Exact Kolmogorov-Type Inequalities with Bounded Leading Derivative in the Case of Low Smoothness

Abstract

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 $x \in L_\infty ^r$:

$$\left\| {x^{\left( k \right)} } \right\|_q \leqslant \frac{{\left\| {{\phi }_{r - k} } \right\|_q }}{{\left\| {{\phi }_r } \right\|_p^\alpha }}\left\| x \right\|_p^\alpha \left\| {x^{\left( k \right)} } \right\|_\infty ^{1 - \alpha } $$

where $\alpha = \min \left\{ {1 - \frac{k}{r},\frac{{r - k + {1 \mathord{\left/ {\vphantom {1 q}} \right. \kern-\nulldelimiterspace} q}}}{{r + {1 \mathord{\left/ {\vphantom {1 p}} \right. \kern-\nulldelimiterspace} p}}}} \right\}$ and ϕ _r is the perfect Euler spline of order r.