2016
DOI: 10.1002/mana.201500017
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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 hom… Show more

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Cited by 5 publications
(13 citation statements)
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“…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.…”
Section: Consider the Setmentioning
confidence: 99%
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“…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.…”
Section: Consider the Setmentioning
confidence: 99%
“…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:…”
Section: Consider the Setmentioning
confidence: 99%
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“…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.…”
Section: Introductionmentioning
confidence: 99%