Calcul fonctionnel a plusieurs variables pour des operateurs pseudodifferentiels dansR n

We study aC ^∞ functional calculus with several variables forv pseudodifferential operatorsP ₁, …,P _v inR ⁿ . Whenf is a function belonging to the classS _1.0 ^r (R ^v ) of Hörmander, we prove that, under some conditions,f(P ₁,…,P _v) is a pseudodifferential operator, and we give an asymptotic formula for its symbol.     

Étude des conditions au bord pour des problemes elliptiques fortement non lineaires

Résumé De l'existence des espaces de traces des espaces de Sobolev. Orliez[19] et de l'étude de propriétés de densité dans les espaces de Sobolev-Orlicz, on déduit des caractérisations des conditions aux bords des solutions de problèmes mixtes d'équations ed d'inéquations elliptiques fortement non linéaires. Entrata in Redazione il 12 febbraio 1975.     

Les operateurs quasi Fredholm: Une generalisation des operateurs semi Fredholm

The present paper is concerned with the study of a new class of linear operators on a Hilbert space: the class of quasi-Fredholm operators, which contains many operators already studied in the litterature (in particular semi-Fredholm operators). An operatorA is said to be quasi-Fredholm of degreed, if the following conditions are satisfied:

1. For alln greater thand, R(A ⁿ )∩N(A)=R(A ^d )∩N(A);
2. N(A)∩R(A ^d ) is closed inH;
3. R(A)+N(A ^d ) is closed inH.

Two characterisations of quasi-Fredholm operators are given:

1. A is quasi-Fredholm iff there exists a direct decomposition ofH into the sum of two subspacesH ₁ andH ₂ which are invariant underA and such that the restriction ofA toH ₁ is quasi-Fredholm of degree 0 and the restriction ofA toH ₂ is nilpotent (Kato decomposition).
2. A is quasi-Fredholm iff there exists a neighborhoodD of 0 in C such that for all λ≠0 in that neighborhoodA?λI has a generalized inverse which is meromorphic inD?{0} (The generalized inverse is holomorphic inD iffA is of degree 0).

The bulk of the paper is devoted to the proofs of these characterizations and of related results, making use of the theory of operators ranges and of generalized inverses. Most of the results extend easily to the Banach case. The rest of the paper deals with the class of quasi-normal operators, which is closely related to the class of spectral operators. Some applications of the first part of the paper are given in this context.