JournalszaaVol. 35, No. 3pp. 285–308

On Superposition Operators in Spaces of Regular and of Bounded Variation Functions

  • Artur Michalak

    Adam Mickiewicz University, Poznan, Poland
On Superposition Operators in Spaces of Regular and of Bounded Variation Functions cover
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Abstract

For a function f:[0,1]×RRf:[0,1]\times \mathbb R \to \mathbb R we define the superposition operator Ψf:R[0,1]R[0,1]\Psi_f: \mathbb R^{[0,1]} \to \mathbb R^{[0,1]} by the formula Ψf(φ)(t)=f(t,φ(t))\Psi_f(\varphi)(t)=f(t,\varphi(t)). First we provide necessary and sufficient conditions for ff under which the operator Ψf\Psi_f maps the space R(0,1)R(0,1), of all real regular functions on [0,1][0,1], into itself. Next we show that if an operator~Ψf\Psi_f maps the space BV(0,1)BV(0,1), of all real functions of bounded variation on [0,1][0,1], into itself, then

(1) it maps bounded subsets of BV(0,1)BV(0,1) into bounded sets if additionally ff is locally bounded,

(2) f=fcr+fdrf= f_{cr}+f_{dr} where the operator Ψfcr\Psi_{f_{cr}} maps the space D(0,1)BV(0,1)D(0,1)\cap BV(0,1), of all right-continuous functions in BV(0,1)BV(0,1), into itself and the operator Ψfdr\Psi_{f_{dr}} maps the space BV(0,1)BV(0,1) into its subset consisting of functions with countable support

(3) lim supnn12f(tn,xn)f(sn,xn)<\limsup_{n\to \infty} n^{\frac{1}{2}}|f(t_n,x_n)-f(s_n,x_n)|<\infty for every bounded sequence (xn)R(x_n)\subset \mathbb R and for every sequence ([sn,tn))([s_n,t_n)) of pairwise disjoint intervals in [0,1][0,1] such that the sequence (f(tn,xn)f(sn,xn))(|f(t_n,x_n)-f(s_n,x_n)|) is decreasing.

Moreover we show that if an operator Ψf\Psi_f maps the space D(0,1)BV(0,1)D(0,1)\cap BV(0,1) into itself, then ff is locally Lipschitz in the second variable uniformly with respect to the first variable.

Cite this article

Artur Michalak, On Superposition Operators in Spaces of Regular and of Bounded Variation Functions. Z. Anal. Anwend. 35 (2016), no. 3, pp. 285–308

DOI 10.4171/ZAA/1566