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FunctionsofboundedJordanvariation
15
Moregenerally,
x
iscalledHöldercontinuous(or
γ
-Lipschitzcontinuous
for0<γ<1)ifthereexistsaconstantL>0suchthat
|x(s)lx(t)|<L|slt|γ
(0<s,t<1).
(1.4)
1
WedenotethesetofallLipschitzcontinuousfunctionson
[
0,1
]
by
Lip
,
andthesetofall
γ
-Lipschitzcontinuousfunctionson
[
0,1
]
by
Lipγ
.
Writing
lip(x)=lip(x;[0,1]):=
0<s<t<1
sup
|x(s)lx(t)|
|slt|
fortheminimalLipschitzconstantLin(1.3)and,for0<γ<1,
lip
γ(x)=lip
γ(x;[0,1]):=
0<s<t<1
sup
|x(s)lx(t)|
|slt|γ
fortheminimalHölderconstant
L
in
(1.4)
,onemayshowthatthe
spacesLip=Lip1andLipγ,equippedwiththenorms
"x"Lip:=|x(0)|+lip(x)(x∈Lip),
and
"x"Lip
γ:=|x(0)|+lip
γ(x)(x∈Lipγ),
respectively,areBanachspaces.Theinclusions
C1⊆Lip⊆Lipα⊆Lipβ⊆C(1>α>β>0)
(1.5)
showthatLipschitzandHöldercontinuityissituated“between”con-
tinuityandcontinuousdifferentiability,andthespace
Lipγ
becomes
smallerif
γ
increases.Inparticular,thefunction
uτ(t):=tτ
belongs
to
Lipτ\Lip
for0
<τ<
1.Allinclusionsin
(1.5)
arecontinuous
imbeddingswhicharestrictfor0
<β<α<
1.Thelastimbedding
in
(1.5)
isevencompactfor
β>
0,sinceaboundedsetin
Lipβ
isclearly
equicontinuous.
Nowwedefineandstudytwoparameter-dependentfamiliesof
functionswhichwillbequitehelpfulinwhatfollowstoillustrateour
abstractresultsandtoconstructexamplesorcounterexamples.
