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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.

##### MSC:
 46G25 (Spaces of) multilinear mappings, polynomials 32A07 Special domains in $${\mathbb C}^n$$ (Reinhardt, Hartogs, circular, tube) (MSC2010)
##### Keywords:
Bohnenblust-Hille inequality; optimal constants
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