On the optimal constants of the Bohnenblust--Hille and Hardy--Littlewood inequalities

Daniel Pellegrino

Published 2015-01-05Version 1

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.