Landau-Kolmogorov inequalities for semigroups and groups

Abstract

Abstract.An elementary functional analytic argument is given showing how inequalities of the form \\f(k)\f < A^ll/H""*!!/00!!*, where/is a real, n-times differentiable function and ||-|| denotes the sup norm on (0, oo) (or (-00,00)), yield corresponding inequalities, \Akx\" < ^¿M"-\A"x\ , for generators of linear contraction semigroups (or groups) on arbitrary Banach spaces with norm ||. 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.

Norm Inequalities for Generators of Analytic Semigroups and Cosine Operator Functions

“…We denote by A the space of analytic functions / from I a to a Banach space X and by ||/ (/:) ||oo the supremum of ||/ (/:) (z)|| in z over I a for * = 0,1,.. Applying the inequality (4) to g a , we get ||/ w (zo)||^5",,(a)||/||'^/">||/<"»||^, where «",*(«) = C njk {r(ln + \)} k /"{rw + I)}" 1 and this yields (3). We recall that a family {T(z) : z G /«} of operators in L(X) is called an analytic semigroup in I a if (i) z -> T(z) is analytic in I a , (ii) T(0) = / and lim 2 _^z 6 / a T(z)x -x for every x G X and (iii) T(zi + zi) -T(z{)T{zi) for zi,Z2 G / a .…”

“…x G /} for k = 0,1,..., n, where || • || denotes the norm in X. Following an idea essentially due to E. M. Stein [11], Z. Ditzian [5] and M. W. Certain and T. G. Kurtz [3] (cf. also P. R. Chernoff [4]) proved the following theorems: THEOREM A. Suppose/ G C (n) (R + ,X).…”

“…In connection with Theorem B, M. W. Certain and T. G. Kurtz [3] raised the question as to whether the constants D n k appearing in the inequalities of Theorem B can be improved if the strongly continuous semigroup is supposed to be also analytic.…”

A Landau–Kolmogorov inequality for generators of families of bounded operators

“…In [7], it is shown that Landau's inequality implies the Kallman-Rota inequality via the Hahn-Banach theorem in a straightforward way (see also [5]). Moreover, the inequality (1.1) allows one to reduce the constant 4/3 to 6/5 in the inequality (1.2) when A generates a strongly continuous contraction cosine function, see [27,Theorem 3].…”

On the Kolmogorov Inequality for M Φ -Norm

“…This result was extended by Stein to L p -norm [7], by Bang to any Orlicz norm [1] and by Bang and Le to N È -norm [2]. The Kolmogorov-Stein inequality and its variants are a problem of interest for many mathematicians and have various applications (see, for example [3,9] and their references).…”