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 $X={\mathbb R}_+^m× {\mathbb R}^{d-m}$, $0\le m\le d$, we obtain inequalities that estimate the uniform norm of a mixed derivative of a function using the uniform norm of the function and the $\|·\|_{H^ω}$-norm of its mixed derivative.