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 ${L}^{1}$ norm: for anisotropic homogeneous differential operators ${T}_{1},\dots ,{T}_{\ell }$, we study the conditions under which the inequality

$\parallel {T}_{1}f{\parallel }_{{L}_{1}\left({ℝ}^{d}\right)}\lesssim \sum _{j=2}^{\ell }\parallel {T}_{j}f{\parallel }_{{L}_{1}\left({ℝ}^{d}\right)}$

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