Burkholder integrals, Morrey’s problem and quasiconformal mappings

Astala, Kari¹ ; Iwaniec, Tadeusz² ; Prause, István¹ ; Saksman, Eero¹

¹ Department of Mathematics and Statistics, University of Helsinki, P. O. Box 68, Gustaf Hällströmin katu 2b, FI-00014, Helsinki, Finland
² Department of Mathematics, Syracuse University, Syracuse, New York 13244, USA, and Department of Mathematics and Statistics, University of Helsinki, P. O. Box 68, Gustaf Hällströmin katu 2b, FI-00014, Helsinki, Finland

Journal of the American Mathematical Society, Tome 25 (2012) no. 2, pp. 507-531

Résumé

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 $2\leqslant p \leqslant 1+1/\|\mu \|_\infty$.

Détail

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

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