Деякі точні нерівності типу Ландау–Колмогорова–Надя у просторах Соболєва функцій багатьох змінних

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

doi:10.3842/umzh.v75i10.7680

V. Babenko Dnipro National University named after Oles Honchar
V. Babenko Drake University, Des Moines, USA
O. Kovalenko Dnipro National University named after Oles Honchar
N. Parfinovych Dnipro National University named after Oles Honchar

DOI: https://doi.org/10.3842/umzh.v75i10.7680

Keywords: Nagy and Landau -- Kolmogorov type inequality, charge, gradient, mixed derivative

Abstract

UDC 517.5

For a function $f$ from the Sobolev space $W^{1,p}(C),$ where $C\subset R^d$ is an open convex cone, we establish a sharp inequality estimating $\| f\|_{L_{\infty}}$ via the $L_{p}$-norm of its gradient and a seminorm of the function. With the help of this inequality, we prove a sharp inequality estimating the ${L_{\infty}}$-norm of the Radon-Nikodym derivative of a charge defined on Lebesgue measurable subsets of $C$ via the $L_p$-norm of the gradient of this derivative and the seminorm of the charge. In the case where $C=R_+^m\times R^{d-m},$ $0\le m\le d,$ we obtain inequalities estimating the ${L_{\infty}}$-norm of a mixed derivative of the function $f\colon C\to R$ via its ${L_{\infty}}$-norm and the $L_p$-norm of the gradient of mixed derivative of this function.

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Some sharp Landau-Kolmogorov–Nagy-type inequalities in Sobolev spaces of multivariate functions

Abstract

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