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Proceedings of the American Mathematical Society

ISSN 1088-6826(online) ISSN 0002-9939(print)

 

 

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
DOI: https://doi.org/10.1090/S0002-9939-1977-0458242-2
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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DOI: https://doi.org/10.1090/S0002-9939-1977-0458242-2
Keywords: Semigroup generators, group generators, Kallman-Rota Inequality
Article copyright: © Copyright 1977 American Mathematical Society