This paper is concerned with the Riemann boundary value problem \[ (\partial /\partial \bar z)\Phi (z)=0,\quad z\in {\mathbb{C}}\setminus \gamma,\quad \Phi^+(t)=G(t)\Phi^-(t)+g(t),\quad t\in \gamma, \] where \(\gamma\) is a closed Lyapunov curve. Under some assumptions it is shown that there exists a solution \(\Phi \in W_{1,p}(D^+)\), \(1<p<\infty\), and the explicit formula of the solution is given as well for the both cases that the index \(G\geq 0\) and the index \(G<0\). Moreover, an a priori estimate of the solution is obtained for both cases.