]}\) and the optimal constants on the left hand side of Khinchine's inequality. In this paper the authors prove that the asymptotic growth of this sequence is optimal in the sense that \(\lim_{n\to\infty}{C_n\over C_{n-1}}=1\). From this it follows that if \((K_n)_n\) is the sequence of optimal constants for the Bohnenblust-Hille inequality, then it also has optimal asymptotic growth (i.e., \(\lim_{n\to\infty}{K_n\over K_{n-1}}\) will also be equal to \(1\)).