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

Abstract

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