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 ${mathbb R}_{+}:=[0,infty)$ and the class $mathfrak{M}downarrow $ ($mathfrak{M}uparrow $) consisting of all non-increasing (non-decreasing) Borel functions $fcolon{mathbb R}_{+} ightarrow[0,+infty ]$. In paper A some two-sided inequalities for Hardy operators on the cones of monotone functions are proved. The idea to study such equivalences follows from the Hardy inequality $$ left( int_{[0,infty)}f^pdlambda ight)^{frac{1}{p}}le left( int_{[0,infty)} left( frac{1}{Lambda(x)} int_{[0,x]} f(t)dlambda(t) ight)^p dlambda(x) ight)^{frac{1}{p}} $$ $$ leq frac{p}{p-1}left( int_{[0,infty)}f^pdlambda ight)^{frac{1}{p}}, $$ which holds for any $fin mathfrak{M}downarrow$ and $1
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