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 ) , 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.