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On some sharp Landau--Kolmogorov--Nagy type inequalities in Sobolev spaces of multivariate functions

V. F. Babenko, V. V. Babenko, O. V. Kovalenko, N. V. Parfinovych

TL;DR

This work develops sharp Nagy-type inequalities for functions in $W^{1,p}(C)$ on open convex cones $C\subset \mathbb{R}^d$, providing additive and multiplicative bounds on the $L_\infty$-norm in terms of the $L_p$-norm of the gradient and a boundary seminorm, with explicit sharp constants. It then extends these Nagy-type results to the Radon–Nikodym derivatives of charges and to bounds on mixed derivatives over cones $C=\mathbb{R}_+^m\times\mathbb{R}^{d-m}$, proving sharpness for $m=0,1$. The main method pairs a sharp Ostrowski-type estimate with scale-smoothing operators $S_h$ and discrete difference constructions to derive both additive and multiplicative inequalities, with extremal functions $f_{e,h}$ (and their derivatives) achieving equality. These results yield a unified framework for bounding unbounded operators by bounded ones and have potential applications in approximation problems in functional analysis, particularly for charges and mixed-derivative operators.

Abstract

For a function $f$ from the Sobolev space $W^{1,p}(C)$ ($C\subset\mathbb{R}^d$ is an open convex cone), a sharp inequality that estimates $\| f\|_{L_{\infty}}$ via the $L_{p}$-norm of its gradient and a seminorm of the function is obtained. With the help of this inequality, a sharp inequality is proved, which estimates the ${L_{\infty}}$-norm of the Radon--Nikodym derivative of a charge defined on Lebesgue measurable subsets of $C$ via the $L_p$-norm of the gradient of this derivative and a seminorm of the charge. In the case, when $C=\mathbb{R}_+^m\times \mathbb{R}^{d-m}$, $0\le m\le d$, we obtain inequalities that estimate the ${L_{\infty}}$-norm of a mixed derivative of a function $f\colon C\to \mathbb{R}$ using its ${L_{\infty}}$-norm and the $L_p$-norm of the gradient of the function's mixed derivative.

On some sharp Landau--Kolmogorov--Nagy type inequalities in Sobolev spaces of multivariate functions

TL;DR

This work develops sharp Nagy-type inequalities for functions in on open convex cones , providing additive and multiplicative bounds on the -norm in terms of the -norm of the gradient and a boundary seminorm, with explicit sharp constants. It then extends these Nagy-type results to the Radon–Nikodym derivatives of charges and to bounds on mixed derivatives over cones , proving sharpness for . The main method pairs a sharp Ostrowski-type estimate with scale-smoothing operators and discrete difference constructions to derive both additive and multiplicative inequalities, with extremal functions (and their derivatives) achieving equality. These results yield a unified framework for bounding unbounded operators by bounded ones and have potential applications in approximation problems in functional analysis, particularly for charges and mixed-derivative operators.

Abstract

For a function from the Sobolev space ( is an open convex cone), a sharp inequality that estimates via the -norm of its gradient and a seminorm of the function is obtained. With the help of this inequality, a sharp inequality is proved, which estimates the -norm of the Radon--Nikodym derivative of a charge defined on Lebesgue measurable subsets of via the -norm of the gradient of this derivative and a seminorm of the charge. In the case, when , , we obtain inequalities that estimate the -norm of a mixed derivative of a function using its -norm and the -norm of the gradient of the function's mixed derivative.
Paper Structure (6 sections, 4 theorems, 33 equations)

This paper contains 6 sections, 4 theorems, 33 equations.

Table of Contents

  1. Introduction
  2. Notations and some auxiliary results
  3. Nagy type inequality
  4. Landau--Kolmogorov type inequalities for charges
  5. Inequalities that contain the $L_p$-norm of the gradient of the mixed derivative
  6. Some applications

Key Result

Theorem 1

Let $p\in (d,\infty]$, $h>0$ and $f\in W^{1,p}(h K\cap C)$. Then The inequality is sharp. It becomes equality for the function $\alpha\cdot f +\beta$, where $\alpha,\beta\in\mathbb R$ and

Theorems & Definitions (6)

  • Theorem 1
  • Theorem 2
  • proof
  • Theorem 3
  • Theorem 4
  • proof