Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

A new pointwise inequality for rough operators and applications

Diego Chamorro, Anca-Nicoleta Marcoci, Liviu-Gabriel Marcoci

TL;DR

The paper studies pointwise control for rough singular integral operators $T_Ω$ with kernels on the sphere, establishing a sharp bound that scales with $\|Ω\|_{L^ρ(\mathbb{S}^{n-1})}$ and combines the Hardy–Littlewood maximal operator with Morrey norms. This pointwise estimate is then used to derive Sobolev-type inequalities across Lebesgue, weighted, Orlicz, and Lorentz spaces, including refined bounds with explicit parameters $r$, $σ$, and $θ$ (e.g., $r=\frac{n\mathfrak{q}}{n-\mathfrak{q}}$, $θ=\frac{\mathfrak{q}}{n}$). The authors extend previous $L^{n,\infty}$-based results by allowing $Ω\in L^ρ(\mathbb{S}^{n-1})$ with $1<ρ<n$, and they treat multiple function-space frameworks (Orlicz and Lorentz) with corresponding weight structures. These contributions broaden the applicability of Sobolev-type estimates for rough operators in harmonic analysis and PDE contexts, providing a versatile toolkit for pointwise control and norm bounds in diverse function spaces.

Abstract

We study in this article a new pointwise estimate for ''rough'' singular integral operators. From this pointwise estimate we will derive Sobolev type inequalities in a variety of functional spaces.

A new pointwise inequality for rough operators and applications

TL;DR

The paper studies pointwise control for rough singular integral operators with kernels on the sphere, establishing a sharp bound that scales with and combines the Hardy–Littlewood maximal operator with Morrey norms. This pointwise estimate is then used to derive Sobolev-type inequalities across Lebesgue, weighted, Orlicz, and Lorentz spaces, including refined bounds with explicit parameters , , and (e.g., , ). The authors extend previous -based results by allowing with , and they treat multiple function-space frameworks (Orlicz and Lorentz) with corresponding weight structures. These contributions broaden the applicability of Sobolev-type estimates for rough operators in harmonic analysis and PDE contexts, providing a versatile toolkit for pointwise control and norm bounds in diverse function spaces.

Abstract

We study in this article a new pointwise estimate for ''rough'' singular integral operators. From this pointwise estimate we will derive Sobolev type inequalities in a variety of functional spaces.
Paper Structure (7 sections, 6 theorems, 75 equations)

This paper contains 7 sections, 6 theorems, 75 equations.

Table of Contents

  1. Introduction
  2. Some functional spaces and classical inequalities
  3. Proof of the Theorem \ref{['Theo_Pointwise']}
  4. Proof of the Theorem \ref{['Theo_RefinedSobolevInequalities']}
  5. Weighted inequalities
  6. Refined Sobolev inequalities in the framework of Orlicz spaces
  7. Inequalities in classical Lorentz spaces

Key Result

Theorem 1

Over the space $\mathbb{R}^n$ with $n\geq 2$, consider $\Omega$ a function such that $\Omega \in L^1(\mathbb{S}^{n-1})$, ${\int_{\mathbb{S}^{n-1}}\Omega \ d\sigma=0}$ and such that $\Omega\in L^\rho(\mathbb{S}^{n-1})$ with $1<\rho<n$ and consider the operator $T_{\Omega}$ associated to the function

Theorems & Definitions (6)

  • Theorem 1: Pointwise inequality
  • Theorem 2: Refined Sobolev inequalities
  • Corollary 5.1: One weighted inequality
  • Theorem 3
  • Theorem 4
  • Lemma 7.1