A note on estimates for elliptic systems with \(L^1\) data (Q2280087)

The authors study elliptic systems \[ {\mathbb{A}} u = f \qquad \text{subject to} \quad {\mathcal{C}} f = 0 \quad\text{in}\quad {\mathbb{R}}^n \] for \({\mathbb{A}} : C^\infty_c({\mathbb{R}}^n,V) \to C^\infty_c({\mathbb{R}}^n,E)\) a \(k\,\)th order homogeneous linear elliptic differential operator, \({\mathcal{C}}: C^\infty_c({\mathbb{R}}^n,E) \to C^\infty_c({\mathbb{R}}^n,F)\) an \(l\,\)th order homogeneous linear differential operator, and \(V,E,F\) finite dimensional inner product spaces, in order to obtain characterizations of the conditions on \({\mathbb{A}}\) and \({\mathcal{C}}\) such that the estimates \[ \|D^{k-j}u\|_{L^{\frac{n}{n-j}}({\mathbb{R}}^n)}\leq c\,\|f\|_{L^1({\mathbb{R}}^n)} \] hold for \(j\in \{1,\ldots,\min\{k,n-1\}\}\) or \[ \|D^{k-n}u\|_{L^{\infty}({\mathbb{R}}^n)}\leq c\|f\|_{L^1({\mathbb{R}}^n)} \] if \(k \geq n\).