Remarks on Ornstein's non-inequality in $\mathbb{R}^{2\times 2}$

TL;DR

This paper provides a concise proof of Ornstein's $L^1$ non-inequality for first- and second-order operators in two dimensions, utilizing a simple two-dimensional laminate supported on three points.

Contribution

It introduces a minimalistic proof technique for Ornstein's non-inequality in two dimensions using a three-point laminate, simplifying previous approaches.

Findings

Concise proof of Ornstein's $L^1$ non-inequality in 2D.

Utilizes a two-dimensional laminate supported on three points.

Simplifies understanding of Ornstein's inequality in low dimensions.

Abstract

We give a very concise proof of Ornstein's $L^1$ non-inequality for first- and second-order operators in two dimensions. The proof just needs a two-dimensional laminate supported on three points.