Sharp Higher Integrability Theory, Part I: The $L^p$-norm of the Beurling--Ahlfors transform

We confirm a conjecture posed by T. Iwaniec in 1982: the $L^p$ norm of the Beurling–Ahlfors transform is $p-1$ if $p\ge 2$, and $(p-1{)}^{-1}$ if $1<p\le 2$.