This PhD thesis is devoted to boundedness of some classical linear operators in various function spaces. We prove boundedness of weighted Hardy type operators and the weighted Riesz potential in Morrey—Orlicz spaces. Furthermore, we consider central Morrey—Orlicz spaces and prove boundedness of the Riesz potential in these spaces. We also present results concerning boundedness of Hardy type operators in Hölder type spaces. The thesis consists of four papers (Papers A—D), two complementary appendices (A₁, B₁) and an introduction.

The introduction is divided into three parts. In the first part we give main definitions and properties of Morrey spaces, Orlicz spaces and Morrey—Orlicz spaces. In the second part we consider boundedness of the Riesz potential and Hardy type operators in various Banach ideal spaces. These operators have lately been studied in Lebesgue spaces, Morrey spaces and Orlicz spaces by many authors. We briefly describe this development and thereafter we present how these results have been extended to Morrey—Orlicz spaces (Paper A) and central Morrey—Orlicz spaces (Paper B). Finally, in the third part, we introduce Hölder type spaces and present our main results from Paper C and Paper D, which concern boundedness of Hardy type operators in Hölder type spaces.

 In Paper A we prove boundedness of the Riesz fractional integral operator between distinct Morrey—Orlicz spaces, which is a generalization of the Adams type result. Moreover, we investigate boundedness of some weighted Hardy type operators and weighted Riesz fractional integral operator between distinct Morrey—Orlicz spaces. The Appendix A₁ contains detailed calculations of some examples, which illustrate one of our main results presented in Paper A.

In Paper B we prove strong and weak boundedness of the Riesz potential in central Morrey—Orlicz spaces. We also give some examples, which illustrate the main theorem. Detailed calculations connected to one of the examples are described in the Appendix B₁.

 In Paper C we consider n-dimensional Hardy type operators and prove that these operators are bounded in Hölder spaces.

 In Paper D we develop the results from paper C and derive necessary and sufficient conditions for the boundedness of n-dimensional weighted Hardy type operators in Hölder type spaces. We also obtain necessary and sufficient conditions for the boundedness of weighted Hardy operators in Hölder spaces on compactification of Rⁿ.