Some sharp Landau-Kolmogorov–Nagy-type inequalities in Sobolev spaces of multivariate functions
UDC 517.5 For a function f from the Sobolev space W^1,p(C), where CR^d is an open convex cone, we establish a sharp inequality estimating \| f\|_L_∞ 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_∞ -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× R^d-m, 0md, we obtain inequalities estimating the L_∞ -norm of a mixed derivative of the function fC→ R via its L_∞ -norm and the L_p-norm of the gradient of mixed derivative of this function. , number=10, journal=Ukrains’kyi Matematychnyi Zhurnal, author=Babenko, V. and Babenko, V. and Kovalenko, O. and Parfinovych, N., year=2023, month=Oct., pages=1347–1353