Anisotropic Ornstein non inequalities

TL;DR

This paper explores conditions for a priori estimates of anisotropic homogeneous differential operators in L^1 norm, extending to martingale transforms, and investigates when certain inequalities hold.

Contribution

It provides new insights into the existence of inequalities for anisotropic differential operators and their relation to martingale transforms.

Findings

Identifies conditions under which the L^1 inequalities hold for anisotropic operators.

Establishes connections between differential operator inequalities and martingale transform problems.

Provides criteria for the validity of a priori estimates in anisotropic settings.

Abstract

We investigate existence of a priori estimates for differential operators in $L^1$ norm: for anisotropic homogeneous differential operators $T_1, \ldots , T_{\ell}$, we study the conditions under which the inequality $$ \|T_1 f\|_{L_1(\mathbb{R}^d)} \lesssim \sum\limits_{j = 2}^{\ell}\|T_j f\|_{L_1(\mathbb{R}^d)} $$ holds true. We also discuss a similar problem for martingale transforms.