Some new Hardy-type Inequalities for integral operators with kernels

Abstract: This Licentiate 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 kernel. The kernels we consider belong to the new classes O+ n and O-n , n = 0; 1; :::, which are wider than co-called Oinarov class of kernels. The thesis consists of three papers (papers A, B and C), an appendix to paper A and an introduction, which gives an overview to this specific field of functional analysis and also serves to put the papers in this thesis into a more 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 < q < p <∞ is considered and the involved kernels are from the classes O+1 and 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. Paper B deals with Hardy-type inequalities restrictedto the cones of monotone functions. The case 1 < p ≤ q <∞ 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 paper C even the most 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. In all these papers and the introduction a number of open questions are pointed out. Some of these questions will be investigated in the further PhD studies.