Sharp inequalities for the Beurling–Ahlfors transform on radial functions

Abstract

For $1\le p\le 2$, we prove sharp weak-type $(p,p)$-estimates for the Beurling–Ahlfors operator acting on the radial function subspaces of $L^p\left(\mathbb{C}\right)$. A similar sharp $L^p$-result is proved for $1<p\le 2$. The results are derived from martingale inequalities which are of independent interest.