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.
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