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

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 $\mathrm{\parallel }f{\mathrm{\parallel }}_{L_{\mathrm{\infty }}}$ 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 $L_{\mathrm{\infty }}$-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 $L_{\mathrm{\infty }}$-norm of a mixed derivative of a function $f\text{\hspace{0.17em}⁣}:C\to \mathbb{R}$ using its $L_{\mathrm{\infty }}$-norm and the $L_p$-norm of the gradient of the function’s mixed derivative.