Some Sharp Landau–Kolmogorov–Nagy-Type Inequalities in Sobolev Spaces of Multivariate Functions

Abstract

For a function f from the Sobolev space W^1,p(C), where C ⊂ ℝ^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, 0 ≤ m ≤ d, we obtain inequalities estimating the L_∞-norm of a mixed derivative of the function f : C → ℝ via its L_∞-norm and the L_p-norm of the gradient of mixed derivative of this function.