L^p estimates on a Cauchy type transform

We first show that the fact that the Beurling transform preserves the L^2 norm for smooth functions of compact support allows us to define the Beurling transform on any L^2 function. We the introduce linear transform P that differs from the Cauchy transform by a constant constant depending on the function. Thus form smooth functions, the conformal part of the gradient of P is the Beurling transform and the anti-conformal part of the gradient of P is the identity. With a view to establishing version of these identities that hold for L^p functions (and weak gradients) we establish that P is bounded f \in L^p for p bigger than 2 and that P(f) is 1-2/p Holder continuous.