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

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 $\frac{{\mathrm{\partial }}^{d}f}{\mathrm{\partial }{x}_1\dots \mathrm{\partial }{x}_d}$ belongs to the Sobolev space $W^{1,\mathrm{\infty }}$.