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

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 $${\int }_{{\mathbb{R}}^n}\frac{|D^{k-1}u|}{|x|}\mathrm{d}x\le C{\int }_{{\mathbb{R}}^n}|A(D)u|$$ and the Sobolev inequality $$\Vert D^{k-n}u{\Vert }_{L^{\infty }({\mathbb{R}}^n)}\le C{\int }_{{\mathbb{R}}^n}|A(D)u|$$ when $A(D)$ is elliptic and satisfies a recently introduced cancellation property. We also study the necessity of these two conditions.

22 pages, final version with minor changes

References in corpus (3)

• Limiting Sobolev inequalities for vector fields and canceling linear differential operators
• Subelliptic Bourgain-Brezis Estimates on Groups
• Estimates for differential operators of vector analysis involving $L^1$-norm

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