On the Closure of Smooth Compactly Supported Functions in Weighted Hölder Spaces

The closure of the set of smooth compactly supported functions in a weighted Hölder space on $\mathbb R^n$, $n\ge 1$, with a weight controlling the behavior at the point at infinity is described. As an application, a solvability criterion for operator equations generated by de Rham differentials both in these spaces and on the closure of the set of smooth compactly supported functions in them for $n\ge 2$ is obtained.