Landau and Hadamard Inequalities in W ¹ H ^ω(ℝ+) and W ¹ H ^ω(ℝ)

Abstract

In this chapter we describe the extremal functions of the Kolmogorov-Landau problem

$$ {{\left\| {f'} \right\|}_{{\mathbb{L}\infty \left( I \right)}}} \to \sup ,\quad f \in {{W}^{1}}{{H}^{\omega }}(I),\quad {{\left\| f \right\|}_{{\mathbb{L}\infty \left( I \right)}}} \leqslant B, $$

((0.0))

for all concave moduli of continuity ω and I = ℝ or ℝ₊. These results generalize the solution of the problem (0.0) for ω(t) = t by E. Landau [54] in the case I = ℝ₊ and J. Hadamard [31] in the case I = ℝ. A number of other elementary cases of the Kolmogorov-Landau problem for ω(t) = t are discussed by I. J. Schoenberg in [72].