This is an announcement for the paper "The Bohnenblust-Hille inequality for real homogeneous polynomials is hypercontractive and this result is optimal" by D. Pellegrino and J.B. Seoane-Sepulveda.

Abstract: It was recently shown by A. Montanaro that the low growth of the constants of the multilinear Bohnenblust-Hille inequality, for real scalars, plays a crucial role in Quantum Information Theory. In this paper, among other results, we show that the polynomial Bohnenblust--Hille inequality, for real scalars, is hypercontractive; the case of complex scalars was recently proved in the paper "The Bohnhenblust-Hille inequality for homogeneous polynomials is hypercontractive" , by Defant, Frerick, Ortega-Cerd'{a}, Ouna"{\i}es, and Seip (Annals of Mathematics, 2011). Our proof is presented in a simple form, by making use of a deep result that dates back to Erd"os (Bull. Amer. Math. Soc., 1947). We also show, in strong contrast to what happens in the case of multilinear mappings, that the hypercontractive growth of these constants cannot be improved. The complex version of this result remains still open.

Archive classification: math.FA

Mathematics Subject Classification: 46G25, 30B50

Submitted from: jseoane@mat.ucm.es

The paper may be downloaded from the archive by web browser from URL

http://front.math.ucdavis.edu/1209.4632

or

http://arXiv.org/abs/1209.4632