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Multiplication in Sobolev and Besov spaces. (English) Zbl 0894.46026
Ambrosetti, A. (ed.) et al., Nonlinear analysis. A tribute in honour of Giovanni Prodi. Pisa: Scuola Normale Superiore, Quaderni. Universitá di Pisa. 27-50 (1991).
Let $$E_{1},E_{2},\dots, E_{m}$$ and $$E$$ be Banach spaces. Suppose that $$\phi :\Pi E_{j} \to E$$ is a continuous $$m$$-linear map. The map $$\phi$$ defines naturally an $$m$$-linear map $$\Phi : \Pi E^{\Omega}_{j} \to E^{\Omega}$$ as $$\Phi(u_{1},\dots,u_{m})(x) = \phi (u_{1}(x),\dots,u_{m}(x))$$, where $$\Omega$$ is a nonempty open subset of $${\mathbb{R}}^n$$. The map $$\Phi$$ is called the pointwise multiplication induced by $$\phi$$. In this paper the author studies the continuity of $$\Phi$$ on various subspaces of the spaces occurring in this product. Namely, he studies pointwise multiplications on Sobolev and Besov spaces.
For the entire collection see [Zbl 0830.00011].

##### MSC:
 4.6e+36 Sobolev spaces and other spaces of “smooth” functions, embedding theorems, trace theorems