The paper deals with inequalities for homogeneous linear differential operators. More exactly, let \(A(D)\) be a homogeneous linear differential operator of order \(k\in {\mathbb N}\) on \({\mathbb R}^n\) from a vector space \(V\) to a vector space \(E\). Such an operator is called canceling if \[ \bigcup_{\xi \in {\mathbb R}^n\setminus \{0\}} A(\xi) [V] = \{0\}.\tag{1}\] Then it is proved that the estimate \[ \| D^{k-1} u \|_{L^{n/(n-1)}} \leq \| A(D) \|_{L^1}\] holds if and only if \(A(D)\) is elliptic and canceling. This result covers some classical inequalities like the Gagliardo-Nirenberg-Sobolev inequality, the Korn-Sobolev inequality and the Hodge-Sobolev estimates for differential forms due to J. Bourgain and H. Brezis. Extensions of (1) like \[ \| D^{k-1} u \|_{\dot{F}^s_{p,q}} \leq \| A(D) \|_{L^1}, \] where \(\dot{F}^s_{p,q}\) is the homogeneous Triebel-Lizorkin space, and \[ \| D^{k-1} u \|_{\dot{B}^s_{p,q}} \leq \| A(D) \|_{L^1}, \] where \(\dot{B}^s_{p,q}\) is the homogeneous Besov space, are discussed as well.

Reviewer: Winfried Sickel (Jena)

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