Domains of type $ 1\text{,}1$ operators: a case for Triebel–Lizorkin spaces

Jon Johnsen

doi:10.1016/j.crma.2004.05.008

Mathematical Analysis/Partial Differential Equations

Domains of type

1, 1

operators: a case for Triebel–Lizorkin spaces

Presented by: Jean-Michel Bony

Jon Johnsen ¹

¹ Department of Mathematics, Aalborg University, Fredrik Bajers Vej 7G, DK-9220 Aalborg Øst, Denmark

Comptes Rendus. Mathématique, Volume 339 (2004) no. 2, pp. 115-118.

Abstract
Résumé

Pseudo-differential operators of type $1, 1$ are proved continuous from the Triebel–Lizorkin space $F_{p, 1}^{d}$ to $L_{p}$ , $1 ⩽ p < \infty$ , when of order d, and this is, in general, the largest possible domain among the Besov and Triebel–Lizorkin spaces. Hörmander's condition on the twisted diagonal is extended to this framework, using a general support rule for Fourier transformed pseudo-differential operators.

On démontre que les opérateurs pseudo-différentiels de type $1, 1$ et d'ordre d sont continus de l'espace $F_{p, 1}^{d}$ de Triebel–Lizorkin dans $L_{p}$ , $1 ⩽ p < \infty$ , et que parmi les espaces de Besov et Triebel–Lizorkin, ces domaines sont, en général, les plus grand possible. La condition de Hörmander sur la diagonale–miroir est établie pour ce cadre, en utilisant un résultat général sur le support de la transformation de Fourier d'un opérateur pseudo-différentiel.

Received: 2004-05-07
Accepted: 2004-05-10
Published online: 2004-07-15

DOI: 10.1016/j.crma.2004.05.008

Author's affiliations:

Jon Johnsen ¹

¹ Department of Mathematics, Aalborg University, Fredrik Bajers Vej 7G, DK-9220 Aalborg Øst, Denmark

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     pages = {115--118},
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Jon Johnsen. Domains of type $ 1\text{,}1$ operators: a case for Triebel–Lizorkin spaces. Comptes Rendus. Mathématique, Volume 339 (2004) no. 2, pp. 115-118. doi : 10.1016/j.crma.2004.05.008. https://comptes-rendus.academie-sciences.fr/mathematique/articles/10.1016/j.crma.2004.05.008/

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