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There exist multilinear Bohnenblust-Hille constants \((C_n)^\infty_{n=1}\) with \(\lim_{n\to \infty}(C_{n+1} -C_n)=0\). (English) Zbl 1264.46033
In [Ann. Math. (2) 32, 600–622 (1931; Zbl 0001.26901, JFM 57.0266.05 )], H. F. Bohnenblust and E. Hille showed that, for every positive integer \(n\), there is \(C_n>0\) such that \[ \left(\sum_{i_1,\dots,i_n=1}^N|U(e_{i_1},e_{i_2},\dots, e_{i_m})|^{{2n\over n+1}} \right)^{{n+1\over2n}}\leq C_n\sup_{|z_j|<1}|U(z_1,\dots,z_n)| \] for every positive integer \(N\) and for every \(n\)-linear mapping \(U: {\mathbb C}^N\times\cdots\times {\mathbb C}^N\to {\mathbb C}\), where \((e_i)_{i=1}^N\) denotes the canonical basis for \({\mathbb C}^N\). In this paper, the authors show that, if \((K_n)_{n=1}^\infty\) is the sequence of optimal constants for the multilinear Bohnenblust-Hille inequality and there is \(M\in [-\infty, \infty]\) with \(\lim_{n\to\infty}(K_{n+1}-K_n)=M\), then \(\lim_{n\to\infty}(K_{n+1}- K_n)=0\). Upper bounds for these optimal constants are also provided with these bounds differing in the real and complex cases. The authors also provide a new proof of the fact that the exponent of \({2n}\over {n+1}\) in both the Bohnenblust-Hille multilinear and polynomial inequality is sharp.

46G25 (Spaces of) multilinear mappings, polynomials
32A07 Special domains in \({\mathbb C}^n\) (Reinhardt, Hartogs, circular, tube) (MSC2010)
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