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Equi alen De ini ions o Ve y S ic CD(K,N) -spaces
© The Au ho (s) 2022
Published e sion
Schul z, Timo
Schul z, T. (2023). Equi alen De ini ions o Ve y S ic CD(K,N) -spaces. Jou nal o Geome ic
Analysis, 33(3), A icle 108. h ps://doi.o g/10.1007/s12220-022-01068-x
2023
The Jou nal o Geome ic Analysis (2023) 33:108
h ps://doi.o g/10.1007/s12220-022-01068-x
Equi alen De ini ions o Ve y S ic CD(K,N) -spaces
Timo Schul z1,2,3
Recei ed: 20 Sep embe 2019 / Accep ed: 5 Sep embe 2022
© The Au ho (s) 2022
Abs ac
We show he equi alence o he de ini ions o e y s ic CD(K,N) -condi ion de ined,
on one hand, using (only) he en opy unc ionals, and on he o he , he ull dis-
placemen con exi y class DCN. In pa icula , we show ha assuming he con exi y
inequali ies o he c i ical exponen implies i o all he g ea e exponen s. We also
es ablish he exis ence o op imal anspo maps in e y s ic CD(K,N) -spaces wi h
ini e N.
Keywo ds Op imal anspo ·Ricci cu a u e ·Me ic measu e spaces
Ma hema ics Subjec Classi ica ion P ima y 53C23
1 In oduc ion
Syn he ic no ions o cu a u e (bounds) ha e es ablished hei posi ion in geome ic
analysis bo h as a ool o s udy geome ic and analy ic p ope ies o non-smoo h
spaces, and as a new app oach o a ack p oblems e en in he smoo h se ing. The
amewo k p esen in his pape is he gene alisa ion o Ricci cu a u e lowe bounds
o me ic measu e spaces, mo e p ecisely he se ing o CD(K,N) -spaces in oduced
in he seminal pape s o Lo –Villani [8] and S u m [15,16] based on a concep o
displacemen con exi y o ce ain en opy unc ionals in oduced by McCann [13].
The de ini ions o S u m and Lo –Villani o CD(K,N) -spaces bo h sha e wo
no able p ope ies, namely hey a e ue gene alisa ions o he no ion o Ricci cu a u e
lowe bounds o (weigh ed) Riemannian mani olds, and, keeping in mind he G omo ’s
p ecompac ness heo em o Riemannian mani olds sha ing a common Ricci lowe
BTimo Schul z
[email p o ec ed]
1Depa men o Ma hema ics and S a is ics, Uni e si y o Jy askyla, P.O.Box 35, 40014
Jy askyla, Finland
2Facul y o Ma hema ics, Biele eld Uni e si y, Pos ach 10 01 31, 33501 Biele eld, Ge many
3Ins i u e o Applied Ma hema ics, Uni e si y o Bonn, Endeniche Allee 60, 53115 Bonn, Ge many
0123456789().: V,- ol 123
108 Page 2 o 20 T. Schul z
bound, hey a e s able unde sui able con e gence o me ic measu e spaces. The de -
ini ions o CD(K,N) -spaces by S u m, and by Lo and Villani a e di e en , bu unde
an addi ional (essen ial) non-b anching assump ion o he spaces in ques ion, hese
wo no ions o CD(K,N) -spaces ag ee. Howe e , he non-b anching p ope y, while
gi ing many desi ed esul s o CD(K,N) -spaces [3–6,9,12,16], is no s able unde
any easonable con e gence e en when coupled wi h he CD(K,N) -condi ion.
In his pape , we s udy con exi y p ope ies o a poin wise densi y o anspo plans
in (possibly) b anching CD(K,N) -spaces gi ing an equi alen de ini ion (P oposi ion
4.2) o he so-called e y s ic CD(K,N) -condi ion in oduced in [14](seealso
[1]), analogous o he known cha ac e isa ion o essen ially non-b anching CD(K,N)
-spaces, see [3]. Ha ing he poin wise de ini ion in hand, we p o e Theo em 4.4, he
equi alence o e y s ic CD(K,N) -condi ion and i s Lo –Villani ype analogue (see
Sec . 2.2 o he p ecise de ini ions).
The main di e ence in he de ini ions by S u m and by Lo –Villani is ha while
S u m equi es con exi y o hold only o ce ain speci ic en opy unc ionals, namely
he Rényi en opies, Lo and Villani equi e i o hold o all unc ionals in he so-
called displacemen con exi y class. Using he de ining con exi y p ope ies o he
unc ionals in he displacemen con exi y class, we deduce easily he equi alence
o he wo de ini ions o e y s ic CD(K,N) -spaces om he poin wise con exi y
inequali y.
To ob ain he poin wise condi ion, we use Theo em 3.1, he exis ence o op imal
anspo maps be ween wo measu es absolu ely con inuous wi h espec o he e -
e ence measu e p o en in [14] in he in ini e dimensional case. Fo comple eness, we
p esen he e he p oo in he ini e dimensional case. In ac , we need a bi mo e han
jus he exis ence o anspo map. We need he plan o be gi en by a map no only
om he endpoin s, bu also om he in e media e poin s.
As a byp oduc , we p o e Theo em 3.3, he exis ence o op imal anspo map
om a (boundedly suppo ed) absolu ely con inuous measu e o a singula one. We
cons uc he plan gi en by a map by gluing oge he plans ob ained be ween (abso-
lu ely con inuous) in e media e poin s o he endpoin s. We p o e, in simila ashion
o wha is done in [11], ha he esul ing plan sa is ies he con exi y inequali y o
educed cu a u e dimension condi ion be ween any h ee poin s o he uni in e al.
2 P elimina ies
S anding assump ions o his pape o a me ic measu e space (X,d,m)a e com-
ple eness and sepa abili y o he me ic d, and local ini eness o he Bo el
measu e m.
A me ic space (X,d)is said o be a leng h space, i he dis ance be ween any wo
poin s xand yis ob ained by in imising he leng h o cu es connec ing xand y.A
cons an speed cu e pa ame ised on he uni in e al wi h leng h equal o he dis ance
be ween he endpoin s is called a (cons an speed) geodesic. The se o all cons an
speed geodesics endowed wi h he sup emum me ic is deno ed by Geo(X).
123
Equi alen De ini ions o Ve y S ic CD(K,N)-Spaces Page 3 o 20 108
2.1 Op imal Mass T anspo a ion
We conside he Monge–Kan o o ich o mula ion o he op imal anspo p oblem
wi h quad a ic cos . Deno e by P(X) he se o all Bo el p obabili y measu es on X.
We de ine he Wasse s ein 2-dis ance W2be ween wo Bo el p obabili y measu es
μ, ν ∈P(X)as he in imum
W2(μ, ν) := in
σ∈A(μ,ν) X×X
d2(x,y)dσ(x,y)1
2
,
whe e A(μ, ν) := {σ∈P(X×X):P1
#σ=μ, P2
#σ=ν}is he se o admissible
anspo plans be ween μand ν. The exis ence o an admissible plan ha ealises
he in imum is ue in a he gene al se ing, including ou s [17]. Such a minimising
admissible plan is called an op imal plan, and he se o op imal plans be ween measu es
μand νis deno ed by Op (μ, ν).
Deno e by P2(X) he se o all Bo el p obabili y measu es wi h ini e second
momen , ha is, hose μ∈P(X)which a e o ini e W2-dis ance om a Di ac mass.
Mo eo e , deno e by Pac
2(X)a u he subse o P2(X)o measu es absolu ely con-
inuous wi h espec o he e e ence measu e m.
We ecall, ha he Wasse s ein dis ance W2de ines an ac ual me ic on he se
P2(X). The space (P2(X), W2)inhe i s also some p ope ies om he base space X,
namely he space (P2(X), W2)is comple e and sepa able leng h space, i (X,d)is. In
he case o leng h spaces, we ha e he ollowing use ul cha ac e isa ion o Wasse s ein
geodesics. A cu e → μ ∈P2(X)is geodesic, i and only i he e exis s a measu e
π∈P(Geo(X)) so ha (e0,e1)#π∈Op (μ0,μ
1), and μ =(e )#π o all ∈[0,1],
whe e γ→ e (γ ) := γ is he e alua ion map [7]. Such a p obabili y measu e πis
called op imal dynamical plan, o jus op imal plan o sho , and he se o all op imal
dynamical plans om μ0 o μ1is deno ed by Op Geo(μ0,μ
1).
Recall, ha o π∈Op Geo(μ0,μ
1),weha e ha ( es 2
1)#(Fπ) is s ill an op i-
mal plan o all 1, 2∈[0,1], 1< 2, and o all Fwi h Fdπ=1, whe e
es 2
1:Geo(X)→Geo(X),( es 2
1)(γ )( )=γ( 2+(1− ) 1).Fo π∈P(Geo(X)),
we deno e by π−1 he push o wa d measu e o πunde he map γ→ γ−1,
γ−1( ):= γ(1− ).
2.2 Syn he ic Ricci Cu a u e Lowe Bounds
Based on he no ion o displacemen con exi y, in oduced by McCann [13], o sui able
en opy unc ionals, S u m [15], and independe ly Lo and Villani [8] in oduced
no ions o Ricci cu a u e lowe bounds o gene al (non-smoo h) me ic measu e
spaces.
We ecall he de ini ion o a mo e es ic i e e sion o cu a u e dimension
condi ion— he so-called e y s ic CD(K,N) -condi ion—and, mo i a ed by he exis-
ence esul o op imal maps in he con ex o such spaces, we in oduce a Lo –Villani
ype analogue o he e y s ic CD(K,N) -condi ion.
123
108 Page 4 o 20 T. Schul z
Fo he de ini ions, we need o in oduce some auxilia y no a ion. As building
blocks, we de ine, o K∈Rand N∈(0,∞], coe icien s [0,1]×R+→R∪{∞},
( ,θ)→ σ( )
K,N(θ) as
σ( )
K,N(θ) :=
⎧
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎨
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎩
,i N=∞
∞,i Kθ2≥Nπ2
sin θK
N
sinθK
N,i 0 <Kθ2<Nπ2
,i K=0
sinh θ−K
N
sinhθ−K
N,i K<0.
Using hese coe icien s we u he de ine, o N∈(1,∞], coe icien s β( )
K,N(θ) and
τ( )
K,N(θ) as
β( )
K,N(θ) := 1−Nσ( )
K,N−1(θ)N−1
,and
τ( )
K,N(θ) := 1
Nσ( )
K,N−1(θ)N−1
N.
To be p ecise, we de ine o >0,N>1
β( )
K,N(θ) :=
⎧
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎨
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎩
eK
6(1− 2)θ2,i N=∞
∞,i N<∞,Kθ2>(N−1)π2
sin θK
N−1
sinθK
N−1N−1
,i 0 <Kθ2≤(N−1)π2
1,i N<∞,K=0
sinh θ−K
N−1
sinhθ−K
N−1N−1
,i N<∞,K<0,
and β(0)
K,N≡1.
Fo N∈(1,∞], de ine he en opy unc ionals En N:P2(X)→R∪{±∞}as
En N(μ) := − ρ−1
Ndμ,
o N<∞, and
En ∞(μ) := log ρdμ+∞dμ⊥.
123

Equi alen De ini ions o Ve y S ic CD(K,N)-Spaces Page 5 o 20 108
He e μ=ρm+μ⊥wi h μ⊥⊥m, and μ⊥({ρ>0})=0. Fu he , o anspo plan
π∈P(Geo(X)) wi h (e0)#π=μ0∈P2(X), and o ∈[0,1],K∈R, de ine he
dis o ed en opy
En ( )
N,π (μ0):= − β( )
K,N(d(γ0,γ
1))1
Nρ0(γ0)−1
Ndπ(γ),
o N<∞, and
En ( )
∞,π (μ0):= log ρ0(γ0)
β( )
K,∞(d(γ0,γ
1))dπ(γ) +∞dμ⊥
0.
De ini ion 2.1 We say ha a me ic measu e space (X,d,m)is a e y s ic
CD(K,N) -space, i o all μ0,μ
1∈Pac
2(X)wi h bounded suppo s, he e exis s
π∈Op Geo(μ0,μ
1)such ha o all non-nega i e and bounded Bo el unc ions
F:Geo(X)→Rwi h Fdπ=1, and o all 1, 2∈[0,1], 1< 2,weha e
En N(˜μ )≤(1− )En (1− )
N,˜π(˜μ0)+ En ( )
N,˜π−1(˜μ1)(1)
o all ∈[0,1], whe e ˜μ := (e )#˜π:= (e )#( es 2
1)#Fπ.
Rema k 2.2 The de ini ion would make sense also wi hou he assump ion on he
boundedness o he suppo s. In ha case, he unc ionals En ∞and En ∞,π a e no a
p io i well-de ined o all μ∈P2(X), due o he ac ha (ρ log ρ)−dmmigh be
−∞. Howe e , a e equi ing (1) o hold o μi,i∈{0,1}, wi h (ρilog ρi)+∈L1(m),
we know by [15, Theo em 4.24], ha ( o ixed x0∈X)m(B(x0, )) ≤Ae(B 2)holds
o all >1, and hus (ρ log ρ)−∈L1(m) o all μ=ρm∈P2(X),see[1].
We will also use he de ini ion o e y s ic CD∗(K,N)-condi ion, which one ge s
by modi ying he abo e de ini ions (see [2] o he de ini ion o educed cu a u e
dimension condi ion). Mo e p ecisely, one eplaces he con exi y inequali y (1)by
inequali y
En N(˜μ )≤−σ(1− )
K,N(d(γ0,γ
1))ρ−1
N
0(γ0)+σ( )
K,N(d(γ0,γ
1))ρ−1
N
1(γ1)d˜π. (2)
De ini ion 2.1 is a (possibly) mo e es ic i e e sion o he s ic CD(K,N) -
condi ion in oduced in [1], and is gi en in he spi i o S u m’s o iginal de ini ion
o cu a u e dimension condi ion. To de ine Lo –Villani ype analogue o he con-
di ion, we need o in oduce he so-called displacemen con exi y classes, in oduced
by McCann in [13].
We say, ha a con inuous and con ex unc ion U:R+→Ris in he displacemen
con exi y class DCN(o dimension N∈(1,∞]), i U(0)=0, and i he unc ion
s→ u(s)is con ex, whe e uis de ined as
u:(0,∞)→R,s→ sNU(s−N),
123
108 Page 6 o 20 T. Schul z
i N<∞, and
u:R→R,s→ esU(e−s),
i N=∞.
Rema k 2.3 We ecall, ha he displacemen classes a e nes ed. Indeed, i N<N,
we ha e ha DCN⊂DCN. This can be seen o example by w i ing
uN(s):= sNU(s−N)=sN
NN
UsN
N−N=: uNsN
N
as a composi ion o a con ex and dec easing unc ion uNand conca e unc ion s→
sN
N.I N=∞, one w i es
uN(s)=eNlog sU(e−Nlog s),
and concludes again, by conca i y o s→ log s, ha uNis con ex.
Fo U∈DCN, de ine he (en opy) unc ional Um:P2(X)→R∪{∞}as
Um(μ) := U◦ρdm+U(∞)dμ⊥,
whe e U(∞):= lim
s→∞
U(s)
s∈R∪{∞}. Fu he mo e, o π∈P(Geo(X)),K∈R
and ∈[0,1], de ine he unc ional U( )
π,m:P2(X)→R∪{∞}as
U( )
π,m(μ) := XGeo(X)
Uρ(γ0)
β( )
K,N(γ0,γ
1)β( )
K,N(γ0,γ
1)dπx(γ ) dm(x)
+X
U(∞)dμ⊥,
whe e {πx}is a disin eg a ion o πwi h espec o he e alua ion map e0.
Rema k 2.4 The unc ional U( )
π,mis no well-de ined in gene al due o he non-
uniqueness o he disin eg a ion. Howe e , he de ini ion will be used only o
π∈Op Geo(μ, ν), in which case he disin eg a ion is unique up o μ-measu e ze o
se . Ano he cause o being ill-de ined is he possible in eg abili y issue, which may
appea bo h o he posi i e and o he nega i e pa o U◦ρ(and βU(ρ/β)), c ea ing
∞−∞si ua ions. This can be seen by aking U(s)=slog s−s1−1
Nin he hype bolic
space. Because o hese issues, we will use he abo e de ini ions only o measu es
wi h bounded suppo , in which case he unc ionals a e well-de ined, see e.g. [17,
Theo em 17.28] o he p oo .
123
Equi alen De ini ions o Ve y S ic CD(K,N)-Spaces Page 7 o 20 108
De ini ion 2.5 A me ic measu e space is said o sa is y he e y s ic CD(K,N) con-
di ion in he spi i o Lo –Villani, i o all μ0,μ
1∈Pac
2(X)wi h bounded suppo s,
he e exis s π∈Op Geo(μ0,μ
1)such ha o all bounded non-nega i e Bo el unc-
ions F:Geo(X)→Rwi h Fdπ=1, and o all 1, 2∈[0,1], 1< 2,we
ha e
Um(˜μ )≤(1− )U(1− )
˜π,m(˜μ0)+ U( )
˜π−1,m(˜μ1)
o all ∈[0,1]and o all U∈DCN, whe e ˜μ := (e )#˜π:= (e )#( es 2
1)#Fπ.
Rema k 2.6 By choosing UN(s)=−s1−1
N, o N<∞, and U∞(s)=slog s, one
immedia ely sees ha spaces sa is ying De ini ion 2.5 also sa is y De ini ion 2.1.
3 Exis ence o Op imal Maps
In p o ing ou main esul s in Sec . 4, we will use he ac ha he plan gi en by he
de ini ion o e y s ic CD(K,N) -spaces is induced by a map. The case N=∞is
co e ed in [14], and he p oo o he ini e dimensional case ollows along he same
lines. Fo comple eness, we will ou line he p oo o he ini e dimensional case he e.
I should be poin ed ou , ha wi h ou de ini ion o e y s ic CD(K,N) -spaces, we
do no a p io i know ha e y s ic CD(K,N) -condi ion o ini e Nimplies he e y
s ic CD(K,∞)-condi ion.
Theo em 3.1 (Exis ence o op imal maps) Le (X,d,m)be a e y s ic C D∗(K,N)
(CD(K,N)) -space, and le μ0,μ
1∈Pac
2(X)wi h bounded suppo s. Le π∈
Op Geo(μ0,μ
1)be he op imal plan gi en by he e y s ic C D∗(K,N)(CD(K,N))
-condi ion. Then πis induced by a Bo el map T :X→Geo(X), i.e. π=T#μ0wi h
e0◦T=id.
Rema k 3.2 I we emo e in De ini ion 2.1 he assump ion o he boundedness o he
suppo s o μ0and μ1, we may emo e i also om Theo em 3.1.
P oo Le N<∞, and μ0,μ
1∈Pac
2(X). Fu he mo e, le π∈Op Geo(μ0,μ
1)be
he op imal plan gi en by he de ini ion o e y s ic CD∗(K,N)-space. Suppose
ha πis no induced by a map. Towa ds a con adic ion, we will show ha he e exis
plans π1,π2π, and imes 1and 2su icien ly close o each o he so ha μ1
i=μ2
i
and μ1
i+1⊥μ2
i+1.
We begin by doing some educ ions. Fi s o all, by w i ing he whole space Xas a
union o bounded se s, we may assume ha he leng h o he geodesics in he suppo
o πis bounded by some cons an C, and since sp mis p ope , we may also assume
ha sp μ0is compac . Fu he mo e, by di iding he in e al [0,1]in o su icien ly
small subin e als Ij, and looking a he es ic ion measu es ( es Ij)#π,wemay
assume ha
σ( )
K,N(θ) ∈[(1−ε) ,(1−ε)−1 ](3)
123
108 Page 8 o 20 T. Schul z
o all ∈[0,1]and θ≤C.He eε>0 is chosen so ha (1−ε)421
N>1.
Nex , as was done in [14], we ind imes T,S∈(0,1),T<S, and op imal plans
π1,π2πso ha μ1
T=μ2
Tand μ1
S⊥μ2
S, whe e μ := (e )#π o all ∈[0,1].
We e e o [14] o he a gumen s and he cons uc ion. Le hen n∈Nbe such ha
+1
n1−( +1
n)
1− ≥1
(1−ε)42−1
N,(4)
o ∈[T,S]. Again, by he a gumen s used in [14], we ind imes 1, 2∈[T,S],
1< 2, wi h | 2− 1|<1
n, and op imal plans ¯π1,¯π2such ha ¯μ1
1=¯μ1
2and μ1
2⊥μ2
2.
Now we a e eady o a i e o a con adic ion by simila compu a ions as was done
in [10]. We i s use he con exi y o he en opy along 1
2(¯π1+¯π2)be ween poin s
0, 1and 2, hen along ¯π1and ¯π2sepa a ely be ween poin s 1, 2and 1. Also he
inequali y (4) is used bo h imes wi h he con exi y inequali y. Then we use he bound
(3) and inally a i e o a con adic ion.
(¯ρ1
1)1−1
Ndm≥(1−ε)2 2− 1
2
21
N−1(( ¯ρ1
0)1−1
N+(¯ρ2
0)1−1
N)dm
+(1−ε)2 1
2
21
N−1(¯ρ1
2)1−1
Ndm+(¯ρ2
2)1−1
Ndm
>(1−ε)4 1
2
(1− 2)
(1− 1)21
N(¯ρ1
1)1−1
Ndm≥(¯ρ1
1)1−1
Ndm.
He e ¯ρi
is he densi y o (e )#¯πiwi h espec o m. In he case o e y s ic CD(K,N)
-space, he p oo is exac ly he same a e eplacing σ( )
K,Nby τ( )
K,Nin he condi ion
(3). 
As a co olla y, we ge he exis ence o an op imal map om absolu ely con inuous
measu e o singula one, by app oaching he singula endpoin wi h absolu ely con in-
uous in e media e poin s. Combined wi h cons uc ion simila o he one used in [11],
we a i e o he ollowing heo em.
Theo em 3.3 Le (X,d,m)be a e y s ic C D∗(K,N)-space wi h N <∞, and
μ0∈Pac
2(X)and μ1∈P2(X),sp μ1⊂sp m, p obabili y measu es wi h bounded
suppo . Then he e exis s π∈Op Geo(μ0,μ
1)along which he con exi y inequali y
(2)holds be ween any poin s 1< 2< 3(wi h ˜μ =μ =(e )#π) o he en opy
En N. Mo eo e , πis induced by a map om μ0.
Rema k 3.4 We do no claim, ha he con exi y would hold along Fπ, whe e Fis
a bi a y bounded non-nega i e Bo el unc ion wi h Fdπ=1. In ac , he p oo
below will in some cases p oduce a geodesic (μ )such ha o any li πo (μ ) his
is known o be alse.
The idea o he p oo o he abo e heo em is ai ly simple. Fi s o all, by app oxi-
ma ing he possibly singula measu e μ1by absolu ely con inuous ones, one ob ains a
geodesic μ wi h μ mdue o he lowe semi-con inui y o he en opy En N. Then,
123
Equi alen De ini ions o Ve y S ic CD(K,N)-Spaces Page 15 o 20 108
alence o he de ini ion o e y s ic CD(K,N) and Lo –Villani- ype analogous o he
de ini ion.
We will need he ollowing simple lemma.
Lemma 4.1 Le (X,d,m)be a e y s ic CD(K,N) -space, μ0,μ
1∈Pac
2(X)abso-
lu ely con inuous measu es wi h espec o he e e ence measu e and wi h bounded
suppo s, and π∈Op Geo(μ0,μ
1)gi en by he de ini ion o e y s ic CD(K,N)
-condi ion. Then μ ∈Pac
2(X) o all ∈(0,1).
P oo Suppose he claim is no ue. Then he e exis s π∈Op Geo(μ0,μ
1)as in
De ini ion 2.1 wi h μ := (e )#π=ρm+μ⊥,μ⊥⊥m. Thus, he e exis s a Bo el se
A⊂Xso ha μ⊥(A)>0 and m(A)=0. Le A:= e−1
(A), and de ine ˜π:= π|A.
In he case N=∞we ge a con adic ion a e es ic ing he plan π u he so ha
ρ0and ρ1a e bounded, and hence he en opies En ∞(μ0)and En ∞(μ1)a e ini e.
In he case N<∞ he a gumen goes as ollows. Fo π-a.e. γ∈A, we ha e ha
d(γ0,γ
1)>0 and hus τ( )
K,N(d(γ0,γ
1)) > 0. Thus,
0<τ(1− )
K,N(d(γ0,γ
1))ρ0(γ0)+τ( )
K,N(d(γ0,γ
1))ρ1(γ1)dπ(γ) ≤En N(μ )=0
gi ing he con adic ion. 
P oposi ion 4.2 Le (X,d,m)be a me ic measu e space. Then (X,d,m)is e y
s ic CD(K,N) -space, i and only i o all absolu ely con inuous measu es μ0,μ
1∈
Pac
2(X)wi h bounded suppo , he e exis s an op imal plan π∈Op Geo(μ0,μ
1),
wi h μ := (e )#π∈Pac
2(X), o which he ollowing wo condi ions hold:
(i) Fo all ∈(0,1), he e exis s a Bo el map T :X→Geo(X) o which π=
(T )#μ , and e ◦T =id.
(ii) I N =∞, hen o e e y 1< 2< 3, he inequali y
log ρ 2(γ 2)≤( 3− 2)
( 3− 1)log ρ 1(γ 1)+( 2− 1)
( 3− 1)log ρ 3(γ 3)
−K
2
( 3− 2)
( 3− 1)
( 2− 1)
( 3− 1)d2(γ 1,γ
3)(10)
holds o π-almos e e y γ, whe e ρ is he densi y o μ wi h espec o he
e e ence measu e m.
I N <∞, hen o e e y 1< 2< 3, he inequali y
ρ−1
N(γ 2)≥τ
( 3− 2)
( 3− 1)
K,N(d(γ 1,γ
3))ρ−1
N
1(γ 1)+τ
( 2− 1)
( 3− 1)
K,N(d(γ 1,γ
3))ρ−1
N
3(γ 3)(11)
holds o π-almos e e y γ.
Mo eo e , i πis he plan gi en by he de ini ion o e y s ic CD(K,N) -space,
hen o π-almos e e y γ, he inequali y (10)(N =∞)o (11)(N <∞) holds o
L3-almos e e y ( 1, 2, 3)∈[0,1]wi h 1< 2< 3.
123

108 Page 16 o 20 T. Schul z
Rema k 4.3 I we emo e in De ini ion 2.1 he assump ion o he boundedness o he
suppo s o μ0and μ1, we may emo e i also om P oposi ion 4.2.
P oo We will p o e only he case N=∞. The p oo o he ini e dimensional case is
he same wi h ob ious modi ica ions. Le (X,d,m)be a e y s ic CD(K,∞)-space.
Le μ0,μ
1∈Pac
2(X), and le πbe he op imal plan gi en by he de ini ion o e y s ic
CD(K,∞)-space. We will p o e ha he condi ions (i) and (ii) hold o π. By Lemma
4.1 we ha e ha μ is absolu ely con inuous wi h espec o m. Mo eo e , he plan
( es
0)#π∈Op Geo(μ0,μ
)is such as in he de ini ion o e y s ic CD(K,∞).
Thus, by Theo em 3.1, i is induced by a map T om he in e media e measu e μ .
Hence we ha e ha π=(S◦e0◦T)#μ =: (T )#μ , whe e Sis he map gi en by
Theo em 3.1 o which π=S#μ0, p o ing he claim (i).
Fo (ii), suppose o he con a y, ha he e exis 1, 2, 3∈[0,1], 1< 2< 3, and
ase A⊂Geo(X)wi h π(A)>0, so ha he inequali y (10) does no hold o any
γ∈A. De ine ˜π:= π|A, and u he de ine ˜μ := (e )#˜π=˜ρ m, o all ∈[0,1].
W i ing Aas union
A=
i∈N
{γ∈A:max
j=1,2,3ρ j(γ j)≤i},
we may assume ha ˜ρ jis bounded om abo e, and so in pa icula ha (˜ρ jlog ˜ρ j)+
is in eg able o j∈{1,2,3}.Le {π
x}be he disin eg a ion o πwi h espec o he
e alua ion map e . Then we ha e, o all non-nega i e Bo el unc ions :X→R,
ha
X
(x)˜ρ (x)dm(x)=Geo(X)
(γ )χA(γ ) dπ(γ)
=XGeo(X)
(γ )χA(γ ) dπ
x(γ ) dμ (x)
=X
(x)Geo(X)
χA(γ ) dπ
x(γ ) dμ (x)
=X
(x)Geo(X)
χA(γ ) dπ
x(γ )ρ (x)dm(x),
whe e ρ is he densi y o μ := (e )#πwi h espec o he e e ence measu e m. Thus
˜ρ (x)=χA(T (x))ρ (x) o m-almos e e y x∈X. In pa icula , we ha e ha
˜ρ (γ )=χA(T (γ ))ρ (γ )=χA(γ )ρ (γ ),
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Equi alen De ini ions o Ve y S ic CD(K,N)-Spaces Page 17 o 20 108
o π-almos e e y γ, and o all ∈{ 1, 2, 3}. Hence, we ge
En ∞˜μ 2=X
˜ρ 2log ˜ρ 2dm=X
log ˜ρ 2d˜μ 2=Geo(X)
log ˜ρ 2(γ 2)d˜π
=A
log ρ 2(γ 2)dπ
>( 3− 2)
( 3− 1)A
log ρ 1(γ 1)dπ+( 2− 1)
( 3− 1)A
log ρ 3(γ 3)dπ
−K
2
( 3− 2)
( 3− 1)
( 2− 1)
( 3− 1)A
d2(γ 1,γ
3)dπ
=( 3− 2)
( 3− 1)En ∞˜μ 1+( 2− 1)
( 3− 1)En ∞˜μ 3
−K
2
( 3− 2)
( 3− 1)
( 2− 1)
( 3− 1)W2
2(˜μ 1,˜μ 3),
which con adic s he assump ion o πbeing he plan gi en by he de ini ion o e y
s ic CD(K,∞)-space. Hence (ii) holds.
Fo he o he di ec ion, suppose ha π∈Op Geo(μ0,μ
1)is such ha condi ions
(i) and (ii) hold. Le F:Geo(X)→Rbe a bounded non-nega i e Bo el unc ion o
which Fdπ=1, and le 1, 2, 3∈[0,1], 1< 2< 3. Deno e μF
:= (e )#Fπ.
As p e iously, by (i), we ge ha
ρF
(x):= F(T (x))ρ (x),
is he densi y o μF
wi h espec o m.He eρ is he densi y o μ wi h espec o he
e e ence measu e m. In pa icula , we ha e ha along geodesics he densi y is, up o
a mul iplica i e cons an , he same as he o iginal densi y. Mo e p ecisely, we ha e
ρF
(γ )=F(γ )ρ (γ ),
o π-almos e e y γ, and o e e y ∈{ 1, 2, 3}. Thus, by (ii) we ha e ha
ρF
2log ρF
2dm=log ρF
2dμF
2=log ρF
2(γ 2)F(γ ) dπ
=log ρ 2(γ 2)F(γ ) dπ+log F(γ )F(γ ) dπ
≤( 3− 2)
( 3− 1)log ρ 1(γ 1)F(γ ) dπ+( 2− 1)
( 3− 1)log ρ 3(γ 3)F(γ ) dπ
−K
2
( 3− 2)
( 3− 1)
( 2− 1)
( 3− 1)d2(γ 1,γ
3)F(γ ) dπ+log F(γ )F(γ ) dπ
=( 3− 2)
( 3− 1)ρF
1log ρF
1dμF
1+( 2− 1)
( 3− 1)ρF
3log ρF
3dμF
3
−K
2
( 3− 2)
( 3− 1)
( 2− 1)
( 3− 1)W2
2(μF
1,μF
3),
gi ing he claim.
123
108 Page 18 o 20 T. Schul z
Fo he las claim, de ine o all γ∈Geo(X)
Iγ:= {( 1, 2, 3)∈J:(ii) ails along γa ( 1, 2, 3)},
whe e J:= {( 1, 2, 3)∈[0,1]: 1< 2< 3}, and he se
I:= 
γ
{γ}×Iγ.
Then by (ii)
0=J
π({γ: ∈Iγ})dL3( )=χId(π ⊗L3)
=L3(Iγ)dπ(γ).
Hence Iγhas Lebesgue measu e ze o o π-almos e e y γ∈Geo(X).
Theo em 4.4 Le (X,d,m)be a me ic measu e space. Then he ollowing a e equi -
alen :
(i) The space (X,d,m)is a e y s ic CD(K,N) -space (see De ini ion 2.1).
(ii) The space (X,d,m)is a e y s ic CD(K,N) -space in he spi i o Lo −−Villani
(see De ini ion 2.5).
P oo Clea ly condi ion (ii) implies condi ion (i). Fo he o he implica ion, assume
ha μ0,μ
1∈Pac
2(X), and π∈Op Geo(μ0,μ
1)gi en by he de ini ion o e y s ic
CD(K,N) -condi ion. Le U∈DCN, and F:Geo(X)→Rnon-nega i e, bounded
Bo el unc ion wi h Fdπ=1.
We i s p o e he claim, when N<∞. De ine u(s):= sNU(s−N). Then uis
a dec easing and con ex unc ion, since U∈DCN. Hence, by Theo em 4.2 condi-
ion (ii),
U(μF
2)=U◦ρF
2dm=u((ρF
2)−1
N)ρF
2dm=u((ρF
2)−1
N(γ 2))F(γ ) dπ
=u(F(γ )ρ−1
N
2(γ 2))F(γ ) dπ
≤u(F(γ )(τ
( 3− 2)
( 3− 1)
K,N(d(γ 1,γ
3))ρ−1
N
1(γ 1)
+τ
( 2− 1)
( 3− 1)
K,N(d(γ 1,γ
3))ρ−1
N
3(γ 3)))F(γ ) dπ
≤( 3− 2)
( 3− 1)u(F(γ )( 3− 1)
( 3− 2)τ
( 3− 2)
( 3− 1)
K,N(d(γ 1,γ
3))ρ−1
N
1(γ 1))F(γ ) dπ
123
Equi alen De ini ions o Ve y S ic CD(K,N)-Spaces Page 19 o 20 108
+( 2− 1)
( 3− 1)u(F(γ )( 3− 1)
( 2− 1)τ
( 2− 1)
( 3− 1)
K,N(d(γ 1,γ
3))ρ−1
N
3(γ 3))F(γ ) dπ
=( 3− 2)
( 3− 1)Uβ
( 3− 2)
( 3− 1)
(K,N)
π,m(μF
1)+( 2− 1)
( 3− 1)Uβ
( 2− 1)
( 3− 1)
(K,N)
π−1,m(μF
3),
gi ing he claim.
I N=∞, we ha e ha he unc ion u:s→ esU(e−s)is con ex and dec easing
by assump ion. Hence, by P oposi ion 4.2 condi ion (ii)
U(μF
2)=u(−log(F(γ )ρ 2(γ 2)))F(γ ) dπ
≤( 3− 2)
( 3− 1)u⎛
⎝−log ⎛
⎝
F(γ )ρ 1(γ 1)
β( 3− 2)
( 3− 1)
(γ0,γ
1)⎞
⎠⎞
⎠F(γ ) dπ
+( 2− 1)
( 3− 1)u⎛
⎝−log ⎛
⎝
F(γ )ρ 3(γ 3)
β( 2− 1)
( 3− 1)
(γ0,γ
1)⎞
⎠⎞
⎠F(γ ) dπ
=( 3− 2)
( 3− 1)Uβ
( 3− 2)
( 3− 1)
(K,N)
π,m(μF
1)+( 2− 1)
( 3− 1)Uβ
( 2− 1)
( 3− 1)
(K,N)
π−1,m(μF
3),
which comple es he p oo . 
Recall, ha in ou de ini ion o e y s ic CD(K,N) -spaces, we only equi e he
con exi y o he en opy o hold o he c i ical exponen N, opposed o he de ini ion o
gene al CD(K,N) -spaces. The e o e, he ollowing immedia e co olla y is a non- i ial
ac in his se ing.
Co olla y 4.5 A me ic measu e space sa is ying e y s ic CD(K,N)(CD∗(K,N)) -
condi ion, sa is ies e y s ic CD(K,N)(CD∗(K,N)) -condi ion o any N>N.
Acknowledgemen s The au ho would like o hank En ico Pasquale o o sugges ions and discussions
ha led o he p esen pape . The au ho also acknowledges he suppo by he Academy o Finland, p ojec s
#314789 and #312488.
Funding Open Access unding p o ided by Uni e si y o Jy ¨askyl¨a (JYU).
Open Access This a icle is licensed unde a C ea i e Commons A ibu ion 4.0 In e na ional License, which
pe mi s use, sha ing, adap a ion, dis ibu ion and ep oduc ion in any medium o o ma , as long as you gi e
app op ia e c edi o he o iginal au ho (s) and he sou ce, p o ide a link o he C ea i e Commons licence,
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108 Page 20 o 20 T. Schul z
Re e ences
1. Amb osio, L., Gigli, N., Sa a é, G.: Me ic measu e spaces wi h Riemannian Ricci cu a u e bounded
om below. Duke Ma h. J. 163(7), 1405–1490 (2014)
2. Bache , K., S u m, K.-T.: Localiza ion and enso iza ion p ope ies o he cu a u e-dimension condi-
ion o me ic measu e spaces. J. Func . Anal. 259(1), 28–56 (2010)
3. Ca alle i, F., Milman, E.: The globaliza ion heo em o he cu a u e dimension condi ion. In en .
Ma h. 226(1), 1-137 (2021). h ps://doi.o g/10.1007/s00222-021-01040-6
4. Ca alle i, F., Mondino, A.: Sha p and igid isope ime ic inequali ies in me ic-measu e spaces wi h
lowe Ricci cu a u e bounds. In en . Ma h. 208(3), 803–849 (2017)
5. Ca alle i, F., S u m, K.-T.: Local cu a u e-dimension condi ion implies measu e-con ac ion p ope y.
J. Func . Anal. 262(12), 5110–5127 (2012)
6. Gigli, N., Rajala, T., S u m, K.-T.: Op imal maps and exponen ia ion on ini e-dimensional spaces wi h
Ricci cu a u e bounded om below. J. Geom. Anal. 26(4), 2914–2929 (2016)
7. Lisini, S.: Cha ac e iza ion o absolu ely con inuous cu es in Wasse s ein spaces. Calc. Va . Pa ial
Di e . Equ. 28(1), 85–120 (2007)
8. Lo , J., Villani, C.: Ricci cu a u e o me ic-measu e spaces ia op imal anspo . Ann. Ma h. 169(3),
903–991 (2009)
9. Oh a, S.-I.: P oduc s, cones, and suspensions o spaces wi h he measu e con ac ion p ope y. J. Lond.
Ma h. Soc. 76(1), 225–236 (2007)
10. Rajala, T.: Local Poinca é inequali ies om s able cu a u e condi ions on me ic spaces. Calc. Va .
Pa ial Di e . Equ. 44(3–4), 477–494 (2012)
11. Rajala, T.: Imp o ed geodesics o he educed cu a u e-dimension condi ion in b anching me ic
spaces. Disc e . Con in. Dyn. Sys . 33(7), 3043–3056 (2013)
12. Rajala, T.: Failu e o he local- o-global p ope y o CD(K,N)spaces. Ann. Sc. No m. Supe . Pisa
Cl. Sci. 15, 45–68 (2016)
13. McCann, R.J.: A con exi y p inciple o in e ac ing gases. Ad . Ma h. 128(1), 153–179 (1997)
14. Schul z, T.: Exis ence o op imal anspo maps in e y s ic CD(K,∞)-spaces. Calc. Va . Pa ial
Di e . Equ. 57(5), 139, 11 (2018)
15. S u m, K.-T.: On he geome y o me ic measu e spaces, I. Ac a Ma h. 196(1), 65–131 (2006)
16. S u m, K.-T.: On he geome y o me ic measu e spaces, II. Ac a Ma h. 196(1), 133–177 (2006)
17. Villani, C.: Op imal T anspo . Old and New,. G undleh en de Ma hema ischen Wissenscha en [Fun-
damen al P inciples o Ma hema ical Sciences], ol. 338. Sp inge -Ve lag, Be lin (2009)
Publishe ’s No e Sp inge Na u e emains neu al wi h ega d o ju isdic ional claims in published maps
and ins i u ional a ilia ions.
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