Endpoint Sobolev inequalities for vector fields and cancelling operators

Abstract

The injectively elliptic vector differential operators $A(\mathrm{D})$ from $V$ to $E$ on ${\mathbb{R}}^n$ such that the estimate \[ \Vert D^\ell u\Vert_{L^{n/(n - \ell)} (\mathbb{R}^n)} \le \Vert A (\mathrm{D}) u\Vert_{L^1 (\mathbb{R}^n)} \] holds can be characterized as the operators satisfying a cancellation condition \[ \bigcap_{ξ\in \mathbb{R}^n \setminus \{0\}} A (ξ)[V] = \{0\}\;. \] These estimates unify existing endpoint Sobolev inequalities for the gradient of scalar functions (Gagliardo and Nirenberg), the deformation operator (Korn-Sobolev inequality by M.J. Strauss) and the Hodge complex (Bourgain and Brezis). Their proof is based on the fact that $A(\mathrm{D})u$ lies in the kernel of a cocancelling differential operator.

8 pages

References in corpus (4)

• Critical ${\mathrm{L}}^p$-differentiability of ${\mathrm{BV}}^{\mathbb{A}}$-maps and canceling operators
• Bourgain-Brezis Estimates on Symmetric Spaces of Non-compact Type
• Variations on a proof of a borderline Bourgain-Brezis Sobolev embedding theorem
• Limiting Sobolev and Hardy inequalities on stratified homogeneous groups