Prabhu Janakiraman
University of Illinois, Urbana-Champaign

Abstract: Let $ B$ denote the Beurling-Ahlfors transform defined on $ L^p(R^2)$, $ 1<p<\infty$ by

$\displaystyle Bf(z) = -\frac{1}{\pi}\textrm{p.v.}\int_{R^2}\frac{f(w )}{(z-w)^2}dm(w),$ (1)

The celebrated conjecture of T. Iwaniec states that its $ L^p$ norm $ \Vert B\Vert _p=p^*-1$ where $ p^*= \max\{p,\frac{p}{p-1}\}$. In this paper, the new upper estimate

$\displaystyle \Vert B\Vert _p\leq 1.575\,(p^*-1), \hspace{3mm} 1<p<\infty$

is found using martingale methods and further analysis. This is joint work with Rodrigo Ba$ \tilde{n}$uelos.