This Licentiate thesis is devoted to the study of mapping properties of different operators (Hardy type, singular and potential) between various function spaces.

The main body of the thesis consists of five papers and an introduction, which puts these papers into a more general frame.

In paper A we prove the boundedness of the Riesz Fractional Integration Operator from a Generalized Morrey Space to a certain Orlicz-Morrey Space, which covers the Adams resultfor Morrey Spaces. We also give a generalization to the case of Weighted Riesz Fractional Integration Operators for a class of weights.

In paper B we study the boundedness of the Cauchy Singular Integral Operator on curves in complex plane in Generalized Morrey Spaces. We also consider the weighted case with radial weights. We apply these results to the study of Fredholm properties of Singular Integral Operators in Weighted Generalized Morrey Spaces.

In paper C we prove the boundedness of the Potential Operator in Weighted Generalized Morrey Spaces in terms of Matuszewska-Orlicz indices of weights and apply this result to the Hemholtz equation with a free term in such a space. We also give a short overview of some typical situations when Potential type Operators arise when solving PDEs.

In paper D some new inequalities of Hardy type are proved. More exactly, the boundedness of multidimensional Weighted Hardy Operators in Hölder Spaces are proved in cases with and without compactification.

In paper E the mapping properties are studied for Hardy type and Generalized Potential type Operators in Weighted Morrey type Spaces.