On Ulyanov inequalities in Banach spaces and semigroups of linear operators (Q1041628)

An equibounded semigroup \(\{T_X(t): t\geq0\}\) of class \(C_0\) given on a Banach space \((X, \|\cdot\|_X)\) is supposed to be compatible with \(\{T_Y(t): t\geq0\}\) given on a Banach space \((Y, \|\cdot\|_Y)\). The fractional power \((-A)^\alpha\), \(\alpha>0\), of the semigroup generator is introduced according to the second author [\textit{U.\,Westphal}, Compos.\ Math.\ 22, 67--103 (1970; Zbl 0194.15401); in: Applications of fractional calculus in physics (World Scientific, Singapore), 131--170 (2000; Zbl 0989.44002)]: \[ (-A)^\alpha f=\lim_{\varepsilon\to 0} C_{\alpha, r}\int_\alpha^\infty u^{-\alpha,-1}(I-T(u)^r f \,du , \quad 0<\alpha<r, \] and an associated \(K\)-functional is defined as in [\textit{W.\,Trebels} and \textit{U.\,Westphal}, Constr.\ Approx.\ No.\ 3, 355--371 (2003; Zbl 1030.41021); Rend.\ Circ.\ Mat.\ Palermo (2) Suppl.\ 76, 603--620 (2005; Zbl 1144.47035)]. Under the assumption that \(\{T(t): t\geq0\}\) satisfies the Nikolskii type inequality with respect to compatible \(X\) and \(Y\): \(\|T(t)(T(t)f)\|_Y\leq c\varphi(t)\|T(t)f\|_X \) for all \(f\in X\) and \(\varphi\) such that \(\varphi(t)\geq\varphi(s)\), \(s>t>0,\) an abstract Ulyanov type inequality is derived between the (modified) \(K\)-functionals with respect to \((X, D_X((-A)^\alpha))\) and \((Y, D_Y((-A)^\alpha))\), where \(A\) is the generator of the semigroup \(\{T(t): t\geq0\}\). Known characterization of the \(K\)-functionals lead to certain Ulyanov type inequalities.