Analysis & PDE

  • Anal. PDE
  • Volume 10, Number 2 (2017), 351-366.

Anisotropic Ornstein noninequalities

Krystian Kazaniecki, Dmitriy M. Stolyarov, and Michał Wojciechowski

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We investigate the existence of a priori estimates for differential operators in the L1 norm: for anisotropic homogeneous differential operators T1,,T, we study the conditions under which the inequality

T1fL1(d) j=2T jfL1(d)

holds true. Properties of homogeneous rank-one convex functions play the major role in the subject. We generalize the notions of quasi- and rank-one convexity to fit the anisotropic situation. We also discuss a similar problem for martingale transforms and provide various conjectures.

Article information

Anal. PDE, Volume 10, Number 2 (2017), 351-366.

Received: 26 January 2016
Revised: 6 October 2016
Accepted: 14 November 2016
First available in Project Euclid: 16 November 2017

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Digital Object Identifier

Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 26B35: Special properties of functions of several variables, Hölder conditions, etc. 26B25: Convexity, generalizations

Ornstein noninequalities Bellman function martingale transform


Kazaniecki, Krystian; Stolyarov, Dmitriy M.; Wojciechowski, Michał. Anisotropic Ornstein noninequalities. Anal. PDE 10 (2017), no. 2, 351--366. doi:10.2140/apde.2017.10.351.

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