Bohnenblust-Hille inequalities for Lorentz spaces via interpolation

TL;DR

This paper establishes the optimality of Lorentz space $\,\ell_{\frac{2m}{m+1},1}$ for Bohnenblust-Hille inequalities, introduces interpolation methods for Lorentz spaces, and applies these results to Dirichlet series.

Contribution

It develops new interpolation techniques for Lorentz spaces to prove Bohnenblust-Hille inequalities with subpolynomial constants, advancing the understanding of these inequalities in functional analysis.

Findings

Lorentz space $\,\ell_{\frac{2m}{m+1},1}$ is optimal for Bohnenblust-Hille inequalities.

Established multilinear and polynomial inequalities with subexponential constants.

Applied results to improve the theory of Dirichlet series.

Abstract

We prove that the Lorentz sequence space $\ell_{\frac{2m}{m+1},1}$ is, in a~precise sense, optimal among all symmetric Banach sequence spaces satisfying a Bohnenblust-Hille type inequality for $m$-linear forms or $m$-homogeneous polynomials on $\mathbb{C}^n$. Motivated by this result we develop methods for dealing with subtle Bohnenblust-Hille type inequalities in the setting of Lorentz spaces. Based on an interpolation approach and the Blei-Fournier inequalities involving mixed type spaces, we prove multilinear and polynomial Bohnenblust-Hille type inequalities in Lorentz spaces with subpolynomial and subexponential constants. Improving a remarkable result of Balasubramanian-Calado-Queff\'elec, we show an application to the theory of Dirichlet series.