Functions of Generalized Bounded Variation
2013 (English)Doctoral thesis, monograph (Other academic)
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 Lp([0,1]) in terms of its Lp-modulus of continuity ω(f;δ)p. These estimates are sharp for any rate of decay of ω(f;δ)p. Moreover, the constant coefficients in them depend on parameters in an optimal way.
Inspired by these results, we consider the relationship between the Riesz type generalized variation vp,α(f) (1<p<∞, 0≤α≤1-1/p) and the modulus of p-continuity ω1-1/p(f;δ). These functionals generate scales of spaces that connect the space of functions of bounded p-variation and the Sobolev space Wp1. We prove sharp estimates of vp,α(f) in terms of ω1-1/p(f;δ).
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 ω1-1/p(f;δ) in terms of ω1-1/q(f;δ). The inequality is sharp for any order of decay of ω1-1/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 ΛBV 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 ΛBV. 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 Lp([0,1]2). Further, we investigate Fubini-type properties of the space Hp(2) of functions of bounded Hardy-Vitali p-variation. This leads us to consider the symmetric mixed norm space Vp[Vp]sym of functions of bounded iterated p-variation. For p>1, we prove that Hp(2) is not embedded into Vp[Vp]sym, and that Vp[Vp]sym is not embedded into Hp(2). In other words, Fubini-type properties completely fail in the class of functions of bounded Hardy-Vitali type p-variation for p>1.
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 Lp-modulus of continuity, and Fubini-type properties.
Place, publisher, year, edition, pages
Karlstad: Karlstads universitet, 2013. , 136 p.
Karlstad University Studies, ISSN 1403-8099 ; 2013:11
bounded p-variation, generalized bounded variation, modulus of continuity, function spaces, embeddings
Research subject Mathematics
IdentifiersURN: urn:nbn:se:kau:diva-26342ISBN: 978-91-7063-486-4OAI: oai:DiVA.org:kau-26342DiVA: diva2:605177
2013-05-17, 21A 342, Karlstads universitet, Karlstad, 14:00 (English)
Oswald, Peter, Professor
Kolyada, Viktor, Professor