Decompositions in doubling weighted Besov Triebel Lizorkin spaces and applications In this work we deal with atomic and wavelet decompositions in weighted Besov Triebel Lizorkin spaces where the weight functions are so called doubling weights which naturally extend the Muckenhoupt weights. We consider smooth atoms and smooth Daubechies wavelets both with compact supports and introduce a new approach to come in general from an atomic representation to a wavelet isomorphism. In Chapter 3 we prove sharp conditions for continuous and compact embeddings for doubling weighted Besov spaces. In Chapter 4 we study the growth envelope functions for these spaces and obtain partial results.