This is an announcement for the paper "On the optimal constants of the Bohnenblust--Hille and inequalities" by Daniel Pellegrino.
Abstract: We find the optimal constants of the generalized Bohnenblust--Hille inequality for $m$-linear forms over $\mathbb{R}$ and with multiple exponents $ \left( 1,2,...,2\right)$, sometimes called mixed $\left( \ell _{1},\ell _{2}\right) $-Littlewood inequality. We show that these optimal constants are precisely $\left( \sqrt{2}\right) ^{m-1}$ and this is somewhat surprising since a series of recent papers have shown that the constants of the Bohnenblust--Hille inequality have a sublinear growth, and in several cases the same growth was obtained for the constants of the generalized Bohnenblust--Hille inequality. This result answers a question raised by Albuquerque et al. (2013) in a paper published in 2014 in the Journal of Functional Analysis. We also improve the best known constants of the generalized Hardy--Littlewood inequality in such a way that an unnatural behavior of the old estimates (that will be clear along the paper) does not happen anymore.
Archive classification: math.FA
Submitted from: pellegrino@pq.cnpq.br
The paper may be downloaded from the archive by web browser from URL
http://front.math.ucdavis.edu/1501.00965
or
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alspach＠math.okstate.edu