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We prove that the Lorentz sequence space $\ell_{\frac{2m}{m+1},1}$ is, ina~precise sense, optimal among all symmetric Banach sequence spaces satisfyinga Bohnenblust-Hille type inequality for $m$-linear forms or $m$-homogeneouspolynomials on $\mathbb{C}^n$. Motivated by this result we develop methods fordealing with subtle Bohnenblust-Hille type inequalities in the setting ofLorentz spaces. Based on an interpolation approach and the Blei-Fournierinequalities involving mixed type spaces, we prove multilinear and polynomialBohnenblust-Hille type inequalities in Lorentz spaces with subpolynomial andsubexponential constants. Improving a remarkable result ofBalasubramanian-Calado-Queff\'elec, we show an application to the theory ofDirichlet series.

Bohnenblust-Hille inequalities for Lorentz spaces via interpolation