# Carleman type inequalities and Hardy type inequalities for monotone functions

Sammanfattning: This Ph.D. thesis deals with various generalizations of the inequalities by Carleman, Hardy and Polya-Knopp. 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 we consider Carleman's inequality, which may be regarded as a discrete version of Polya-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. In Chapter 3 we give some sharpenings and generalizations of Carleman's inequality. We discuss and comment on these results and put them into the frame presented in the previous chapter. We also include some new proofs and results. In Chapter 4 we prove a multidimensional Sawyer duality formula for radially decreasing functions and with general weights. We also state the corresponding result for radially increasing functions. In particular, these results imply that we can describe mapping properties of operatorsdefined on cones of such monotone functions. Moreover, we point out that these results can also be used to describe mapping properties of operators between some corresponding general weighted multidimensional Lebesgue spaces. In Chapter 5 we give a weight characterization of the weighted Hardy inequality for decreasing functions and we use this results to give a new weight characterization of the weighted Polya-Knopp inequality for decreasing functions and we also give a new scale of weightconditions for the Hardy inequality for decreasing functions. In Chapter 6 we make a unified approach to Hardy type inequalitits for non-increasing functions and prove a result which covers both the Sinnamon result with one condition and Sawyer's result with two independent conditions for the case when one weight is non-decreasing. In all cases we point out that this condition is not unique and can even be chosen among some (infinte) scales of conditions. In Chapter 7 we obtain the characterization of the general Hardy operator restricted to monotone functions. In Chapter 8 we present some new integral conditions characterizing the embedding between some Lorentz spaces. Only one condition is necessary for each case which means that our conditions are different and simpler than other corresponding conditions in the literature. We even prove our results in a more general frame. In our proof we use a technique of discretization and anti-discretization developed by A. Gogatishvili and L. Pick, where they considered the opposite embedding.

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