Operators in Sobolev Morrey spaces

Morrey spaces were introduced by Charles Morrey in 1938. They are a useful tool in the regularity theory of partial differential equations, in real analysis and in mathematical physics. In the nineties of the XX century an active study of general Morrey-type spaces characterized by a functional parameter has started to develop. A number of results on boundedness of classical operators in general Morrey-type spaces were obtained. At the beginning of the XXI century there were new active developments in this area. In the last decade many mathematicians do research on smoothness spaces related to Morrey spaces. Among these spaces the Sobolev-type spaces play an important role. In the thesis Sobolev spaces built on Morrey spaces are studied, which are also referred to as Sobolev Morrey spaces. These are spaces of functions which have derivatives up to certain order in Morrey spaces. We analyze some basic properties of Morrey spaces and of Sobolev Morrey spaces. Then we consider the embedding and multiplication operators in Sobolev Morrey spaces. Finally, the dissertation provides a study of the composition operator in Sobolev Morrey spaces. The results presented in the thesis have been obtained under supervision of Professors V.I. Burenkov and M. Lanza de Cristoforis.