On Landau -- Kolmogorov type inequalities for charges and their applications

Abstract

In this article we prove sharp Landau--Kolmogorov type inequalities on a class of charges defined on Lebesgue measurable subsets of a cone in ${\mathbb{R}}^d$, $d\ge 1$, that are absolutely continuous with respect to the Lebesgue measure. In addition we solve the Stechkin problem of approximation of the Radon--Nikodym derivative of such charges by bounded operators and two related problems. As an application, we also solve these extremal problems on classes of essentially bounded functions $f$ such that their distributional partial derivative $\frac{{\partial }^{d}f}{\partial x_1\dots \partial x_d}$ belongs to the Sobolev space $W^{1,\infty }$.

Cited by in corpus (1)

• On some sharp Landau--Kolmogorov--Nagy type inequalities in Sobolev spaces of multivariate functions