Estimates for Hardy-type integral operators in weighted Lebesgue spaces

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 $1&lt;q&lt;p&lt;\infty$ 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 $1&lt;p\le q&lt;\infty$ 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 $1&lt;p\le q&lt;\infty$ are obtained and discussed. Moreover, some new properties of the classes $\mathcal{O}^+_n$ and $\mathcal{O}^-_n$ are proved.<p>Godkänd; 2013; 20130426 (larare); Tillkännagivande disputation 2013-05-08 Nedanstående person kommer att disputera för avläggande av teknologie doktorsexamen. Namn: Larissa Arendarenko Ämne: Matematik/Mathematics Avhandling: Estimates for Hardy-type Integral Operators in Weighted Lebesgue Spaces Opponent: Professor Massimo Lanza de Cristoforis, Dipartamento di Matematica Universita degli Studi di Padova, Padova, Italy, Ordförande: Professor Peter Wall, Institutionen för teknikvetenskap och matematik, Luleå tekniska universitet Tid: Måndag den 3 juni 2013, kl 10.00 Plats: E246, Luleå tekniska universitet