Change search
ReferencesLink to record
Permanent link

Direct link
Carleman type inequalities and Hardy type inequalities for monotone functions
Luleå University of Technology, Department of Arts, Communication and Education, Education, Language, and Teaching.
2007 (English)Doctoral thesis, comprehensive summary (Other academic)
Abstract [en]

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 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 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.

Place, publisher, year, edition, pages
Luleå: Luleå tekniska universitet, 2007. , 145 p.
Series
Doctoral thesis / Luleå University of Technology 1 jan 1997 → …, ISSN 1402-1544 ; 2007:53
Research subject
Mathematics; Mathematics Education
Identifiers
URN: urn:nbn:se:ltu:diva-18339Local ID: 812f5410-9cf7-11dc-97ff-000ea68e967bOAI: oai:DiVA.org:ltu-18339DiVA: diva2:991346
Note
Godkänd; 2007; 20071127 (ysko)Available from: 2016-09-29 Created: 2016-09-29Bibliographically approved

Open Access in DiVA

fulltext(834 kB)7 downloads
File information
File name FULLTEXT01.pdfFile size 834 kBChecksum SHA-512
3ef5ed397a083534762f1cf31820eb78050b2da70b91a45bf6065ca72f144aa47dd33ee30b767b0536aafed655b0ffa3bca68805035d6da05622d71df0d8899c
Type fulltextMimetype application/pdf

Search in DiVA

By author/editor
Johansson, Maria
By organisation
Education, Language, and Teaching

Search outside of DiVA

GoogleGoogle Scholar
Total: 7 downloads
The number of downloads is the sum of all downloads of full texts. It may include eg previous versions that are now no longer available

Total: 1 hits
ReferencesLink to record
Permanent link

Direct link