Hardy-type inequalities on cones of monotone functions

Sammanfattning: This Licentiate thesis deals with Hardy-type inequalities restricted to cones of monotone functions. The thesis consists of two papers (paper A and paper B) and an introduction which gives an overview to this specific field of functional analysis and also serves to put the papers into a more general frame.We deal with positive $\sigma $-finite Borel measures on ${\mathbbR}_{+}:=[0,\infty)$ and the class $\mathfrak{M}\downarrow $($\mathfrak{M}\uparrow $) consisting of all non-increasing(non-decreasing) Borel functions $f\colon{\mathbbR}_{+}\rightarrow[0,+\infty ]$.In paper A some two-sided inequalities for Hardy operators on thecones of monotone functions are proved. The idea to study suchequivalences follows from the Hardy inequality$$\left( \int_{[0,\infty)}f^pd\lambda\right)^{\frac{1}{p}}\le \left(\int_{[0,\infty)} \left( \frac{1}{\Lambda(x)} \int_{[0,x]}f(t)d\lambda(t)\right)^p d\lambda(x)\right)^{\frac{1}{p}}$$$$\leq \frac{p}{p-1}\left(\int_{[0,\infty)}f^pd\lambda\right)^{\frac{1}{p}},$$which holds for any $f\in \mathfrak{M}\downarrow$ and $1