## Analysis & PDE

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

### Anisotropic Ornstein noninequalities

#### Abstract

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

$∥T1f∥L1(ℝd) ≲∑ j=2ℓ∥T jf∥L1(ℝd)$

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

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

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

Permanent link to this document
https://projecteuclid.org/euclid.apde/1510843425

Digital Object Identifier
doi:10.2140/apde.2017.10.351

Mathematical Reviews number (MathSciNet)
MR3619873

Zentralblatt MATH identifier
1362.35013

#### Citation

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. https://projecteuclid.org/euclid.apde/1510843425

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