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

Abstract

For a function $f$ from the Sobolev space $W^{1,p}(C)$ ($C\subset {\mathbb{R}}^d$ is an open convex cone), a sharp inequality that estimates $\Vert f{\Vert }_{L_{\infty }}$ via the $L_p$-norm of its gradient and a seminorm of the function is obtained. With the help of this inequality, a sharp inequality is proved, which estimates 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 a seminorm of the charge. In the case, when $C={\mathbb{R}}_{+}^m\times {\mathbb{R}}^{d-m}$, $0\le m\le d$, we obtain inequalities that estimate the $L_{\infty }$-norm of a mixed derivative of a function $f:C\to \mathbb{R}$ using its $L_{\infty }$-norm and the $L_p$-norm of the gradient of the function's mixed derivative.

References in corpus (3)

• On approximation of hypersingular integral operators by bounded ones
• On Landau -- Kolmogorov type inequalities for charges and their applications
• Nagy type inequalities in metric measure spaces and some applications