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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. Universit√° 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].

46E35 Sobolev spaces and other spaces of “smooth” functions, embedding theorems, trace theorems