In this paper we derive asymptotic-in-time linear estimates in Hardy spaces Hp(Rn) for the Cauchy problem for evolution operators with structural dissipation. The obtained estimates are a natural extension of the known Lp- Lq estimates, 1 ≤ p≤ q≤ ∞, for these models. Different, standard, tools to work in Hardy spaces, are used to derive optimal estimates

In this paper we present a strong local version of the GagliardoNirenberg estimate that holds for elliptic systems of vector fields with smooth complex coefficients. We also consider L 1 estimates on forms analogous to those known in the case of the de Rham complex on R N .

In this work we show that if A(x,D) is a linear differential operator of order ν with smooth complex coefficients in Ω⊂double-struckRN from a complex vector space E to a complex vector space F, the Sobolev a priori estimate ∥u∥Wν−1,N/(N−1)≤C∥A(x,D)u∥L1holds locally at any point x0∈Ω if and only if A(x,D) is elliptic and the constant coefficient homogeneous operator Aν(x0,D) is canceling in the sense of Van Schaftingen for every x0∈Ω which means that ⋂ξ∈double-struckRN∖{0}aν(x0,ξ)[E]={0}.Here Aν(x,D) is the homogeneous part of order ν of A(x,D) and aν(x,ξ) is the principal symbol of A(x,D). This result implies and unifies the proofs of several estimates for complexes and pseudo‐complexes of operators of order one or higher proved recently by other methods as well as it extends —in the local setup— the characterization of Van Schaftingen to operators with variable coefficients.