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Landau-Kolmogorov inequalities for semigroups and groups

Authors: Melinda W. Certain and Thomas G. Kurtz
Journal: Proc. Amer. Math. Soc. 63 (1977), 226-230
MSC: Primary 47D05
MathSciNet review: 0458242
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Abstract: An elementary functional analytic argument is given showing how inequalities of the form $ {\left\Vert {{f^{(k)}}} \right\Vert^n} \leqslant {K_{n,k}}{\left\Vert f \right\Vert^{n - k}}{\left\Vert {{f^{(n)}}} \right\Vert^k}$, where f is a real, n-times differentiable function and $ \left\Vert \cdot \right\Vert$ denotes the sup norm on $ (0,\infty )$ (or $ ( - \infty ,\infty )$), yield corresponding inequalities, $ \vert{A^k}x{\vert^n} \leqslant {K_{n,k}}\vert x{\vert^{n - k}}\vert{A^n}x{\vert^k}$, for generators of linear contraction semigroups (or groups) on arbitrary Banach spaces with norm $ \vert \cdot \vert$. Since Landau, Kolmogorov, Schoenberg and Cavaretta have established the function inequalities with the best possible constants, this argument gives the generator inequalities with the best possible constants for general Banach spaces extending work of Kallman and Rota, Hille and others. Questions concerning the best possible constants for specific Banach spaces remain open.

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  • [1] A. S. Cavaretta, Jr., An elementary proof of Kolmogorov's theorem, Amer. Math. Monthly 81 (1974), 480-486. MR 0340517 (49:5269)
  • [2] M. Certain, Some theorems on semigroups and groups of operators, Ph.D. thesis, Univ. of Wisconsin, 1974.
  • [3] Z. Ditzian, Some remarks on inequalities of Landau and Kolmogorov, Aequationes Math. 12 (1975), 145-151. MR 0380503 (52:1403)
  • [4] -, Inverse theorems for functions in $ {L_p}$ and other spaces, Proc. Amer. Math. Soc. 54 (1976), 80-82. MR 0393958 (52:14765)
  • [5] H. Gindler and J. Goldstein, Dissipative operator versions of some classical inequalities, J. Analyse Math. 28 (1975), 213-238. MR 0482361 (58:2434)
  • [6] J. Goldstein, On improving the constants in the Kolmogorov inequalities, Tulane Univ., 1976 (preprint).
  • [7] E. Hille, Generalizations of Landau's inequality to linear operators, Linear Operators and Approximation, Birkhäuser, Basel, 1972, pp. 20-32. MR 0402535 (53:6354)
  • [8] J. Holbrook, A Kallman-Rota inequality for nearly Euclidean spaces, Advances in Math. 14 (1974), 335-345. MR 0454732 (56:12980)
  • [9] P. R. Kallman and G.-C. Rota, On the inequality $ {\left\Vert {f'} \right\Vert^2} \leqslant 4\left\Vert f \right\Vert\,\left\Vert {f''} \right\Vert$, Inequalities. II. Academic Press, New York and London, 1970, pp. 187-192. MR 0278059 (43:3791)
  • [10] A. N. Kolmogorov, On inequalities between the upper bounds of the successive derivatives of an arbitrary function on an infinite interval, Amer. Math. Soc. Transl. (1) 2 (1962), 233-243.
  • [11] E. Landau, Einige Ungleichungen für zweimal differenziebare Funktionen, Proc. London Math. Soc. (2) 13 (1913), 43-49.
  • [12] I. J. Schoenberg, The elementary cases of Landau's problem of inequalities between derivatives, Amer. Math. Monthly 80 (1973), 121-158. MR 0315070 (47:3619)
  • [13] I. J. Schoenberg and A. Cavaretta, Solution of Landau's problem concerning higher derivatives on the halfline, MRC T.S.R. 1060, Madison, Wis., 1970.
  • [14] W. Trebels and U. Westphal, A note on the Landau-Kallman-Rota-Hille inequality, Linear Operators and Approximation, Birkhäuser, Basel, 1972, pp. 115-119. MR 0402536 (53:6355)

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Keywords: Semigroup generators, group generators, Kallman-Rota Inequality
Article copyright: © Copyright 1977 American Mathematical Society

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