On Landau-Kolmogorov type inequalities for charges and their applications

V.F. Babenko; V.V. Babenko; O.V. Kovalenko; N.V. Parfinovych

doi:10.15421/242301

On Landau-Kolmogorov type inequalities for charges and their applications

V.F. Babenko (Oles Honchar Dnipro National University), https://orcid.org/0000-0001-6677-1914
V.V. Babenko (Drake University), https://orcid.org/0000-0003-4859-4437
O.V. Kovalenko (Oles Honchar Dnipro National University), https://orcid.org/0000-0002-0446-1125
N.V. Parfinovych (Oles Honchar Dnipro National University), https://orcid.org/0000-0002-3448-3798

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 $$$\mathbb{R}^d$$$, $$$d\geqslant 1$$$, 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 $$$W^{1,\infty}$$$.

Keywords

Landau-Kolmogorov type inequality; Stechkin's problem; gradient; charge; mixed derivative

MSC 2020

26D10; 41A17; 41A44

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References

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