On the Kolmogorov-Stein inequality (Q1288855)

Let \(\Phi: [0,\infty)\to [0,\infty]\) be a non-vanishing, non-decreasing and concave function such that \(\Phi(0)= 0\), and let \(N_\phi\) be the space of measurable functions \(f\) on \(\mathbb{R}\) such that \(\|f\|_{N_\phi}= \int^\infty_0 \Phi(\lambda_f(y)) dy<\infty\), where \(\lambda_f(y)= \text{meas}\{x:|f(y)|> y\}\), \(y\geq 0\). It is proved that if \(f\) and its generalized \(n\)th derivative \(f^{(n)}\) are in \(N_\phi\), then \(f^{(k)}\in N_\phi\) for \(0< k<n\) and \(\|f^{(k)}\|^n_{N_\phi}\leq C_{k,n}\|f\|^{n- k}_{N_\phi}\|f^{(k)}\|^k_{N_\phi}\). An analogous inequality is formulated in case of periodic functions.