A duality approach to gradient H\"older estimates for linear divergence form elliptic equations

TL;DR

This paper introduces a duality-based approach to establish gradient H"older estimates for divergence form elliptic equations, connecting Schauder theory with Hardy and H"older space duality.

Contribution

It develops a novel duality approach using sparse bounds to derive gradient H"older estimates for elliptic PDEs with variable coefficients.

Findings

Established sparse bounds in Schauder theory

Derived gradient reverse H"older inequalities

Connected Schauder theory with Hardy and H"older space duality

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 H\"older inequalities for equations with non-constant coefficients from the theory for constant coefficient equations. We deal with coefficient matrices whose entries are either H\"older 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 H\"older spaces.