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

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.