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.



Iwaniec's conjecture
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>


