Revistes Catalanes amb Accés Obert (RACO)

Trace de Cauchy pour certaines fonctions localement intégrables sur un ouvert borné de $\mathbb{C}$

I. Harlouchet

Resum


Let $\Omega$ be a bounded open subset of $\mathbb{C}$, and let $f$ be a distribution on $\Omega$ such that $\overline{\partial} f$ is a Radon measure of finite total mass. By means of the Cauchy transform, we introduce the "Cauchy trace" of $f$, which takes values in the set of analytic functionals on the boundary $\partial \Omega$ of $\Omega$. The properties of this application are studied in detail. For instance, the characterization of its kernel is discussed according to the properties of the boundary $\partial \Omega$. Roughly speaking, the Cauchy trace allows us to interpret the Cauchy-Pompeiu formula in the same way as the Sobolev trace allows to interpret the Stokes formula.

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