On the hilbert boundary value problem for holomorphic function in sobolev spaces

The holomorphic solution φ of the Hilbert boundary value problem Re (a + ib) φ = g on γ in a disk D bounded by the simple closed curve γ has been solved in the space of Hölder-continuously differentiable functions C₁, C_1,α (D) by many authors. Under the assumptionthat g belongs to the Slobodecky space W_s,p (γ), s = 1 ? 1/p,1 < p < ∞ it is shown here that the problem has a uniquesolution in the Sobolev space W_1,p: (D). An a-priori estimate for the norm of in W_{1 p}(D)is given.