Carleman type inequalities and Hardy type inequalities for monotone functions

This Ph.D. thesis deals with various generalizations of the inequalities by Carleman, Hardy and Pólya-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 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. 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. In particular, we present 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 operators defined 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 new weight characterization of the weighted Hardy inequality for decreasing functions and use this result to give a new weight characterization of the weighted Pólya-Knopp inequality for decreasing functions and we also give a new scale of weightconditions for characterizing the embedding Λp(v) ↪→ Γq(u) for the case 1 < p ≤ q < ∞. In Chapter 6 we make a unified approach to Hardy type inequalitits for decreasing 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 nondecreasing. In all cases we point out that this condition is not unique and can even be chosen among some (infinite) scales of conditions. In Chapter 7 v