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Anisotropic Ornstein noninequalities. (English) Zbl 1362.35013

Summary: 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 \[ \|T_1 f\|_{L_1(\mathbb{R}^d)} \lesssim \sum_{j = 2}^{\ell}\|T_{j} f\|_{L_1(\mathbb{R}^d)} \] 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.

MSC:

35A23 Inequalities applied to PDEs involving derivatives, differential and integral operators, or integrals
26B35 Special properties of functions of several variables, Hölder conditions, etc.
26B25 Convexity of real functions of several variables, generalizations
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