Atomic characterizations of vector-valued function spaces

The rst part of this diploma thesis deals with the topic of nding equivalent norms and characterizations for vector-valued Besov and Triebel-Lizorkin spaces Bs p,q(E) and F s p,q(E). We will deduce general criteria by transferring and extending a theorem of Bui, Paluszy«ski and Taibleson from the scalar to the vector-valued case. By using special norms and characterizations we will derive necessary and su cient conditions for belonging to a vector-valued function spaces Bs p,q(E) or F s p,q(E). It will be shown that an element of S ′(Rn, E) 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 coe cients. Eventually, we will prove theorems 15.8 and 15.11 of [Tri97] dealing with atomic and subatomic representations.