Limiting Sobolev inequalities for vector fields and canceling linear differential operators

TL;DR

This paper characterizes when certain Sobolev inequalities hold for vector fields, showing they are valid if and only if the associated linear differential operator is elliptic and canceling, extending classical inequalities.

Contribution

It provides a necessary and sufficient condition for Sobolev inequalities involving vector fields, introducing the concepts of canceling and cocanceling operators, and extending results to fractional and Lorentz spaces.

Findings

Sobolev inequalities hold iff the operator is elliptic and canceling.

Introduces the class of cocanceling operators and characterizes them.

Extends classical inequalities to fractional and Lorentz spaces.

Abstract

The estimate [\lVert D^{k-1}u\rVert_{L^{n/(n-1)}} \le \lVert A(D)u \rVert_{L^1} ] 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 [\bigcap_{\xi \in \mathbb{R}^n \setminus {0}} A(\xi)[V]={0}.] 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.