We show that by averaging of martingale transforms on the plane one can prove that all weakly quasiregular maps are quasiregular, thus solving a problem of Iwaniec. The more subtle question arises: how good is this averaging procedure. We show that the ``tightest" result can be obtained by averaging special martingale transforms, namely those built on ``french bed" type of tiling the plane.
This is the joint work with Oliver Dragicevic.
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