${\mathrm{L}}^1$-estimates for constant rank operators

Abstract

We show that the inequality $$\Vert D^{k-1}(u-\pi u){\Vert }_{{\mathrm{L}}^{n/(n-1)}({\mathbb{R}}^n)}\le c\Vert \mathbb{B}(D)u{\Vert }_{{\mathrm{L}}^1({\mathbb{R}}^n)}$$ holds for vector fields $u\in {\mathrm{C}}_c^{\infty }$ if and only if $\mathbb{B}$ is canceling. Here $\pi$ denotes the ${\mathrm{L}}^2$-orthogonal projection onto the kernel of the $k$-homogeneous differential operator $\mathbb{B}(D)$ of \emph{constant rank} on ${\mathbb{R}}^n$. Other critical embeddings are established.

10 pages