# A Note on Estimates for Elliptic Systems with $L^1$ Data

## TL;DR

This paper establishes necessary and sufficient conditions for elliptic systems with $L^1$ data to satisfy certain regularity estimates, linking differential operators, constraints, and solution regularity in Euclidean space.

## Contribution

It provides a complete characterization of when solutions to elliptic systems with $L^1$ data meet specific regularity estimates, considering differential constraints and operator compatibility.

## Key findings

- Identifies conditions for $L^1$ data estimates in elliptic systems.
- Characterizes the role of differential constraints in regularity.
- Establishes bounds for derivatives of solutions in Lebesgue spaces.

## Abstract

In this paper we give necessary and sufficient conditions on the compatibility of a $k$th order homogeneous linear elliptic differential operator $\mathbb{A}$ and differential constraint $\mathcal{C}$ for solutions of \begin{align*} \mathbb{A} u=f\quad\text{subject to}\quad \mathcal{C} f=0\quad\text{ in }\mathbb{R}^n \end{align*} to satisfy the estimates \begin{align*} \|D^{k-j}u\|_{L^{\frac{n}{n-j}}(\mathbb{R}^n)}\leq c\|f\|_{L^1(\mathbb{R}^n)} \end{align*} for $j\in \{1,\ldots,\min\{k,n-1\}\}$ and \begin{align*} \|D^{k-n}u\|_{L^{\infty}(\mathbb{R}^n)}\leq c\|f\|_{L^1(\mathbb{R}^n)} \end{align*} when $k\geq n$.