Nagy type inequalities in metric measure spaces and some applications - Babenko, V.F.; Babenko, V.V.; Kovalenko, O.V.; Parfinovych, N.V. (Carpathian Mathematical Publications, 2023)
We obtain a sharp Nagy type inequality in a metric space (X,ρ) with measure μ that estimates the uniform norm of a function using its \|·\|_H^ω-norm determined by a modulus of continuity ω, and a seminorm that is defined on a space of locally integrable functions. We consider charges u that are defined on the set of μ-measurable subsets of X and are absolutely continuous with respect to μ. Using the obtained Nagy type inequality, we prove a sharp Landau-Kolmogorov type inequality that estimates the uniform norm of a Radon-Nikodym derivative of a charge via a \|·\|_H^ω-norm of this derivative, and a seminorm defined on the space of such charges. We also prove a sharp inequality for a hypersingular integral operator. In the case X=R_+^m× R^d-m, 0md, we obtain inequalities that estimate the uniform norm of a mixed derivative of a function using the uniform norm of the function and the \|·\|_H^ω-norm of its mixed derivative.