Anisotropic Ornstein noninequalities

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.