Nagy type inequalities in metric measure spaces and some applications

TL;DR

This paper derives sharp Nagy-type inequalities in metric measure spaces and leverages them to obtain sharp Landau–Kolmogorov type bounds for Radon–Nikodym derivatives of charges, as well as for generalized hypersingular integrals. The authors introduce a modulus-based function class $H^\omega(X)$ and the averaging operator $S_h$, proving tight bounds that link the uniform norm to $H^\omega$-norms and relevant seminorms, with extremal functions constructed to demonstrate sharpness. They extend these results to metric Sobolev spaces via upper gradients, to charges via Radon–Nikodym derivatives, and to hypersingular operators, and finally establish inequalities for mixed derivatives on the plane-like product space $\mathbb{R}_+^m\times\mathbb{R}^{d-m}$, including multiplicative bounds when $\omega(t)=t^\alpha$. Overall, the work provides a unified, sharp framework for estimating intermediate-derivative norms in broad geometric settings with concrete extremals and applications.

Abstract

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 $\|\cdot\|_{H^ω}$ -- norm determined by a modulus of continuity $ω$, and a seminorm that is defined on a space of locally integrable functions. We consider charges $ν$ 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 $\|\cdot\|_{H^ω}$-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 $X=\mathbb{R}_+^m\times \mathbb{R}^{d-m}$, $0\le m\le d$, we obtain inequalities that estimate the uniform norm of a mixed derivative of a function using the uniform norm of the function and the $\|\cdot\|_{H^ω}$-norm of its mixed derivative.

Theorems & Definitions (23)

• Theorem 1
• Remark 1
• Lemma 1
• proof
• Theorem 2
• Remark 2
• proof
• Corollary 1
• proof
• Theorem 3