Iwaniec's Conjecture on The Beurling-Ahlfors Transform

Abstract

Inspired by Astala, Iwaniec, Prause and Saksman's partial result of Morrey's problem regarding rank-one convex and quasiconvex functions on the functionals from Burkholder's martingale theory, we discuss and relate several open problems in different fields of mathematics. In particular, we discuss the theory of Calder\'{o}n and Zygmund regarding the $L^p$-boundedness of the Beurling-Ahlfors transform $\mathscr{B}$ for $1<p<\infty$ to formulate Iwaniec's conjecture regarding the precise operator norms of $\mathscr{B}$. Moreover, we discuss its consequences in the theory of quasiconformal mappings. Finally, we discuss the notions of rank-one convexity and quasiconvexity, motivated by their role in the theory of calculus of variations, and show how a positive answer to Morrey's conjecture implies quasiconvexity of the Burkholder functional, which, in turn, is shown to imply Iwaniec's conjecture.