  We consider the Navier-Stokes equations in the layer ${\mathbb R}^n \times
[0,T]$ over $\mathbb{R}^n$ with finite $T > 0$. Using the standard fundamental
solutions of the Laplace operator and the heat operator, we reduce the
Navier-Stokes equations to a nonlinear Fredholm equation of the form $(I+K) u =
f$, where $K$ is a compact continuous operator in anisotropic normed H\"older
spaces weighted at the point at infinity with respect to the space variables.
Actually, the weight function is included to provide a finite energy estimate
for solutions to the Navier-Stokes equations for all $t \in [0,T]$. On using
the particular properties of the de Rham complex we conclude that the Fr\'echet
derivative $(I+K)'$ is continuously invertible at each point of the Banach
space under consideration and the map $I+K$ is open and injective in the space.
In this way the Navier-Stokes equations prove to induce an open one-to-one
mapping in the scale of H\"older spaces.

Analysis of PDEs