Local Hardy-Littlewood-Sobolev inequalities for canceling elliptic differential operators

Abstract

Let A(x,D) be an elliptic linear differential operator of order ν with smooth complex coefficients in Ω⊂RN from a complex vector space E to a complex vector space F. In this paper we show that if ℓ∈R satisfies 0<ℓ<N and ℓ≤ν, then the estimate(∫RN|Pν−ℓ(x,D)u(x)|q|x|−N+(N−ℓ)qdx)1/q≤C‖A(x,D)u‖L1 holds locally for every u∈Cc∞(U;E) and 1≤q<N/(N−ℓ) assuming A(x,D) is canceling, i.e. ⋂ξ∈RN∖{0}A(x0,ξ)[E]={0} for each x0∈Ω. Here Pν−ℓ(x,D) is a properly supported pseudo-differential operator in Hörmander's class OpS1,δν−ℓ(Ω), 0≤δ<1. This statement is inspired in a new characterization of Hardy-Littlewood-Sobolev inequalities for elliptic and canceling homogeneous operators A(D) with constant coefficients that extends and unifies several results stemming from the classical Hardy-Sobolev estimates. Variants and applications are presented with focus on operators associated to elliptic systems of complex vector fields.