Schindler, Ian and Tintarev, Cyril (2013) The limiting {S}obolev-{D}irac inequality. Rostocker Mathematisches Kolloquium (n° 68).

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Abstract

We prove the critical Dirac-Sobolev inequality for $p\in(1,3)$.
It follows that the Dirac Sobolev spaces are equivalent to classical
Sobolev spaces if and only if $p\in(1,3)$. We prove the cocompactness
of $L^{p^{*}}(\mathbb{R}^{3})$ in }{\normalsize{$\dot{\mathbf{H}}^{1,p}(\mathbb{R}^{3})$.
As an application, we prove the existence of minimizers to a class
of isoperimetric problems.