Approxımatıon of the Ahlfors-Beurlıng transform in the Lebesgue spaces

The Ahlfors-Beurling transform is one of the important operators in complex analysis. It is the "Hilbert transform" on complex plane. This transform plays an essential role in applications to the theory of quasiconformal mappings and to the Beltrami equation with discontinuous coeffcients. Therefore, approximations of the Ahlfors-Beurling transform are of great interest. This article is devoted to the approximation of the Ahlfors- Beurling transform in the Lebesgue spaces. We prove that the approximating operators are bounded maps in the Lebesgue spaces and converges strongly to the Ahlfors-Beurling transform