In the first part of this talk we discuss an atomic decomposition of Besov and Triebel-Lizorkin spaces with Muckenhoupt weights. We improve a result of S. Roudenko on matrix-weighted Besov spaces byproving an atomic decomposition also in the case 0 < p < 1. First results in the theory of weighted spaces under consideration are due to H.Q. Bui et al. The theory of the classical function spaces and their atomic decompositions has been developed by M. Frazier and B. Jawerth and independently by H. Triebel. As a consequence we present examples of Muckenhoupt weights with connection to fractal analysis. In particular we consider atomic decompositions of function spaces with a weight which measures the distance from a point to a certain fractal set.
The second purpose of this talk is to present a solution of the trace problem for these spaces. The corresponding trace operator shall map weighted function spaces of Besov and Triebel-Lizorkin type into suitable function spaces on a fractal set. In particular we characterize traces on n - 1 dimensional hyperplanes of Sobolev spaces with the special weight function that measures a distance from a point to a hyperplane.