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We consider the de Rham complex over scales of weighted isotropic andanisotropic H\"older spaces with prescribed asymptotic behaviour at theinfinity. Starting from theorems on the solvability of the system of operatorequations generated by the de Rham differential $d$ and the operator $d^*$formally adjoint to it, a description of the cohomology groups of the de Rhamcomplex over these scales was obtained. It was also proved that in theisotropic case the cohomology space is finite-dimensional, and in theanisotropic case the general form of an element from the cohomology space ispresented.

On the cohomologies of the de Rham complex over weighted isotropic and anisotropic H\"older spaces