Preprint

Inserted: 6 jun 2019
Last Updated: 6 jun 2019

Pages: 8 pages
Year: 2019

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 \[ \mathbb{A} u=f\quad\text{subject to}\quad\mathcal{C} f=0\quad\text{ in }\mathbb{R}^n \] to satisfy the estimates \[ | |D^{k-j}u | |_{L^{\frac{n}{n-j}}(\mathbb{R}^n)}\leq c| |f| |_{L^1(\mathbb{R}^n)} \] for $j\in \{1,\ldots,\min\{k,n-1\}\}$ and \[ | |D^{k-n}u| |_{L^{\infty}(\mathbb{R}^n)}\leq c| |f| |_{L^1(\mathbb{R}^n)} \] when $k\geq n$.