Generalized polynomials and first steps towards multifractal analysis on Lie groups

In the first part, we study the Fréchet functional equations extended to Lie groups. The smooth solutions of these are what we call here generalized polynomials. We study the regularity of the solutions if we assume very weak conditions on the solution space. A few concrete examples are treated to show how these solutions look like. The second part of this thesis is about the study of Hölder pointwise and local regularity of functions defined on Lie groups. The function spaces we define here are characterized through various results, each of those being useful depending of the context. Some examples are treated and we make a link with wavelet transforms that are adapted in this context.