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

Zbl 0495.35024
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### References:

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