This is an announcement for the paper "There exist multilinear Bohnenblust--Hille constants $(C_{n})_{n=1}^{\infty}$ with $\displaystyle \lim_{n\rightarrow \infty}(C_{n+1}-C_{n}) =0.$" by Daniel Pellegrino, Juan Seoane-Sepulveda and Diana M. Serrano-Rodriguez.
Abstract: After almost 80 decades of dormancy, the Bohnenblust--Hille inequalities have experienced an effervescence of new results and sightly applications in the last years. The multilinear version of the Bohnenblust--Hille inequality asserts that for every positive integer $m\geq1$ there exists a sequence of positive constants $C_{m}\geq1$ such that% [ \left( \sum\limits_{i_{1},\ldots,i_{m}=1}^{N}\left\vert U(e_{i_{^{1}}}% ,\ldots,e_{i_{m}})\right\vert ^{\frac{2m}{m+1}}\right) ^{\frac{m+1}{2m}}\leq C_{m}\sup_{z_{1},\ldots,z_{m}\in\mathbb{D}^{N}}\left\vert U(z_{1},\ldots ,z_{m})\right\vert ] for all $m$-linear forms $U:\mathbb{C}^{N}\times\cdots\times\mathbb{C}% ^{N}\rightarrow\mathbb{C}$ and positive integers $N$ (the same holds with slightly different constants for real scalars). The first estimates obtained for $C_{m}$ showed exponential growth but, only very recently, a striking new panorama emerged: the polynomial Bohnenblust--Hille inequality is hypercontractive and the multilinear Bohnenblust--Hille inequality is subexponential. Despite all recent advances, the existence of a family of constants $\left( C_{m}\right) _{m=1}^{\infty}$ so that [ \lim_{n\rightarrow\infty}\left( C_{n+1}-C_{n}\right) =0 ] has not been proved yet. The main result of this paper proves that such constants do exist. As a consequence of this, we obtain new information on the optimal constants $\left( K_{n}\right) _{n=1}^{\infty}$ satisfying the multilinear Bohnenblust--Hille inequality. Let $\gamma$ be Euler's famous constant; for any $\varepsilon>0$, we show that [ K_{n+1}-K_{n}\leq\left( 2\sqrt{2}-4e^{\frac{1}{2}\gamma-1}\right) n^{\log_{2}\left( 2^{-3/2}e^{1-\frac{1}{2}\gamma}\right) +\varepsilon}, ] for infinitely many $n$. Numerically, choosing a small $\varepsilon$, [ K_{n+1}-K_{n}\leq0.8646\left( \frac{1}{n}\right) ^{0.4737}% ] for infinitely many $n.$
Archive classification: math.FA
Submitted from: dmpellegrino@gmail.com
The paper may be downloaded from the archive by web browser from URL
http://front.math.ucdavis.edu/1207.0124
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