A Landau-Kolmogorov inequality for Lorentz spaces (Q1890453)

The Landau-Kolmogorov inequality for differentiable functions on the half line asserts that if \(f,f^{(n)}\in L_{\infty}(\mathbb{R}_{+})\) then \(f^{(k)}\in L_{\infty}(\mathbb{R}_{+})\) and \[ \| f^{(k)}\| _{\infty}\leq C_{k,n}^{+}\| f\| _{\infty}^{1-k/n}\| f^{(n)}\| _{\infty}^{k/n} \] for \(k\in\{1,\dots,n-1\}\). Here \(C_{k,n}^{+}\) denote the best constants. It is proved that a similar result (with the same constants \(C_{k,n}^{+})\) works when \(L_{\infty}(\mathbb{R}_{+})\) is replaced by a Lorentz space \(N_{\Phi}(\mathbb{R}_{+}).\)