Weight characterizations of Hardy and Carleman type inequalities

Sammanfattning: This PhD thesis deals with some generalizations of the Hardy and Carleman type inequalities and the relations between them. In Chapter 1 we give an introduction and overview of the area that serves as a frame for the rest of the thesis. In particular, a fairly complete description of the development of Hardy and Carleman type inequalities in one and more dimensions can be found in this chapter. In Chapter 2 we consider some scales of weight characterizations for the one-dimensional discrete Hardy inequality for the case 1In Chapter 3 we present and discuss a new scale of weight characterizations for a n-dimensional discrete Hardy type inequality for the case 1In Chapter 4 we introduce the study of the general Hardy type inequality with a ``discrete kernel'' d= d(n,k), where n,k=1,2,... for the case 1In Chapter 5 a non-negative triangular matrix operator is considered in weighted Lebesgue spaces of sequences. Under some additional conditions on the matrix, some new weight characterizations for discrete Hardy type inequalities with matrix operator are proved for the case 1In Chapter 6 we proved that, besides the usual Muckenhoupt condition, there exist four different scales of conditions for characterizing the Hardy type inequality with general measures for the case 1In Chapter 7 some scales of equivalent weight characterizations for the Hardy type inequality with general measures are proved. The conditions are valid in the range of indices 01. We also include a reduction theorem for transferring a three-measure Hardy inequality to the case with two measures.