Bohnenblust–Hille inequalities for Lorentz spaces via interpolation

Abstract

We prove that the Lorentz sequence space ${\ell }_{2m∕\left(m+1\right),1}$ 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 ${ℂ}^n$. 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. An application to the theory of Dirichlet series improves a deep result of Balasubramanian, Calado and Queffélec.