Remarques sur le calcul symbolique dans certains espaces de Besov à valeurs vectorielles

Allaoui, Salah Eddine

doi:10.5802/ambp.273

Cited by 8 publications

(7 citation statements)

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“…In (r1) we have chosen B s p,∞ (R n ) since the embedding B s p,q (R n ) → B s p,∞ (R n ) is satisfied for all q ≥ 1. Result (r1) is proved in [16] or [1] for instance; the proof of (r2) can be found in 5.3.1-Theorem 2 of [16]. Now (r1) and (r2) lead to the following conjecture: Conjecture 1.1.…”

Section: Introductionmentioning

confidence: 91%

Composition operators on Besov algebras

Moussai¹

2012

Rev. Mat. Iberoam.

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We study the composition operator T f (g) := f • g on Besov spaces B s p,q (R). In the case 1 < p < +∞, p ≤ q ≤ +∞ and s > 1 + (1/p) we will prove that the operator T f takes B s p,q (R) into itself if and only if f (0) = 0 and f belongs locally to B s p,q (R). • If 1 < p < +∞, s > n/p and f ∈ C ∞ (R), then T f takes the Bessel potential spaces H s p (R n) (resp. B s p,q (R n)) into itself, see e.g. Meyer [12], (resp. Peetre [13]). • Dahlberg [9] proved that for 1 ≤ p ≤ +∞ and 1 + (1/p) < m < n/p (m integer), if T f maps the Sobolev spaces W m p (R n) into itself, then f is a linear function. Also, after the work of Marcus and Mizel [11] on W 1 p (R n), Bourdaud [2] proved that for either m ≥ 2 and mp > n, or m = n ≥ 2 and p = 1, the operator T f maps W m p (R n) into itself if and only if f ∈ W m, oc p (R). • The result of Dahlberg has been extended to B s p,q (R n). More precisely, for 1 ≤ p, q ≤ +∞ and either 1 + (1/p) < s < n/p, or 1 + (1/p) = s < n/p and 1 < q ≤ +∞, if the operator T f acts on B s p,q (R n) then f (t) = ct for some real c, see e.g. [15] and [16]. The characterization of those functions f such that T f acts on B s p,q (R n) requires to investigate the necessary conditions. In this sense, we recall the following two results:

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Section: Introductionmentioning

confidence: 91%

Composition operators on Besov algebras

Moussai¹

2012

Rev. Mat. Iberoam.

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“…. If = , the Hölder continuity condition of Theorem 4.5 implies that is Lipschitz-continuous; this condition is well-known to be necessary [2,13,14,41,43].…”

Section: Concrete Uniform Boundedness Principlesmentioning

confidence: 99%

Uniform boundedness principles for Sobolev maps into manifolds

Monteil

Schaftingen

2019

Ann. Inst. H. Poincaré C Anal. Non Linéaire

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Given a connected Riemannian manifold  , an -dimensional Riemannian manifold  which is either compact or the Euclidean space, ∈ [1, +∞) and ∈ (0, 1], we establish, for the problems of surjectivity of the trace, of weak-bounded approximation, of lifting and of superposition, that qualitative properties satisfied by every map in a nonlinear Sobolev space , (,  ) imply corresponding uniform quantitative bounds. This result is a nonlinear counterpart of the classical Banach-Steinhaus uniform boundedness principle in linear Banach spaces.The space, the energy and the topology on this space are independent of the embedding and can be defined intrinsically [30].1.1. Extension of traces. We first consider relationships between a qualitative and quantitative properties for the problem of surjectivity of the trace. In the setting of linear Sobolev spaces, given ∈ (0, 1), ∈ (1, +∞) and a manifold  which is either compact or the Euclidean space, the classical trace theory states that the restriction of continuous functions in ( × [0, +∞), ℝ) has a linear continuous extension to the trace operator tr ∶ +1∕ , ( × (0, +∞), ℝ) → , (, ℝ) and that the latter trace operator is surjective [1, Theorem 7.39;32, Chapter 10;61, Theorem 2.7.2]. A. Monteil is a postdoctoral researcher (chargé de recherches) by the Fonds de la Recherche Scientifique-FNRS; J. Van Schaftingen was supported by the Mandat d'Impulsion Scientifique F.4523.17, "Topological singularities of Sobolev maps" of the Fonds de la Recherche Scientifique-FNRS.

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“…voir [8,12,15,11,1]. Et n'oublions pas que le phénomène de trivialité de Dahlberg apparaît dans la zone 1 + 1 p s < n p , voir [4,7,1].…”

Section: Introductioǹunclassified

“…Salah Eddine Allaoui (1) , Gérard Bourdaud ABSTRACT. -We deal with the composition operators T f (g) := f • g acting on vector valued Besov and Lizorkin-Triebel spaces.…”

Section: Composition Dans Les Espaces De Besov Critiquesmentioning

confidence: 99%