Sobolev estimates for (pseudo)‐differential operators and applications

Abstract

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.

On local continuous solvability of equations associated to elliptic and canceling linear differential operators

“…Indeed, from the fact that L is elliptic we easily see that ∇ L is elliptic as well. Furthermore, [14,Lemma4.1] together with the assumption that the system L be linearly independent, shows that ∇ L is canceling.…”

“…Here the operator d L,−1 = d * L,−1 is understood to be zero. The operator A( ⋅ , D) is ellipitic and canceling for k ∉ {1, n − 1} (see Section 4 [14] for details), so that for each x 0 ∈ Ω there exists an neighborhood U ⊂ Ω of x 0 and C > 0 such that the inequality:…”

“…A series of results concerning on local L 1 estimates for linear differential operators has been studied by J. Hounie and T. Picon in the setting of elliptic systems of complex vector fields, complexes and pseudocomplexes ( [12], [13]). The following characterization of local L 1 estimates for operators A(x, D) was proved in [14], namely: Theorem 1.1. Assume, as before, that A(⋅, D) is a linear differential operator of order ν between the spaces E and F .…”

“…Examples of canceling operators satisfying (⋆) can be founded in [14]; this is the case in particular for operators associated to elliptic system of complex vector fields (see [12], [13]). The canceling property for linear differential operators was originally defined by Van Schaftingen [19] in the setup of homogeneous operators with constant coefficients A(D) and stands out by several applications (and characterizations) in the theory of a priori estimates in L 1 norm (see for instance [20] for a brief description).…”

Estimates for Solutions to Fokker–Planck–Kolmogorov Equations with Integrable Drifts

“…In the recent years, there has been a growing interest to various L 1 -estimates for second order partial differential operators, see, e.g., [1], [11], [12], [18], [21], [24], [27], and [29]- [32], where additional references can be found. Their main feature is that classical L p -estimates for solutions to second order elliptic equations valid for p > 1 do not extend directly to the case p = 1 (see [26], [15], [23], [14], [11], and [19, with some number C(p, d) provided that p > 1, but there is no such estimate for p = 1 if d > 1.…”