A new fractional form of the Landau-Kolmogorov inequality
Let T(t) be a semigroup of contractions on a Banach space, and put \(M(n,\alpha;f)=\sup_{h}\| h^{-\alpha}(T(h)-1)^ nf\|\) for \(n\geq \alpha >0\). Among several results, the author shows the inequality \[ \| h^{-\beta}(T(h)-1)^ kf\| \leq C(\alpha,\beta,n,k)\| f\|^{1-(\beta /\alpha)}M(n,\alpha;f)^{\beta /\alpha}, \] for \(\beta <\alpha\), \(k\geq \beta >0\), and some constant C(\(\alpha\),\(\beta\),n,k). Reviewer's remark: Regrettably, the article does not seem to be well- written. I fail to find in it a definite expression of C(\(\alpha\),\(\beta\),n,k), and Lemma 2.1 cited in the proof of Lemma 2.4 should be read (probably) as Lemma 2.2.
semigroup of contractions on a Banach space
scientific article; zbMATH DE number 4165768