Local means and atoms in vector-valued function spaces

The first part of this paper deals with the topic of finding equivalent norms and characterizations for vector-valued Besov and Triebel-Lizorkin spaces. We will deduce general criteria by transferring and extending a theorem of Bui, Paluszynski and Taibleson from the scalar to the vector-valued case. By using special norms and characterizations we will derive necessary and sufficient conditions for belonging to a vector-valued function spaces. It will be shown that an element of the Schwartz space belongs to a function space if and only if it can be written as a linear combination of harmonic atoms resp. quarks with suitable conditions for the coefficients.