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6
A.Acosta,W.Aziz,J.MatkowskiandN.Merentes
2.Auxiliaryresults
ByFwedenotethefamilyofallcontinuousconvexfunctionso:[0j+∞)→
[0j+∞)suchthat
o(0)10j
o(t)>0fort>0;
t→∞
lim
o(t)
t
1+∞.
Obviously,everyo∈Fisstrictlyincreasing.
Inthesequelo∈Fisfixed.
LetI1[Ijb]⊂Rbeaninterval.ByP(I)wedenotethefamilyofall
partitionsτoftheintervalI;i.e.τ∈P(I)ifandonlyif,τ1(τi)
m
il1
for
somem∈N,and
I1τo<τ1<···<τm1b.
Forf∈XIandτ∈P(I)weput
RVł(fjτ):1
Σ
il1
m
o(|f(τi)−f(τi11)|
τi−τi11
)(τi−τi11)j
andwedefine
RVł(f):1sup{RVł(fjτ):τ∈P(I)}
thatiscalledtheRieszo–variationoffinI.
IfRVł(f)<∞jwesaythatfhasaboundedRieszo–variationonI.
Theset
RV
ł(f):1{f∈XI:RVł(f)<+∞}
∗
isconvex(cf.[2])ifoisconvex,but,ingeneral,itisnotalinearspace.It
isknown(cf.[3])that
RVł(IjX):1{f∈XI:∃A>0RVł(Af)<+∞}
isalinearspaceifandonlyifosatisfiesthe∆2condition,i.e.thereexists
aconstantρo≥0andC>0suchthato(2ρ)≤Co(ρ)forallρ≥ρoand
alsoitisanormedspacewiththenorm
"f"ł:1|f(I)|+pł(f)j
where
f∈RVł(IjX)j
pł(f):1inf{ć>0:RVł(
f
ć)≤1}.
Moreover,if(Xj|·|)isaBanachspace,thensois(RVł(IjX)j"·"ł).
