Burkholder integrals, Morrey’s problem and quasiconformal mappings

Inspired by Morrey’s Problem (on rank-one convex functionals) and the Burkholder integrals (of his martingale theory) we find that the Burkholder functionals B p B_p , p ⩾ 2 p 2 , are quasiconcave, when tested on deformations of the identity f ∈ I d + C ∘ ∞ ( Ω ) f∈ Id + { C}^∞ _∘ (Ω ) with B p ( D f ( x ) ) ⩾ 0 B_p\,(Df(x)) 0 pointwise, or equivalently, deformations such that | D f | 2 ⩽ p p − 2 J f |D f |^2 p/p-2J_f . In particular, quasiconcavity holds in explicit neighbourhoods of the identity map. Among the many immediate consequences, this gives the strongest possible L p L^p -estimates for the gradient of a principal solution to the Beltrami equation f z ¯ = μ ( z ) f z