Limiting Sobolev inequalities for vector fields and canceling linear differential operators

Abstract

The estimate

is shown to hold if and only if $A(D)$ is elliptic and canceling. Here $A(D)$ is a homogeneous linear differential operator $A(D)$ of order $k$ on ${\mathbb{R}}^n$ from a vector space $V$ to a vector space $E$. The operator $A(D)$ is defined to be canceling if

This result implies in particular the classical Gagliardo–Nirenberg–Sobolev inequality, the Korn–Sobolev inequality and Hodge–Sobolev estimates for differential forms due to J. Bourgain and H. Brezis. In the proof, the class of cocanceling homogeneous linear differential operator $L(D)$ of order $k$ on ${\mathbb{R}}^n$ from a vector space $E$ to a vector space $F$ is introduced. It is proved that $L(D)$ is cocanceling if and only if for every $f\in L^1({\mathbb{R}}^n;E)$ such that $L(D)f=0$, one has $f\in {\dot{W}}^{-1,n/(n-1)}({\mathbb{R}}^n;E)$. The results extend to fractional and Lorentz spaces and can be strengthened using some tools of J. Bourgain and H. Brezis.