Landau–Kolmogorov inequality

In mathematics, the Landau–Kolmogorov inequality, named after Edmund Landau and Andrey Kolmogorov, is the following family of interpolation inequalities between different derivatives of a function f defined on a subset T of the real numbers:^[1]

${\displaystyle \|f^{(k)}\|_{L_{\infty }(T)}\leq C(n,k,T){\|f\|_{L_{\infty }(T)}}^{1-k/n}{\|f^{(n)}\|_{L_{\infty }(T)}}^{k/n}{\text{ for }}1\leq k<n.}$

On the real line[]

For k = 1, n = 2 and T = [c,∞) or T = R, the inequality was first proved by Edmund Landau^[2] with the sharp constants C(2, 1, [c,∞)) = 2 and C(2, 1, R) = √2. Following contributions by Jacques Hadamard and Georgiy Shilov, Andrey Kolmogorov found the sharp constants and arbitrary n, k:^[3]

${\displaystyle C(n,k,\mathbb {R} )=a_{n-k}a_{n}^{-1+k/n}~,}$

where a_n are the Favard constants.

On the half-line[]

Following work by Matorin and others, the extremising functions were found by Isaac Jacob Schoenberg,^[4] explicit forms for the sharp constants are however still unknown.

Generalisations[]

There are many generalisations, which are of the form

${\displaystyle \|f^{(k)}\|_{L_{q}(T)}\leq K\cdot {\|f\|_{L_{p}(T)}^{\alpha }}\cdot {\|f^{(n)}\|_{L_{r}(T)}^{1-\alpha }}{\text{ for }}1\leq k<n.}$

Here all three norms can be different from each other (from L₁ to L_∞, with p=q=r=∞ in the classical case) and T may be the real axis, semiaxis or a closed segment.

The Kallman–Rota inequality generalizes the Landau–Kolmogorov inequalities from the derivative operator to more general contractions on Banach spaces.^[5]

Notes[]

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