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Functions of Generalized Bounded VariationPrimeFaces.cw("AccordionPanel","widget_formSmash_some",{id:"formSmash:some",widgetVar:"widget_formSmash_some",multiple:true}); PrimeFaces.cw("AccordionPanel","widget_formSmash_all",{id:"formSmash:all",widgetVar:"widget_formSmash_all",multiple:true});
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PrimeFaces.cw("AccordionPanel","widget_formSmash_responsibleOrgs",{id:"formSmash:responsibleOrgs",widgetVar:"widget_formSmash_responsibleOrgs",multiple:true}); 2013 (English)Doctoral thesis, monograph (Other academic)
##### Abstract [en]

##### Abstract [en]

##### Place, publisher, year, edition, pages

Karlstad: Karlstads universitet, 2013. , 136 p.
##### Series

Karlstad University Studies, ISSN 1403-8099 ; 2013:11
##### Keyword [en]

bounded p-variation, generalized bounded variation, modulus of continuity, function spaces, embeddings
##### National Category

Mathematical Analysis
##### Research subject

Mathematics
##### Identifiers

URN: urn:nbn:se:kau:diva-26342ISBN: 978-91-7063-486-4OAI: oai:DiVA.org:kau-26342DiVA: diva2:605177
##### Public defence

2013-05-17, 21A 342, Karlstads universitet, Karlstad, 14:00 (English)
##### Opponent

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#####

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Available from: 2013-04-03 Created: 2013-02-13 Last updated: 2013-04-03Bibliographically approved

This thesis is devoted to the study of different generalizations of the classical conception of a function of bounded variation.

First, we study the functions of bounded *p*-variation introduced by Wiener in 1924. We obtain estimates of the total *p*-variation (1<*p*<∞) and other related functionals for a periodic function* f* in *L ^{p}*([0,1]) in terms of its

Inspired by these results, we consider the relationship between the Riesz type generalized variation *v _{p,α}(f)* (1<

In the same direction, we study relations between moduli of *p*-continuity and *q*-continuity for 1<*p*<*q*<*∞*. We prove an inequality that estimates *ω*_{1-1/p}(*f;δ*) in terms of* ω*_{1-1/q}(*f;δ*). The inequality is sharp for any order of decay of *ω*_{1-1/q}(*f;δ*).

Next, we study another generalization of bounded variation: the so-called bounded Λ-variation, introduced by Waterman in 1972. We investigate relations between the space Λ*BV* of functions of bounded Λ-variation, and classes of functions defined via integral smoothness properties. In particular, we obtain the necessary and sufficient condition for the embedding of the class Lip(*α;p*) into Λ*BV*. This solves a problem of Wang (2009).

We consider also functions of two variables. Applying our one-dimensional result, we obtain sharp estimates of the Hardy-Vitali type *p*-variation of a bivariate function in terms of its mixed modulus of continuity in *L ^{p}*([0,1]

**Baksidestext**

The classical concept of the total variation of a function has been extended in several directions. Such extensions find many applications in different areas of mathematics. Consequently, the study of notions of generalized bounded variation forms an important direction in the field of mathematical analysis.

This thesis is devoted to the investigation of various properties of functions of generalized bounded variation. In particular, we obtain the following results:

- sharp relations between spaces of generalized bounded variation and spaces of functions defined by integral smoothness conditions (e.g., Sobolev and Besov spaces);
- optimal properties of certain scales of function spaces of frac- tional smoothness generated by functionals of variational type;
- sharp embeddings within the scale of spaces of functions of bounded
*p*-variation; - results concerning bivariate functions of bounded
*p*-variation, in particular sharp estimates of total variation in terms of the mixed*L*-modulus of continuity, and Fubini-type properties.^{p}

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