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 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.