A Landau-Kolmogorov inequality for generators of families of bounded operators
A Landau-Kolmogorov type inequality for generators of a class of strongly continuous families of bounded linear operators on Banach space is investigated. In particular, the authors establish a Landau-Kolmogorov inequality when \(A\) is a generator of certain \((a,k)\) regularized resolvents defined on a Banach space \(X\). Specifically, they show that, if \((a,k)\) is a pair satisfying the \(CP\)-condition and \[ C_{a,k}:=\sup_{t>0}\frac{(a*a*k)(t)k(t)}{(k*a)^ 2 (t)} < \infty, \] and if \(A\) is a generator of an \((a,k)\)-regularized resolvent \(\{R(t)\}_{t\geq 0}\) such that \(\| R(t)\| \leq Mk(t)\), \(t\geq0\), with \(M\geq 1\), then the Landau-Kolmogorov inequality \[ \| Ax\|^ 2 \leq 8M^ 2 C_{a,k}\| x\| \| A^ 2 x\|, \] holds for all \(x\in D(A^ 2)\), where \(a\in L^ 1_{loc}(\mathbb R_ +)\) and \(k\in C(\mathbb R_ +)\) are positive functions. Examples are given to illustrate the obtained results.
scientific article; zbMATH DE number 5777335
scientific article