Nagy type inequalities in metric measure spaces and some applications

We obtain a sharp Nagy type inequality in a metric space $(X,\rho )$ with measure $\mu$ that estimates the uniform norm of a function using its $\mathrm{\parallel }\cdot {\mathrm{\parallel }}_{H^{\omega }}$ – 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 $\mathrm{\parallel }\cdot {\mathrm{\parallel }}_{H^{\omega }}$-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 $\mathrm{\parallel }\cdot {\mathrm{\parallel }}_{H^{\omega }}$-norm of its mixed derivative.