The Composition Operator and the Space of the Functions of Bounded Variation in Schramm-Korenblum's Sense

Abstract

We show that the composition operatorH, associated withh:[a,b]→ℝ, maps the spacesLip[a,b]on to the spaceκBVϕa,bof functions of bounded variation in Schramm-Korenblum's sense if and only ifhis locally Lipschitz. Also, verify that if the composition operator generated byh:[a,b]×ℝ→ℝmaps this space into itself and is uniformly bounded, then regularization ofhis affine in the second variable.

Locally Lipschitz Composition Operators in Space of the Functions of BoundedκΦ-Variation

“…From (19) we obtain, by taking limits of rational points in inequality (18), that satisfies (18) for all pairs of positive real numbers; that is, is Φ-decreasing with constant on [0, 1]. By Theorem 13 is of bounded Φ-variation and is continuous.…”

“…Using (17) we can, by means of the standard Cantor diagonalization technique, find a sequence of functions in F which converges pointwise at each rational point of [0, 1], to a function . Since each satisfies (18), so does , for all rational numbers , ∈ [0, 1].…”

“…Lemma 12 (see [18] Theorem 13 (see [15] or [17]). If a function is Φ-decreasing on [ , ], then one has the following properties.…”

The Space of Bounded p(·)-Variation in the Sense Wiener-Korenblum with Variable Exponent

“…The following lemma, established in [38], will be useful in the proof of our main Theorem (Theorem 4.2).…”