#### Vol. 9, No. 5, 2016

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Bohnenblust–Hille inequalities for Lorentz spaces via interpolation

### Andreas Defant and Mieczysław Mastyło

Vol. 9 (2016), No. 5, 1235–1258
##### 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.

##### Keywords
Bohnenblust–Hille inequality, Dirichlet polynomials, Dirichlet series, homogeneous polynomials, interpolation spaces, Lorentz spaces
##### Mathematical Subject Classification 2010
Primary: 46B70, 47A53