Bohnenblust-Hille inequalities for Lorentz spaces via interpolation

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-Queffélec, we show an application to the theory of Dirichlet series.

References in corpus (1)

• The Bohnenblust--Hille inequality for homogeneous polynomials is hypercontractive