On the de Rham complex on a scale of anisotropic weighted Hölder spaces

We obtain a solvabilty criterion for the operator equations induced by de Rham differentials on a scale of anisotropic weighted Hölder spaces on the strip $\mathbb{R}^n \times [0,T]$, $n\geq 1$, where the weight controls the behavior of elements at the infinity point with respect to the space variables. Besides, we give a description of the closures in these space of the set of infinitely differentiable functions on the strip $\mathbb{R}^n \times [0,T]$ that are compactly supported with respect to the space variables. The results are applied to study the properties of the famous Leray-Helmholtz projection from the theory of the Navier-Stokes equations on the scale of these weighted spaces for $n\geq 2$.