Bernstein inequality and inverse approximation theorem in modular function spaces (Q1266945)

This is an interesting survey on Landau-Hadamard-Kolmogorov type inequalities for norms of intermediate derivatives: \[ \| f^{(k)}\| _{L_q(G)}\leq A\| f\| _{L_p(G)}+B\| f^{(n)}\| _{L_r(G)} \quad\text{or else}\quad \| f^{(k)}\| _{L_q(G)}\leq K\| f\| ^\alpha_{L_p(G)}\| f^{(n)}\| ^\beta_{L_r(G)} \] where \(G\in\{\mathbb R,\mathbb R_+,[a,b]\}\) and \(1\leq p,q,r\leq\infty\), \(k,n\in\mathbb Z_+,\;0\leq k\leq n\). The author discusses the problem of finding exact constants in the above inequalities. Analogous questions are also discussed for some multivariate Kolmogorov type inequalities.