This is an announcement for the paper "On the hypercontractivity of the polynomial Bohnenblust--Hille inequality" by Daniel Pellegrino and Juan B. Seoane-Sepulveda.
Abstract: Recently, it was proved that the polynomial Bohnenblust--Hille inequality is hypercontractive, i.e., there is a constant $C>1$ (from now on called constant of hypercontractivity) so that $\frac{D_{m}}{D_{m-1}}=C$ for every $m$, where $D_{m}$ are constants satisfying the polynomial Bohnenblust--Hille inequality. For the case of multilinear mappings a recent result shows that $\lim _{m\rightarrow\infty}\frac{C_{m}}{C_{m-1}}=1$, where $C_{m}$ are constants satisfying the multilinear Bohnenblust--Hille inequality. So it is natural to wonder if there exist constants $D_{m}$'s such that $\lim_{m\rightarrow\infty}\frac{D_{m}% }{D_{m-1}}=1$. In this note we provide lower estimates for the polynomial Bohnenblust--Hille inequality with strong numerical evidence supporting that it is not possible to obtain such $D_{m}.$ Besides the qualitative information, and to the best of our knowledge, this is the first time in which non-trivial lower bounds for the constants of the polynomial Bohnenblust--Hille inequality are presented. We also show that the constant of hypercontractivity $C$ is so that $1.1542\leq C\leq1.8529$, providing as well explicit formulae to improve the lower estimate $1.1542.$ It is our belief that variations of the ideas introduced in this paper can be used for the search of the optimal constants for the polynomial Bohnenblust--Hille inequality.
Archive classification: math.FA
Remarks: 2 figures
Submitted from: dmpellegrino@gmail.com
The paper may be downloaded from the archive by web browser from URL
http://front.math.ucdavis.edu/1201.3873
or
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alspach＠math.okstate.edu