Nagy type inequalities in metric measure spaces and some applications

Keywords:
Nagy type inequality, Landau-Kolmogorov type inequality, Stechkin's problem, charge, modulus of continuity, mixed derivativeAbstract
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 ‖-norm determined by a modulus of continuity \omega, and a seminorm that is defined on a space of locally integrable functions. We consider charges \nu that are defined on the set of \mu-measurable subsets of X and are absolutely continuous with respect to \mu. 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 \|\cdot\|_{H^\omega}-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={\mathbb R}_+^m\times {\mathbb R}^{d-m}, 0\le m\le d, we obtain inequalities that estimate the uniform norm of a mixed derivative of a function using the uniform norm of the function and the \|\cdot\|_{H^\omega}-norm of its mixed derivative.