Iwaniec's conjecture Does anyone know whether there is any geometric applications of the Iwaniec's conjecture on $ l^p $ bound of Beurling Alfhors transform (or the complex Hilbert transform). One application could have been was Behrling's conjecture (solved). Is there is any thing else that might be of geometric significance.
For p=2, it is known that $ l^p $ bound is 1 and it coincides with the conjecture.It is already known to be $ l^p $ bounded by Calderon, Zygmund and even more. Is the exact bound is of any geometric significance.
 A: Iwaniec conjecture is closely connected to the important Morrey's conjecture on the relationship between rank one convexity and quasiconvexity.
You will find a nice discussion in the Section 5 of the following survey by Banuelos
http://arxiv.org/pdf/1012.4850v2.pdf
A: As already pointed out, the Iwaniec conjecture has close links with calculus of variations. It has already been settled for "stretch" functions (with a rotation $e^{i\theta}$ introduced), ie, functions of the form 
\begin{equation*}
f(re^{i\theta})=g(r)e^{-i\theta}
\end{equation*}
for positive and smooth $g$ on the upper-half real plane with end points equal to $0$. 
The Iwaniec conjecture is closely linked with rank one and quasiconvexity, specifically to Morrey's and Sverak's conjecture. Note that if the Banuelos-Wang conjecture is true, then the Iwaniec conjecture will be true. If the Banuelos-Wang conjecture is not true, then Morrey's conjecture would be settled for the case $m=n=2$. 
The truth of the Iwaniec conjecture would have consequences for QC mappings in $\mathbb{R}^n$. If the Iwaniec conjecture does hold, then it would be a stronger variation of Astala's area  distortion theorem on QC mappings. 
The Beurling-Ahlfors transform already has links with algebra/geometry, eg, research has been conduced by Banuelos and Lindeman on an operator $B_n$ (for $n=2$) that can be identified with $B$ for $f:\mathbb{R}^n\to\Gamma$ with Grassmann algebra $\Gamma$. So the truth of the conjecture may well open up new unforeseen research possibilities in areas such as this (see links below).
The truth of the Iwaniec conjecture would tell us how the Cauchy-Riemann operators $\partial f,~\overline{\partial}f$ are like differentially subordinate harmonic functions, or differentially subordinate martingales (which is in fact the key to settling the conjecture). 
Here are a couple of papers I have studied that may help with your question:


*

*Ahlfors-Beurling Operator on Radial Function by Alexander Volberg

*Nonlinear Cauchy-Riemann Operators in $\mathbb{R}^n$ by Tadeusz Iwaniec

*A Martingale Study of the Beurling-Ahlfors Transform in $\mathbb{R}^n$.
This is something I am keen on so I will keep an eye on progress of the conjecture. 
