Given martingales and such that is differentially subordinate to , Burkholder obtained the sharp inequality , where . What happens if one of the martingales is also a conformal martingale? Bañuelos and Janakiraman proved that if and is a conformal martingale differentially subordinate to any martingale , then . In this paper, we establish that if , is conformal, and is any martingale subordinate to , then , where is the smallest positive zero of a certain solution of the Laguerre ordinary differential equation. We also prove the sharpness of this estimate and an analogous one in the dual case for . Finally, we give an application of our results. Previous estimates on the -norm of the Beurling–Ahlfors transform give at best as . We improve this to as .
Alexander Borichev. Prabhu Janakiraman. Alexander Volberg. "Subordination by conformal martingales in and zeros of Laguerre polynomials." Duke Math. J. 162 (5) 889 - 924, 1 April 2013. https://doi.org/10.1215/00127094-2081372