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Journal of the American Mathematical Society

Published by the American Mathematical Society, the Journal of the American Mathematical Society (JAMS) is devoted to research articles of the highest quality in all areas of pure and applied mathematics.

ISSN 1088-6834 (online) ISSN 0894-0347 (print)

The 2020 MCQ for Journal of the American Mathematical Society is 4.79.

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Burkholder integrals, Morrey’s problem and quasiconformal mappings
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by Kari Astala, Tadeusz Iwaniec, István Prause and Eero Saksman PDF
J. Amer. Math. Soc. 25 (2012), 507-531 Request permission

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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Additional Information
  • Kari Astala
  • Affiliation: Department of Mathematics and Statistics, University of Helsinki, P. O. Box 68, Gustaf Hällströmin katu 2b, FI-00014, Helsinki, Finland
  • Email: kari.astala@helsinki.fi
  • Tadeusz Iwaniec
  • Affiliation: 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
  • Email: tiwaniec@syr.edu
  • István Prause
  • Affiliation: Department of Mathematics and Statistics, University of Helsinki, P. O. Box 68, Gustaf Hällströmin katu 2b, FI-00014, Helsinki, Finland
  • Email: istvan.prause@helsinki.fi
  • Eero Saksman
  • Affiliation: Department of Mathematics and Statistics, University of Helsinki, P. O. Box 68, Gustaf Hällströmin katu 2b, FI-00014, Helsinki, Finland
  • MR Author ID: 315983
  • Email: eero.saksman@helsinki.fi
  • Received by editor(s): December 5, 2010
  • Received by editor(s) in revised form: August 19, 2011
  • Published electronically: October 6, 2011
  • © Copyright 2011 American Mathematical Society
    The copyright for this article reverts to public domain 28 years after publication.
  • Journal: J. Amer. Math. Soc. 25 (2012), 507-531
  • MSC (2010): Primary 30C62, 30C70, 49K10, 49K30
  • DOI: https://doi.org/10.1090/S0894-0347-2011-00718-2
  • MathSciNet review: 2869025