Vol. 10, No. 2, 2017

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Anisotropic Ornstein noninequalities

Krystian Kazaniecki, Dmitriy M. Stolyarov and Michał Wojciechowski

Vol. 10 (2017), No. 2, 351–366
Abstract

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.

Keywords
Ornstein noninequalities, Bellman function, martingale transform
Mathematical Subject Classification 2010
Primary: 26B35, 26B25
Milestones
Received: 26 January 2016
Revised: 6 October 2016
Accepted: 14 November 2016
Published: 23 February 2017
Authors
Krystian Kazaniecki
Institute of Mathematics
University of Warsaw
Warsaw
Poland
Dmitriy M. Stolyarov
Institute of Mathematics
Polish Academy of Sciences
Warsaw
Poland
P. L. Chebyshev Research Laboratory
St. Petersburg State University
St. Petersburg
Russia
St. Petersburg Department of Steklov Mathematical Institute
Russian Academy of Sciences
St. Petersburg
Russia
Michał Wojciechowski
Institute of Mathematics
Polish Academy of Sciences
Warsaw
Poland
Institute of Mathematics
University of Warsaw
Warsaw
Poland