As a consequence of the identities we have established previously for the Cauchy transform and Beurling transform, we show that the Beurling transform send the anticonformal part of the gradient to the conformal part of the gradient. We also show that for a smooth function of compact support, the L^2 normal of the conformal part of the gradient is the same and the L^2 norm of the anti-conformal part of the gradient. We use this to show that the Beurling transform preserves the L^2 norm when applied to smooth functions of compact support.
