Exact Kolmogorov-Type Inequalities with Bounded Leading Derivative in the Case of Low Smoothness
We obtain new unimprovable Kolmogorov-type inequalities for differentiable periodic functions. In particular, we prove that, for r = 2, k = 1 or r = 3, k = 1, 2 and arbitrary q, p ∈ [1, ∞], the following unimprovable inequality holds for functions \(x L_∞ ^r \) : \| x^( k ) \|_q \| φ _r - k \|_q