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

TL;DR

This paper establishes a key inequality involving L^1 estimates for constant rank differential operators, characterizing when the inequality holds based on the canceling property of the operator.

Contribution

It provides a necessary and sufficient condition for L^1 estimates of derivatives of vector fields in terms of constant rank operators being canceling.

Findings

The inequality holds if and only if the operator is canceling.

Other critical embeddings related to these operators are also established.

The results characterize the precise conditions for L^1 estimates in this setting.

Abstract

We show that the inequality $$ \|D^{k-1}(u-\pi u)\|_{\mathrm{L}^{n/(n-1)}(\mathbb{R}^n)}\leq c\|\mathbb{B}(D) u\|_{\mathrm{L}^1(\mathbb{R}^n)} $$ holds for vector fields $u\in\mathrm{C}^\infty_c$ 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.