TY - JOUR
AU - V. Babenko
AU - V. Babenko
AU - O. Kovalenko
AU - N. Parfinovych
PY - 2023/10/24
Y2 - 2023/12/01
TI - Some sharp Landau-Kolmogorovâ€“Nagy-type inequalities in Sobolev spaces of multivariate functions
JF - Ukrainsâ€™kyi Matematychnyi Zhurnal
JA - Ukr. Mat. Zhurn.
VL - 75
IS - 10
SE - Research articles
DO - 10.3842/umzh.v75i10.7680
UR - https://umj.imath.kiev.ua/index.php/umj/article/view/7680
AB - UDC 517.5For 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.
ER -