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]

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

On the real line

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

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

where an 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

\|f^{(k)}\|_{L_q(T)} \le K \cdot {\|f\|^\alpha_{L_p(T)}} \cdot {\|f^{(n)}\|^{1-\alpha}_{L_r(T)}}\text{ for }1\le 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

  1. ↑ Weisstein, E.W. "Landau-Kolmogorov Constants". MathWorld--A Wolfram Web Resource.
  2. ↑ Landau, E. (1913). "Ungleichungen für zweimal differenzierbare Funktionen". Proc. London Math. Soc. 13: 43–49.
  3. ↑ Kolmogorov, A. (1962). "On Inequalities Between the Upper Bounds of the Successive Derivatives of an Arbitrary Function on an Infinite Integral". Amer. Math. Soc. Translations. 1–2: 233–243.
  4. ↑ Schoenbergfirst=I.J. (1973). "The Elementary Case of Landau's Problem of Inequalities Between Derivatives.". Amer. Math. Monthly 80: 121–158.
  5. ↑ Kallman, Robert R.; Rota, Gian-Carlo (1970), "On the inequality \Vert f^{\prime} \Vert^{2}\leqq4\Vert f\Vert\cdot\Vert f''\Vert", Inequalities, II (Proc. Second Sympos., U.S. Air Force Acad., Colo., 1967), New York: Academic Press, pp. 187–192, MR 0278059.