Functions of Generalized Bounded Variation

Sammanfattning: 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 ?(f;?)p. These estimates are sharp for any rate of decay of ?(f;?)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,?(f) (1<p<?, 0???1-1/p) and the modulus of p-continuity  ?1-1/p(f;?). 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,?(f) in terms of ?1-1/p(f;?).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 ?1-1/p(f;?) in terms of ?1-1/q(f;?). The inequality is sharp for any order of decay of ?1-1/q(f;?).Next, we study another generalization of bounded variation: the so-called bounded ?-variation, introduced by Waterman in 1972. We investigate relations between the space ?BV of functions of bounded ?-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(?;p) into ?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 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.