Weighted inequalities involving Hardy-type and limiting geometric mean operators

Abstract: This thesis consists of an introduction and five papers, which all are devoted to different questions connected to weighted Hardy-type inequalities and their generalizations. The introduction gives an elementary overview of the area and serves as a frame for the following papers. In particular, it is pointed out where the obtained results fit in. The first paper gives a characterization of second order maximal overdetermined Hardy's inequality in a Hilbert space on a finite interval. The second one gives criteria for several types of overdetermined problems of arbitrary order on the semiaxis and it clearly shows the difference between the finite interval case and the semiaxis case. The third paper is devoted to some necessary and sufficient conditions on the weight functions, which guarantee that a Hardy-type inequality can be characterized by a single constant condition instead of two independent ones in the general case. In the fourth paper a criteria for a partial case of the maximal overdetermined Hardy-type inequality on a finite interval is given. And finally, in the fifth paper some corresponding weighted integral inequalities with the limiting generalized geometric mean operator are characterized for the whole range of parameters. Here we in particular use some results from the third paper in a crucial way.