On Ulyanov inequalities in Banach spaces and semigroups of linear operators

JOURNAL OF APPROXIMATION THEORY

卷:160

Article

Trebels, W.¹  Westphal, U.² 

[1] Tech Univ Darmstadt, Fachbereich Math, AG 5, D-64289 Darmstadt, Germany

[2] Leibniz Univ Hannover, Inst Anal, D-30167 Hannover, Germany

关键词: Ulyanov inequality;    Nikolskii inequality;    K-functionals;    Semigroups of operators;    Fractional powers of generators;   

DOI  :  10.1016/j.jat.2008.04.003

来源: Elsevier

【 摘 要 】

Let X, Y be Banach spaces and {T(t): t >= 0} be a consistent, equibounded semigroup of linear operators on X as well as on Y: it is assumed that {T(t)} satisfies a Nikolskii type inequality with respect to X and Y : parallel to T(2t)f parallel to(Y)less than or similar to phi(t)parallel to T(t)f parallel to(X). Then an abstract Ulyanov type inequality is derived between the (modified) K-functionals with respect to (X, D-X((-A)(x))) and (Y, D-Y((-A)(alpha))), alpha > 0, where A is the infinitesimal generator of {T(t)}. Particular choices of X, Y are Lorentz-Zygmund spaces, of {T(t)} are those connected with orthogonal expansions such as Fourier, spherical harmonics, Jacobi, Laguerre, Hermite series. Known characterizations of the K-functionals lead to concrete Ulyanov type inequalities. In particular, results of Ditzian and Tikhonov in the case X = L-p, Y = L-q, 1 <= p < q <= infinity, are partly covered. (C) 2008 Elsevier Inc. All rights reserved.

【 授权许可】

Free