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

TL;DR

This paper establishes sharp inequalities in Sobolev spaces that relate the supremum norm of functions and their derivatives to their gradient norms and seminorms, extending classical inequalities to multivariate functions and measures.

Contribution

The paper introduces new sharp inequalities in Sobolev spaces for multivariate functions, including estimates for Radon--Nikodym derivatives and mixed derivatives, generalizing classical Landau--Kolmogorov--Nagy inequalities.

Findings

Derived sharp inequalities for functions in Sobolev spaces involving their supremum norms and gradients.

Extended inequalities to Radon--Nikodym derivatives of charges on convex cones.

Provided estimates for mixed derivatives in multivariate Sobolev spaces.

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 $\| f\|_{L_{\infty}}$ 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 $f\colon C\to \mathbb{R}$ using its ${L_{\infty}}$-norm and the $L_p$-norm of the gradient of the function's mixed derivative.