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 $1<p<\infty.$In the paper similar equivalences are found for some otherHardy-type operators for the full range of parameter $p,\, p\neq 0.$ As anapplication of one of the results, we also obtain a newcharacterization of the discrete inequality for one of the mostinteresting cases of parameters, namely when $0<q<p\leq 1.$The equivalences we have proved in paper A are used in paper B to obtainnecessary and sufficient conditions for some other Hardy-typeinequalities on cones of monotone functions. In particular, wegive a complete description for inequalities with Volterra integraloperators involving Oinarov's kernel for the parameters$0<p<\infty,\,\,1\leq q<\infty.$ We also study inequalities of the form$$\left(\int_{[0,\infty)}{\left(\int_{[x,\infty)}fd\lambda\right)}^q d\lambda(x)\right)^\frac{1}{q} \leq C\left(\int_{[0,\infty)}f^p d\mu\right)^\frac{1}{p},\,\,\,f \in \mathfrak{M}\downarrow,\, f\not\equiv 0$$and$$\left(\int_{[0,\infty]}\left(\int_{[0,x]}fd\lambda\right)^qd\lambda(x)\right)^\frac{1}{q}\leq C\left(\int_{[0,\infty)}f^pd\mu\right)^\frac{1}{p},\,\,\,f \in \mathfrak{M}\uparrow,\, f\not\equiv 0$$and find necessary and sufficient conditions not only for positive, but also for negativeparameters of summation.
| Identifer | oai:union.ndltd.org:UPSALLA1/oai:DiVA.org:ltu-17410 |
| Date | January 2011 |
| Creators | Popova, Olga |
| Publisher | Luleå tekniska universitet, Matematiska vetenskaper, Luleå |
| Source Sets | DiVA Archive at Upsalla University |
| Language | English |
| Detected Language | English |
| Type | Licentiate thesis, comprehensive summary, info:eu-repo/semantics/masterThesis, text |
| Format | application/pdf |
| Rights | info:eu-repo/semantics/openAccess |
| Relation | Licentiate thesis / Luleå University of Technology, 1402-1757 ; |
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