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New weighted Riesz-type pointwise inequalities and applications to generalized Sobolev estimates

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

TL;DR

The paper develops a weighted, Riesz-type pointwise framework for rough singular integrals, establishing an $A_\delta$-weighted pointwise bound for the maximal rough operator $T^*_\Omega$ in terms of a weighted Morrey maximal function of the gradient and a Morrey norm of the gradient. It extends previous unweighted and Lorentz-based estimates by allowing $\Omega\in L^\rho(\mathbb{S}^{n-1})$ with $1<\rho<n$ and weights in the broader $A_\delta$ class, under a lower Ahlfors condition. From these pointwise bounds, the authors derive a suite of weighted Sobolev-type inequalities linking $T^*_\Omega(f)$ to $\nabla f$ in Lebesgue and Morrey-type spaces, including an interpolation-style formulation in Morrey spaces. The results provide a general, robust toolkit for obtaining weighted Sobolev estimates in settings with rough kernels and nontrivial weights, with potential applications to PDEs and harmonic analysis in weighted contexts.

Abstract

In this article we study some new pointwise inequalities between rough singular integral operators, weighted maximal functions of the gradient and weighted Morrey spaces. These pointwise estimates will naturally lead us to a new class of weighted Sobolev-type inequalities.

New weighted Riesz-type pointwise inequalities and applications to generalized Sobolev estimates

TL;DR

The paper develops a weighted, Riesz-type pointwise framework for rough singular integrals, establishing an -weighted pointwise bound for the maximal rough operator in terms of a weighted Morrey maximal function of the gradient and a Morrey norm of the gradient. It extends previous unweighted and Lorentz-based estimates by allowing with and weights in the broader class, under a lower Ahlfors condition. From these pointwise bounds, the authors derive a suite of weighted Sobolev-type inequalities linking to in Lebesgue and Morrey-type spaces, including an interpolation-style formulation in Morrey spaces. The results provide a general, robust toolkit for obtaining weighted Sobolev estimates in settings with rough kernels and nontrivial weights, with potential applications to PDEs and harmonic analysis in weighted contexts.

Abstract

In this article we study some new pointwise inequalities between rough singular integral operators, weighted maximal functions of the gradient and weighted Morrey spaces. These pointwise estimates will naturally lead us to a new class of weighted Sobolev-type inequalities.

Paper Structure

This paper contains 4 sections, 3 theorems, 43 equations.

Table of Contents

  1. Introduction and presentation of the results
  2. Some functional spaces, classical inequalities and Muckenhoupt weights properties
  3. Proof of the Theorem \ref{['Theo_Pointwise_Weighted']}
  4. Proof of the Corollaries

Key Result

Theorem 1

Consider the operator $T^*_\Omega$ defined in the expression (Def_OperatorTstar) where the function $\Omega$ is such that $\Omega \in L^1(\mathbb{S}^{n-1})$, ${\int_{\mathbb{S}^{n-1}}\Omega \ d\sigma=0}$ and $\Omega\in L^\rho(\mathbb{S}^{n-1})$ with $1<\rho<n$. Fix $\mathfrak{s}$ a real parameter su If $f: \mathbb{R}^n \longrightarrow \mathbb{R}$ is a regular function such that $\nabla f$ belongs

Theorems & Definitions (3)

  • Theorem 1: $A_\delta$-pointwise inequality
  • Corollary 1.1: Weighted Sobolev estimates
  • Lemma 2.1