This PhD thesis deals with the theory of Hardy-type inequalities in a new situation, namely when the classical Hardy operator is replaced by a more general operator with a kernel. The kernels we consider belong to the new classes $\mathcal{O}^+_n$ and $\mathcal{O}^-_n$, $n=0,1,...$, which are wider than co-called Oinarov class of kernels. This PhD thesis consists of four papers (papers A, B, C and D), two complementary appendixes (A$_1$, C$_1$) and an introduction, which put these publications into a more general frame. This introduction also serves as a basic overview of the field. In paper A some boundedness criteria for the Hardy-Volterra integral operators are proved and discussed. The case $1<q<p<\infty$ is considered and the involved kernels are from the classes $\mathcal{O}^+_1$ and $\mathcal{O}^-_1$. A complete solution of the problem is presented and discussed. The appendix to paper A contains the proof of Theorem 3.1, which is not included in the paper. In paper B even more complicated (than in paper A) case with variable limits on the Hardy operator is investigated. The main results of the paper are proved by applying the block-diagonal method given by Batuev and Stepanov and the results from paper A. Paper C deals with Hardy-type inequalities restricted to the cones of monotone functions. The case $1<p\le q<\infty$ is considered and the involved kernels satisfy conditions, which are less restrictive than the usual Oinarov condition. Also in this case a complete solution is obtained and some concrete applications are pointed out. In particular, in paper C some open questions are raised. These questions are discussed and solved in an appendix to Paper C. In paper D we study superpositions of the Hardy-Volterra integral operator and its adjoint. The boundedness and compactness criteria in the range of parameters $1<p\le q<\infty$ are obtained and discussed. Moreover, some new properties of the classes $\mathcal{O}^+_n$ and $\mathcal{O}^-_n$ are proved.

Some new Hardy-type inequalities for Hardy-Volterra integral operators are proved and discussed. The case 1 < q < p < ∞ is considered and the involved kernels satisfy conditions, which are less restrictive than the usual Oinarov condition.

This Licentiate thesis deals with the theory of Hardy-type inequalities in anew situation, namely when the classical Hardy operator is replaced by amore general operator with kernel. The kernels we consider belong to thenew classes O+ n and O-n , n = 0; 1; :::, which are wider than co-called Oinarovclass of kernels.The thesis consists of three papers (papers A, B and C), an appendix topaper A and an introduction, which gives an overview to this specific fieldof functional analysis and also serves to put the papers in this thesis into amore general frame.In paper A some new Hardy-type inequalities for the case with Hardy-Volterra integral operators involved are proved and discussed. The case 1