Continuité des opérateurs pseudo-différentiels

Texte intégral

(1)ΔϴΒόθϟ΍ Δϴσ΍ήϘϤϳΪϟ΍ Δϳή΋΍ΰΠϟ΍ ΔϳέϮϬϤΠϟ΍ MINISTERE DE L’ENSEIGNEMENT SUPERIEUR ET DE LA RECHERCHE SCIENTIFIQUE UNIVERSITE HADJ LAKHDAR BATNA (ALGERIE) THESE Présentée à la Faculté des Sciences Département de Mathématique Pour l’obtention du diplôme de. DOCTORAT EN SCIENCES Option : Mathématique Par Aissa DJERIOU THÈME. Continuité des opérateurs pseudo-différentielssur certains espaces fonctionnels Soutenue le : 09/02/2012 Devant le jury d’examen : S.E. REBIAI. Prof.. Université de Batna. Président. M. MOUSSAI. Prof.. Université de M’sila. Directeur de Thèse. A. AIBECHE. Prof.. Université de Sétif. Examinateur. R. BENACER. Prof.. Université de Batna. Examinateur. S. DRABLA. Prof.. Université de Sétif. Examinateur. M. NADIR. Prof.. Université de M’sila. Examinateur.

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(25) έ΍Ωϧϭϣέϫ ϑϧλ ϰϟ· ϲϣΗϧΗ ϲΗϟ΍ V ( x, [ ) Γ΍ϭϧϟ΍ϭ m ΔΑΗέϟ΍ . M Fpv,q

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(29) v. M F p ,Pq. vP p ,q " P. : ϕϘΣΗ. ˬ Fpv,q ϡϣόϣϟ΍ ϝΎΑϳέΗ ϥΎϛέϭίϟ Δϳϟ΍Ωϟ΍ Ε΍˯Ύοϔϟ΍ ϰϠϋ P. v6 : ¡0, . ¯ ¢0, . sup t P sup 0t 1. 0 s d1. Δϟ΍Ωϟ΍ Δργ΍ϭΑ Δϓέόϣ Ε΍˯Ύοϔϟ΍ ϩΫϫ. vP s

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(31). f .. :ΔϳΣΎΗϔϣϟ΍ ΕΎϣϠϛϟ΍ ˯Ύοϓ ˬϑϭίϭΑ ˯Ύοϓ ˬϝΎΑϳέΗ ϥΎϛέϭίϟ ˯Ύοϓ ˬΔϳϠοΎϔΗ ϪΑηϟ΍ Ε΍έΛ΅ϣϟ΍ . Δϳϟϭ΃ Γ΍ϭϧ ˬϲϠϳΎΑ ΩϭϭϠΗϟ ϝϳϠΣΗ ˬέ΍Ωϟϭϫ ˯Ύοϓ ˬϑϼΑϭγ. Abstract We study the continuity of some pseudo-differential operators of order m , with a m symbol @x, 8 that belongo to Hörmander's class S 1,- , 0 t - 1 , on n v generalized Triebel-Lizorkin space F p ,q R , generalized localized Triebel-Lizorkin v n space F p ,q R e r , generalized pointwise multipliers Triebel-Lizorkin space MF pv ,q R n and MF pv ,q R n e r . These spaces F pv ,q R n are defined from a positive function v6 : ¡0, . ¯ ¢0, . satisfying the condition:. sup t"6 sup 0t 1. 0st1. v6 s .. v6 ts . Key words: Pseudo-differential operators, Triebel-Lizorkin spaces, Besov spaces, Sobolev spaces, Hölder spaces, Littlewood-Paley decomposition, elementary symbols..

(32) 5. . Résumé Nous étudions la continuité des opérateurs pseudo-différentiels d'ordre m, avec un m symbole @x, 8 , qui appartiennent à la classe de Hörmander S 1,- où 0 t - 1 , v6 v6 v6 sur les espaces de Triebel-Lizorkin généralisé F p ,q , F p ,q e r , MF p ,q et v MF p ,6q e r . Ces espaces sont définis par une fonction positive v6 : ¡0, . ¯ ¢0, . qui vérifie v6 s sup t"6 sup .. 0t 1 0st1 v6 ts . Mot-clés: Opérateurs Pseudo-differentiels, espace de Triebel-Lizorkin, espace de Besov, espace de Sobolev, espace de Hölder, décomposition de Littlewood-Paley, symboles élémentaires..

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