We investigate the existence of a priori estimates for differential operators in the norm: for anisotropic homogeneous differential operators , 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.
"Anisotropic Ornstein noninequalities." Anal. PDE 10 (2) 351 - 366, 2017. https://doi.org/10.2140/apde.2017.10.351