On the growth of the optimal constants of the multilinear Bohnenblust-Hille inequality (Q2435408)

The multilinear Bohnenblust--Hille inequality was proved in 1931, as a fundamental tool for the solution of an important problem in the framework of Dirichlet series. It states that for all positive integers \(n\geq2\) there is a constant \(C=C(n)\geq1\) such that \[ \left( \sum\limits_{i_{1},\dots,i_{n}=1}^{\infty}\left| T(e_{i_{^{1}} },\dots,e_{i_{n}})\right| ^{\frac{2n}{n+1}}\right) ^{\frac{n+1}{2n}}\leq C\left\| T\right\| \] for all continuous \(n\)-linear forms \(T:c_{0}\times\cdots\times c_{0} \rightarrow\mathbb{K}\) (here \(\mathbb{K}\) denotes \(\mathbb{R}\) or \(\mathbb{C} \)). The particular case \(n=2\) is the famous Littlewood \(4/3\) inequality. The exact value (or even the exact growth) of the constants \(C(n)\) is still an important open problem, of crucial importance in different areas. In this paper, among other results, the author proves an interesting dichotomy theorem that sheds some light on the behavior of \(C(n)\).