Differential calculus on multiple products (Q2334356)

The article under review studies mappings on products of locally convex spaces and manifolds with mixed order of differentiability in the components. For example, if \(U_i \subseteq E_i\) are open subsets of locally convex spaces \(E_i\), then a mapping \(f \colon U_1 \times \cdots \times U_n \rightarrow F\) is called a \(C^\alpha\)-differentiable (or \(C^\alpha\))-map with \(\alpha = (\alpha_1, \ldots, \alpha_n)\) a multiindex (\(\alpha_i \in \mathbb{N} \cup \{\infty\}\)) if \(f\) is \(\alpha_1\) times partially differentiable in the first component and each of the resulting mappings is, then a \((\alpha_2,\ldots,\alpha_n)\)-differentiable map of the remaining components. This is a generalisation of the concept of \(C^{r,s}\)-mappings (i.e.,the case \(n=2\)) which was studied in [\textit{H. Alzaareer} and \textit{A. Schmeding}, Expo. Math. 33, No. 2, 184--222 (2015; Zbl 1330.46039)]. Beyond a discussion of the basic properties of \(C^\alpha\)-mappings (e.g., a suitable version of the Schwarz theorem), the main result of the article is a version of the exponential law for \(C^\alpha\)-mappings (Theorem~A). Namely, the author proves that, under certain assumptions on the locally convex spaces (e.g., the occurring iterated products are \(k\)-spaces), there is an isomorphism of topological vector spaces \[C^\alpha (U,C^\beta (V,F)) \cong C^{(\alpha,\beta)} (U\times V , F)\] where \(U = U_1 \times \cdots \times U_n\) and \(V = V_1 \times \cdots \times V_m\) (\(n,m \in \mathbb{N}\)) and the function spaces are topologised with a version of the compact open topology. This result (and the other results of the paper) hold under more general cirumstances than sketched in this review. For example, the theory can handle domains which are manifolds or non-open sets (e.g., manifolds with (rough) boundary), see Theorem~B in the paper under review.