  In this article we prove sharp Landau--Kolmogorov type inequalities on a
class of charges defined on Lebesgue measurable subsets of a cone in
$\mathbb{R}^d$, $d\geq 1$, 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 $\frac{\partial ^d f}{\partial x_1\ldots\partial x_d}$
belongs to the Sobolev space $W^{1,\infty}$.
