Subordination by conformal martingales in L^{p} and zeros of Laguerre polynomials

• Published in: Duke mathematical journal 2013, Vol.162 (no. 5), p.889-924
• Main Authors: Borichev, Alexander, Janakiraman, Prabhu, Volberg, Alexander
• Format: Article
• Subjects: 32A55; 42A15; 42B20; 60G46
• ISSN: 1547-7398
• DOI: 10.1215/00127094-2081372

Given martingales W and Z such that W is differentially subordinate to Z , Burkholder obtained the sharp inequality E|W|^{p}\le(p^{*}-1)^{p}E|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\geq2 and W is a conformal martingale differentially subordinate to any martingale Z , then E|W|^{p}\leq[(p^{2}-p)/2]^{p/2}E|Z|^{p} . In this paper, we establish that if p\geq2 , Z is conformal, and W is any martingale subordinate to Z , then \mathbb{E}|W|^{p}\le[\sqrt{2}(1-z_{p})/z_{p}]^{p}\mathbb{E}|Z|^{p} , where z_{p} 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\lt p\lt 2 . Finally, we give an application of our results. Previous estimates on the L^{p} -norm of the Beurling–Ahlfors transform give at best \|B\|_{p}\lesssim\sqrt{2}p as p\rightarrow\infty . We improve this to \|B\|_{p}\lesssim1.3922p as p\rightarrow\infty .