Local $\mathbf {L}^{1}$ estimates for elliptic systems of complex vector fields

Hounie, Jorge; Picon, Tiago

doi:10.1090/s0002-9939-2014-12219-2

Cited by 7 publications

(14 citation statements)

References 25 publications

(2 reference statements)

Supporting

Mentioning

Contrasting

Order By: Relevance

“…Finally, we define dg to be the Haar measure on G, normalized in such a way that the following two integration formula of Harish-Chandra Here ρ is the half sum of positive roots given by (16), and for each a ∈ A, log a is the element in the Lie algebra a of A for which e log a = a. Indeed, (25) can be deduced from (24) as follows: if F is a compactly supported continuous function on G, then…”

Section: Computation Of the Divergencementioning

confidence: 99%

Bourgain–Brezis inequalities on symmetric spaces of non-compact type

Chanillo

Schaftingen

Yung

2017

Journal of Functional Analysis

View full text Add to dashboard Cite

show abstract

Section: Computation Of the Divergencementioning

confidence: 99%

Bourgain–Brezis inequalities on symmetric spaces of non-compact type

Chanillo

Schaftingen

Yung

2017

Journal of Functional Analysis

View full text Add to dashboard Cite

show abstract

“…In particular, if the system

L_{1}, \dots, L_{n}

is involutive then the pseudo‐complex becomes the complex

d_{L}

and Theorem is reduced to (see details in [, Section 3.3]). We now show how to obtain Theorem from Theorem .…”

Section: Examples and Applicationsmentioning

confidence: 99%

“…It is readily seen that

B (x, D)

is elliptic (cf. [, Lemma 3.1]) and to check that

B (x, D)

is canceling it is convenient to choose appropriate generators of the system

scriptL

. In fact, the vectors

{L_{1}, \dots, L_{n}}

span a subbundle of

C T (Ω)

and estimates express a property of this bundle rather than a property of a specific set of generators.…”

Section: Examples and Applicationsmentioning

confidence: 99%

“…We will refer to d L,k , C ∞ , k R n as the pseudo-complex {d L } associated with L on . Theorem 5.3 can be extended to the pseudo-complex {d L } ( [17]):…”

Section: )mentioning

confidence: 99%

“…Estimates in the setup of constant vector fields (, , , ), nilpotent groups (, ) and CR complexes (), have also been considered recently. For the case of operators with variable coefficients in which the vector fields

\partial_{x_{j}}

1 \leq j \leq N

, are replaced by vector fields with smooth complex coefficients that make up an elliptic system, see and .…”

Section: Introductionmentioning

confidence: 99%

See 2 more Smart Citations

Sobolev estimates for (pseudo)‐differential operators and applications

Hounie

Picon

2016

Mathematische Nachrichten

View full text Add to dashboard Cite

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.

show abstract