# On $C^1$, $C^2$, and weak type-$(1,1)$ estimates for linear elliptic operators: Part II

## TL;DR

This paper improves boundary regularity results for elliptic equations with Dini mean oscillation coefficients and extends weak type-(1,1) estimates and Harnack inequalities to boundary cases, enhancing understanding of solution behavior.

## Contribution

It extends boundary regularity and weak type estimates for elliptic equations with Dini mean oscillation coefficients, including non-divergence form and boundary cases.

## Key findings

- Solutions are continuously differentiable up to the boundary under specified conditions.
- Weak type-(1,1) estimates are extended to boundary cases for elliptic operators.
- A Harnack inequality for non-negative adjoint solutions is established at the boundary.

## Abstract

We extend and improve the results in \cite{DK16}: showing that weak solutions to full elliptic equations in divergence form with zero Dirichlet boundary conditions are continuously differentiable up to the boundary when the leading coefficients have Dini mean oscillation and the lower order coefficients verify certain conditions. Similar results are obtained for non-divergence form equations. We extend the weak type-(1, 1) estimates in \cite{DK16} and \cite{Es94} up to the boundary and derive a Harnack inequality for non-negative adjoint solutions to non-divergence form elliptic equations, when the leading coefficients have Dini mean oscillation.