Paracomposition et opérateurs paradifferentiels. (Paracomposition and paradifferential operators). (French) Zbl 0596.47023

Alinhac, S.

Commun. Partial Differ. Equations 11, 87-121 (1986).

In a paper from 1981 [Ann. Sci. Éc. Norm. Super., IV Sér. 14, 209-246 (1981; Zbl 0495.35024)], J.-M. Bony introduces the notion of the paradifferential operator and the paramultiplication, used in the study of the nonlinear partial differential equations. In a similar context the author introduces and examines the properties of the ”paracomposition” \(X^*:{\mathcal D}'(\Omega_ 2)\to {\mathcal D}'(\Omega_ 1)\), where \(X:\Omega_ 1\to \Omega_ 2\) is a diffeomorphism of class \({\mathbb{C}}^{\rho +1}\) \((\rho >0)\), \(\Omega_ 1\) and \(\Omega_ 2\) are two open sets in \(R^ n\). Among other results, the existence of a linear operator \(X^*\) with the following properties is shown: \(X^*:H^ s_{loc}(\Omega_ 2)\to H^ s_{loc}(\Omega_ 1)\); \(X^*:C^{\sigma}_{loc}(\Omega_ 2)\to C^{\sigma}_{loc}(\Omega_ 1)\); \(u\circ X=X^*u+T_{u'\circ X}X+R\), where \(H^ s\) and \(C^{\rho}\) are the Sobolev and the Hölder function spaces respectively, \(R\in C_{loc}^{\rho +1+\epsilon}\) \((\epsilon =\inf (\sigma -1,\rho +1))\) if \(u\in C^{\sigma}_{loc}\) \((\sigma >1,\rho >0)\), and \(R\in H_{loc}^{r+1+\epsilon}\) \((\epsilon =\inf (s- (n/2)-1,r-(n/2)+1))\) if \(u\in H^ s_{loc}\) \((s>(n/2)+1,r>n/2)\), \(T_{u'\circ X}\) being the Bony’s paramultiplication and \(X^*u\) the new paracomposition.

Reviewer: I.Gottlieb

MSC:

47B38	Linear operators on function spaces (general)
35L75	Higher-order nonlinear hyperbolic equations
46F10	Operations with distributions and generalized functions
46E35	Sobolev spaces and other spaces of “smooth” functions, embedding theorems, trace theorems
47F05	General theory of partial differential operators

Keywords:

Sobolev space; distribution space; paradifferential operator; paramultiplication; nonlinear partial differential equations; paracomposition; Hölder function spaces

Citations:

Zbl 0495.35024

References:

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