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Uniformlycontinuouscomposition...
7
Remark1.Iff∈RVł(IjX),thenfiscontinuousinI.Itisaconse-
quenceofLemma2.1(d)inChistyakov[3].
Wewillneedthefollowingpropertyofpł.
Lemma1(cf.Chistyakov,[4],Lemma3.4).Leto∈Fandf∈
RV∗
ł(IjX).
IfT>0,thenVł(f/T)≤1ifandonlyifpł(f)≤T.
3.Mainresult
ForasetC⊂Xweput
RVł(IjC):1{f∈RVł(IjX):f(I)⊂C}.
ByA(XjY)denotethespaceofalladditivemappingsA:X→Y,and
byL(XjY)denotethespaceofalllinearmappingsfromXintoY.
Themainresultreadsasfollows.
Theorem1.LetI1[Ijb]andojw∈F.Supposethat(Xj|·|)isa
linearrealnormedspace,(Yj|·|)isarealBanachspace,C⊂Xisaclosed
andconvexset,andh:I×C→Yisafunction.Ifthecompositionoperator
Hgivenby
H(f)(t):1h(tjf(t))j
t∈Ijf∈XIj
mapsthesetRVł(IjC)intoRVψ(IjY)andHisuniformlycontinuous,then
therearethefunctionsA:I→A(XjY)andB∈RVψ(IjY)suchthat
h(tjx)1A(t)x+B(t)j
t∈Ijx∈C.
Moreover,if0∈CandInt(C)/1∅,thenA:I→L(XjY)andB∈
RVψ(IjY).
Proof.Foreveryx∈C,theconstantfunctionI3tl→xbelongsto
RVł(IjC).SincetheNemytskiioperatorHmapsthespaceRVł(IjC)into
RVψ(IjY),itfollowsthatforeveryx∈Cthefunctionh(·jx)belongsto
RVψ(IjY).
TheuniformcontinuityofHonRVł(IjC)impliesthat
"H(f1)−H(f2)"ψ≤ω("f1−f2"ł)forf1jf2∈RVł(IjC)j
whereω:R+→R+isthemoduluscontinuityofH,i.e.
ω(ρ):1sup{"H(f1)−H(f2)"ψ:"f1−f2"ł≤ρ;f1jf2∈RVł(IjC)}
forρ>0.
