Burkholder integrals, Morrey’s problem and quasiconformal mappings

Astala, Kari; Iwaniec, Tadeusz; Prause, István; Saksman, Eero

doi:10.1090/S0894-0347-2011-00718-2

Burkholder integrals, Morrey’s problem and quasiconformal mappings
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by Kari Astala, Tadeusz Iwaniec, István Prause and Eero Saksman

J. Amer. Math. Soc. 25 (2012), 507-531

DOI: https://doi.org/10.1090/S0894-0347-2011-00718-2

Published electronically: October 6, 2011

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Abstract:

Inspired by Morrey’s Problem (on rank-one convex functionals) and the Burkholder integrals (of his martingale theory) we find that the Burkholder functionals $\text {B}_p$, $p \geqslant 2$, are quasiconcave, when tested on deformations of the identity $f\in \mathrm {Id} + {\mathscr C}^\infty _\circ (\Omega )$ with $\text {B}_p (Df(x)) \geqslant 0$ pointwise, or equivalently, deformations such that $|D f |^2 \leqslant \frac {p}{p-2}J_f$. In particular, quasiconcavity holds in explicit neighbourhoods of the identity map. Among the many immediate consequences, this gives the strongest possible $\mathscr L^p$-estimates for the gradient of a principal solution to the Beltrami equation $f_{\bar {z}}=\mu (z) f_z$, for any $p$ in the critical interval $2\leqslant p \leqslant 1+1/\|\mu \|_\infty$.

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