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Quelques inégalités pour les potentiels sur un domaine lipschitzien. (Some inequalities for potentials on a Lipschitz domain). (French) Zbl 0604.31002
Let $$\Omega$$ be a domain in $${\mathbb{R}}^ 2$$ delimited by the graph $$\Gamma$$ of a Lipschitzian function, $$V^ 1(\Omega)$$ be the pre- hilbertian space of finite energy functions on $$\Omega$$ and $$V^{1/2}(\Gamma)$$ is the space of ”Lipschitz-square-integrable functions on $$\Gamma$$ ”. It is first proved a Dirichlet type theorem, that is there exists a continuous map $$\gamma$$ from $$V^ 1(\Omega)$$ onto $$V^{1/2}(\Gamma)$$ admitting a continuous harmonic lifting.
Using appropriate definitions of normal and tangential derivatives in $$V^ 1(\Omega)$$ this leads to a Green formula and moreover to the solution of a Neumann problem.
If F(f) is the Cauchy integral of a function f on $$\Gamma$$, the singular Cauchy integral of f is defined by $$I(f)=2\gamma F(f)-f$$ and continuity properties of this map are then investigated.
Reviewer: J.Lacroix
##### MSC:
 31A20 Boundary behavior (theorems of Fatou type, etc.) of harmonic functions in two dimensions 42B20 Singular and oscillatory integrals (Calderón-Zygmund, etc.)