Hardy-Sobolev inequalities for vector fields and canceling linear differential operators

TL;DR

This paper proves Hardy and Sobolev inequalities for vector fields governed by elliptic and canceling linear differential operators, extending classical inequalities to a broader class of operators.

Contribution

It establishes new Hardy and Sobolev inequalities for elliptic, canceling differential operators, and analyzes the necessity of these conditions.

Findings

Hardy inequality holds for elliptic, canceling operators

Sobolev inequality established under similar conditions

Conditions are shown to be necessary for the inequalities to hold

Abstract

Given a homogeneous k-th order differential operator $A (D)$ on $\mathbb{R}^n$ between two finite dimensional spaces, we establish the Hardy inequality $$\int_{\mathbb{R}^n} \frac{\lvert D^{k-1}u\rvert}{\lvert x \rvert} \,\mathrm{d} x \leq C \int_{\mathbb{R}^n} \lvert A(D)u\rvert $$ and the Sobolev inequality $$\lVert D^{k-n} u\rVert_{L^{\infty}(\mathbb{R}^n)}\leq C \int_{\mathbb{R}^n} \lvert A(D)u\rvert $$ when $A(D)$ is elliptic and satisfies a recently introduced cancellation property. We also study the necessity of these two conditions.