# Weighted inequalities of Hardy-type and their limiting inequalities

Sammanfattning: This thesis deals with various generalizations of two famous inequalities namely the Hardy inequality and the Pólya-Knopp inequality and the relation 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 Chapter 2 the idea of using the weighted Hardy inequality to receive the weighted Pólya-Knopp inequality as a natural limiting inequality is investigated and some problems that arises are discussed. In Chapter 3 a new necessary and sufficient condition for the weighted Hardy inequality is proved and also used to give a new necessary and sufficient condition for a corresponding weighted Pólya-Knopp type inequality. In Chapter 4 a new two-dimensional Pólya-Knopp inequality is proved. This inequality may be regarded as a natural endpoint inequality of the famous two-dimensional Hardy inequality by E. Sawyer, which is characterized by three independent integral conditions while our endpoint inequality is characterized by one condition. In Chapter 5 the three necessary and sufficient conditions for the two- dimensional version of the Hardy inequality given by E. Sawyer are investigated and compared with the corresponding conditions in one dimension. Moreover, the corresponding endpoint problems and conditions are pointed out. In Chapter 5 we also prove a new two-dimensional Hardy inequality, where the weightfunction on the right hand side is of product type. In this case we only need one integral inequality to characterize the inequality and, moreover, by performing the natural limiting process we receive the same result as in Chapter 4. In Chapter 6 we prove criteria for boundedness between weighted Rn spaces of a fairly general multidimensional Hardy-type integral operator with an Oinarov kernel. The integrals are taken over cones in Rn with origin as a vertex. In Chapter 7 the related results are proved for the limiting geometric mean operator with an Oinarov kernel. Finally, in Chapter 8 we consider Carleman's inequality, which may be regarded as a discrete version of Pólya-Knopp's inequality and also as a natural limiting inequality of the discrete Hardy inequality. We present several simple proofs of and remarks (e.g. historical) about this inequality. Moreover, we discuss and comment some very new results and put them into this frame. We also include some new proofs and results e.g. a weight characterization of a general weighted Carleman type inequality.

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