Subordination by conformal martingales in Lp and zeros of Laguerre polynomials

Abstract

Given martingales W and Z such that W is differentially subordinate to Z, Burkholder obtained the sharp inequality E|W|p≤(p∗−1)pE|Z|p, where p∗=max {p,p/(p−1)}. What happens if one of the martingales is also a conformal martingale? Bañuelos and Janakiraman proved that if p≥2 and W is a conformal martingale differentially subordinate to any martingale Z, then E|W|p≤[(p2−p)/2]p/2E|Z|p. In this paper, we establish that if p≥2, Z is conformal, and W is any martingale subordinate to Z, then E|W|p≤[2(1−zp)/zp]pE|Z|p, where zp 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 1<p<2. Finally, we give an application of our results. Previous estimates on the Lp-norm of the Beurling–Ahlfors transform give at best ‖B‖p≲2p as p→∞. We improve this to ‖B‖p≲1.3922p as p→∞.