Manifolds of mappings on cartesian products

Given smooth manifolds $M_1,\ldots, M_n$ (which may have a boundary or corners), a smooth manifold $N$ modeled on locally convex spaces and $α\in({\mathbb N}_0\cup\{\infty\})^n$, we consider the set $C^α(M_1\times\cdots\times M_n,N)$ of all mappings $f\colon M_1\times\cdots\times M_n\to N$ which are $C^α$ in the sense of Alzaareer. Such mappings admit, simultaneously, continuous iterated directional derivatives of orders $\leq α_j$ in the $j$th variable for $j\in\{1,\ldots, n\}$, in local charts. We show that $C^α(M_1\times\cdots\times M_n,N)$ admits a canonical smooth manifold structure whenever each $M_j$ is compact and $N$ admits a local addition. The case of non-compact domains is also considered.

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