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A duality approach to gradient Hölder estimates for linear divergence form elliptic equations

Olli Saari, Yuanlin Sun, Hua-Yang Wang, Yuanhong Wei

TL;DR

The paper addresses gradient Hölder and Calderón–Zygmund-type regularity for divergence-form elliptic equations with rough coefficients. It develops a sparse domination framework rooted in duality between local Hardy spaces and Hölder spaces, using constant-coefficient theory as input to drive a Hölder-regularity iteration. A central contribution is a duality-based bound that yields $|\nabla u|$ in $C^{\alpha}$ and a concrete $\varepsilon$-sparse bound for the gradient in the Hölder case, with extensions to uniformly continuous coefficients via a simpler $L^{p}$-based argument. The results unify Schauder and Calderón–Zygmund estimates under minimal coefficient regularity, and illuminate how local Hardy-space duality underpins Schauder theory and related estimates. The approach also provides a flexible path to local weighted and Hardy-space–type bounds through sparse domination.

Abstract

We prove a sparse bound in the context of Schauder theory for divergence form elliptic partial differential equations. In addition, we show how an iteration argument inspired by sparse domination bounds can be used to deduce gradient reverse Hölder inequalities for equations with non-constant coefficients from the theory for constant coefficient equations. We deal with coefficient matrices whose entries are either Hölder continuous or just uniformly continuous, leading to different results. The purpose of the approach is to highlight the connection between Schauder theory and duality of local Hardy spaces and local Hölder spaces.

A duality approach to gradient Hölder estimates for linear divergence form elliptic equations

TL;DR

The paper addresses gradient Hölder and Calderón–Zygmund-type regularity for divergence-form elliptic equations with rough coefficients. It develops a sparse domination framework rooted in duality between local Hardy spaces and Hölder spaces, using constant-coefficient theory as input to drive a Hölder-regularity iteration. A central contribution is a duality-based bound that yields in and a concrete -sparse bound for the gradient in the Hölder case, with extensions to uniformly continuous coefficients via a simpler -based argument. The results unify Schauder and Calderón–Zygmund estimates under minimal coefficient regularity, and illuminate how local Hardy-space duality underpins Schauder theory and related estimates. The approach also provides a flexible path to local weighted and Hardy-space–type bounds through sparse domination.

Abstract

We prove a sparse bound in the context of Schauder theory for divergence form elliptic partial differential equations. In addition, we show how an iteration argument inspired by sparse domination bounds can be used to deduce gradient reverse Hölder inequalities for equations with non-constant coefficients from the theory for constant coefficient equations. We deal with coefficient matrices whose entries are either Hölder continuous or just uniformly continuous, leading to different results. The purpose of the approach is to highlight the connection between Schauder theory and duality of local Hardy spaces and local Hölder spaces.

Paper Structure

This paper contains 7 sections, 16 theorems, 167 equations.

Table of Contents

  1. Introduction
  2. Notational conventions
  3. Local Hardy spaces
  4. Preliminaries on constant coefficient equations
  5. Equations with Hölder continuous coefficients
  6. A sparse estimate
  7. Equations with uniformly continuous coefficients

Key Result

Theorem 1.2

Let $\varepsilon \in (0,1)$. Let $0 < \lambda \le \Lambda< \infty$. Let $\alpha \in (0,1)$ and set $p = n/(n + \alpha)$. Let $Q$ be a cube and let $A \in C^{\alpha}(6Q;{\mathbb R}^{n \times n})$ satisfy eq:intro-matrix in $6Q$. Let $F \in C^{\infty}(6Q;{\mathbb R}^{n})$. Assume that $u \in W_0^{1,2} Then, given $g \in C^{\infty}(6Q)$ and $\varphi_Q \in C^{\infty}_c(2Q)$ with for all $\gamma \in {

Theorems & Definitions (36)

  • Definition 1.1
  • Theorem 1.2
  • Definition 3.1
  • Definition 3.2
  • Theorem 3.3
  • Definition 3.4
  • Corollary 3.5
  • Definition 3.6
  • Lemma 3.7
  • Lemma 3.8
  • ...and 26 more