Behavior of the sequence of norms of primitive of a function in Lorentz spaces (Q2275783)

Let \(\Psi: [0,\infty)\to [0,\infty)\) be a nondecreasing, nonzero concave function with \(\Psi(0)= 0\). Let \(N_\psi(\mathbb{R})\) be the Lorentz space of all measurable functions \(f\) such that \[ \| f\|_{N_\psi}= \int^\infty_0 \Psi(\lambda_f(y))\,dy< \infty, \] where \(\lambda_f(y)= \text{mes}\{x:|f(x)|> y\}\), \(y\geq 0\) Let \(I^nf\) be the \(n\)th primitive of \(f\) in the sense of the theory of distributions. It is proved that if \(f\in N_\psi(\mathbb{R})\) and \(I^nf\in N_\psi(\mathbb{R})\) for \(n= 1,2,\dots\), then \[ \lim_{n\to\infty} \| I^nf\|^{1/n}_{N_\psi}= \sigma^{-1}, \] where \(\sigma= \text{inf}\{|\xi|: \xi\in \text{supp\,}\widehat f\}\), \(\widehat f\) being the Fourier transform of \(f\).