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
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