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

TL;DR

This paper analyzes the cohomology of the de Rham complex within weighted isotropic and anisotropic Hölder spaces, revealing finite-dimensionality in the isotropic case and characterizing cohomology elements in the anisotropic case.

Contribution

It provides a detailed description of the cohomology groups of the de Rham complex over weighted Hölder spaces, including solvability results and explicit forms of cohomology elements.

Findings

Cohomology space is finite-dimensional in isotropic case.

Explicit description of cohomology elements in anisotropic case.

Established solvability conditions for associated operator equations.

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.