Towards sharp Bohnenblust–Hille constants

Abstract

We investigate the optimality problem associated with the best constants in a class of Bohnenblust–Hille-type inequalities for [Formula: see text]-linear forms. While germinal estimates indicated an exponential growth, in this work we provide strong evidences to the conjecture that the sharp constants in the classical Bohnenblust–Hille inequality are universally bounded, irrespectively of the value of [Formula: see text]; hereafter referred as the Universality Conjecture. In our approach, we introduce the notions of entropy and complexity, designed to measure, to some extent, the complexity of such optimization problems. We show that the notion of entropy is critically connected to the Universality Conjecture; for instance, that if the entropy grows at most exponentially with respect to [Formula: see text], then the optimal constants of the [Formula: see text]-linear Bohnenblust–Hille inequality for real scalars are indeed bounded universally with respect to [Formula: see text]. It is likely that indeed the entropy grows as [Formula: see text], and in this scenario, we show that the optimal constants are precisely [Formula: see text]. In the bilinear case, [Formula: see text], we show that any extremum of the Littlewood’s [Formula: see text] inequality has entropy [Formula: see text] and complexity [Formula: see text], and thus we are able to classify all extrema of the problem. We also prove that, for any mixed [Formula: see text]-Littlewood inequality, the entropy do grow exponentially and the sharp constants for such a class of inequalities are precisely [Formula: see text]. In addition to the notions of entropy and complexity, the approach we develop in this work makes decisive use of a family of strongly non-symmetric [Formula: see text]-linear forms, which has further consequences to the theory, as we explain herein.