A Brief Approach to a Riemann-Hilbert Problem on Quasi-circles

The authors introduce the Sobolev space \(H^{{1\over 2}}(\Gamma)\) $$H^{\frac{1}{2}}(\mathrm{\Gamma })$$ on a quasi-circle Γ and give a fast approach to the jump formula which gives a decomposition of an element in \(H^{{1\over 2}}(\Gamma)\) $$H^{\frac{1}{2}}(\mathrm{\Gamma })$$ as the boundary values of two Dirichlet functions in the complementary domains of Γ.