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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.