Hausdorff measures and the Morse-Sard theorem

Publ. Ma . 45 (2001), 149–162
HAUSDORFF MEASURES AND THE MORSE-SARD
THEOREM
Ca los Gus a o T. de A. Mo ei a
Abs ac
Le F:U⊂Rn→Rmbe a diffe en iable unc ion and p<man
in ege . I k≥1 is an in ege , α∈[0,1] and F∈Ck+(α), i we se
Cp(F)={x∈U| ank(D (x)) ≤p} hen he Hausdo ff measu e
o dimension (p+n−p
k+α)o F(Cp(F)) is ze o.
1. In oduc ion
The Mo se-Sa d heo em is a undamen al heo em in analysis ha
is in he basis o ans e sali y heo y and diffe en ial opology. The
classical Mo se-Sa d heo em s a es ha he image o he se o c i ical
poin s o a unc ion F:Rn→Rmo class Cn−m+1 has ze o Lebesgue
measu e in Rm. I was p o ed by Mo se ([M]) in he case m= 1 and by
Sa d ([S1]) in he gene al case.
Due o i s heo e ical impo ance, he Mo se-Sa d heo em was gen-
e alized in many di ec ions. Many o hese gene aliza ions a e ela ed
wi h Hausdo ff measu es and Hausdo ff dimensions.
Gi en a me ic space Xand a posi i e eal numbe α, we define he
Hausdo ff measu e o dimension αassocia ed o a co e ing U=(Uλ)λ∈L
o Xby bounded se s Uλby mα(U)=λ∈L(diam Uλ)α, whe e diam Uλ
deno es he diame e o Uλ, and, i we define he no m o a co e ing U
by ||U|| = supU∈U(diam U), hen he Hausdo ff measu e o dimension α
o Xis mα(X) = lim in
Uco e ing o X
||U||→0
mα(U).
I is no difficul o see ha he e is a unique d∈[0,+∞] such ha i
α>d hen mα(X) = 0 and i α<d hen mα(X)=+∞. This numbe d
is called he Hausdo ff dimension o X. I is easy o see ha i X⊂Rn
hen i s Hausdo ff dimension d=: HD(X) belongs o [0,n].
150 C. G. T. de A. Mo ei a
Sa d himsel p o ed ha i Cp(F)={x∈Rn| ank(DF(x)) ≤p}
hen o any ε>0 he e is k∈Nsuch ha i Fis Ck hen F(Cp(F))
has ze o Hausdo ff measu e o dimension p+ε([S2]). This esul was
made mo e p ecise by Fede e ([F]), who p o ed ha i k∈N hen he
Hausdo ff measu e o dimension p+n−p
ko F(Cp(F)) is ze o. We should
also men ion he wo ks o Chu ch ([Ch1], [Ch2]), which ga e mo e
esul s abou he s uc u e o he se o c i ical alues o diffe en iable
maps. La e , Yomdin ([Y]) p o ed ha he Hausdo ff dimension o
F(Cp(F)) is a mos p+n−p
k+α, p o ided ha F∈Ck+α, whe e k∈N
and 0 ≤α<1. Mo e ecen ly, Ba es ([B2]) p o ed ha i F∈Ck+α
wi h k∈N,0<α≤1 and p+n−p
k+α=m hen F(Cp(F)) has ze o
Lebesgue measu e in Rm( his in pa icula imp o es he hypo hesis o
he classical Mo se-Sa d heo em om F∈Cn−m+1 o F∈Cn−m+Lips.,
i.e., F∈Cn−mand Dn−mFLipschi z).
The aim o his wo k is o gene alize he men ioned esul s by p o -
ing a gene al e sion o he Mo se-Sa d Theo em in ol ing Hausdo ff
measu es. Le k≥1 be an in ege and α∈[0,1]. We say ha a
unc ion F:U⊂Rn→Rmis o class Ck+(α)a a subse Ao Ui F
is Ckin Uand o each x∈A he e a e εx>0, Kx>0 such ha
|y−x|<ε
x⇒|DkF(y)−DkF(x)|≤Kx|y−x|α( his is less es ic i e
han supposing F∈Ck+α). Ou main esul is he ollowing
Theo em. Le F:U⊂RnCk
−→ Rmand le p<mbe an in ege . I
Cp(F):={x∈U| ank(DF(x)) ≤p}and i Fis o class Ck+(α)a
Cp(F) hen he Hausdo ff (p+n−p
k+α)-measu e o F(Cp(F)) is ze o.
In pa icula , i k+α=n−p
m−p, we eco e he esul o [B2], wi h a
weake hypo hesis. We ema k ha i p+n−p
k+α<m, he Hausdo ff (p+
n−p
k+α)-measu e is no he Lebesgue measu e o a p oduc measu e in Rm,
and so we can no use Fubini’s Theo em. This difficul y is sol ed in
he p esen pape by eplacing he use o Fubini’s heo em by a ca e ul
decomposi ion o he c i ical se , combined wi h a pa ame ized s ong
e sion o he main lemma o Mo se’s pape ([M, Theo em 2.1]).
We shall also gi e examples ha show ha ou esul is qui e sha p,
by gi ing coun e examples o sligh changes o he hypo hesis o o he
conclusion.
2. Func ions whose ze os include a gi en se
We shall p o e he e a e sion o Theo em 3.6 o [M] and Lemma 3.4.2
o [F], which will be undamen al o he la e esul s.
Hausdo Measu es and he Mo se-Sa d Theo em 151
Theo em 2.1. Le k≥1,α∈[0,1],n>pand A⊂U⊂Rn, whe e
Uis an open se . Then he e a e se s A1,A
2... ⊂Asuch ha A=
∞
i=1 Ai, whe e o each i=1,2,... he e is a unc ion ψi:Bi×Vi
C1
−→ U
whe e Biis a ball in some R i, i≥0and Viis a ball in Rpsuch ha
ψi(x, y)=(
ψi(x, y),y), and |ψi(x1,y
1)−ψi(x2,y
2)|≥|(x1,y
1)−(x2,y
2)|,
∀(x1,y
1),(x2,y
2)∈Bi×Viand Ai⊂ψi(Bi×Vi), wi h he ollowing
p ope y: We can w i e Ai=A
i∪A
iso ha ψ−1
i(A
i)has measu e ze o
in Bi×Vi, and i :U→R anishes in Aand is Ck+(α)a Awe
ha e:
•lim sup
(x,y0)→(x0,y0)
(ψi(x, y0))
|x−x0|k+α<+∞,∀(x0,y
0)∈Bi×Visuch ha
ψi(x0,y
0)∈Ai,
•lim
(x,y0)→(x0,y0)
(ψi(x, y0))
|x−x0|k+α=0,∀(x0,y
0)∈Bi×Visuch ha
ψi(x0,y
0)) ∈A
i.
P oo : Le us conside fi s he case k= 1 and d (x)· =0∀x∈A,
∈Rn−p×{0}. In his case we ake A=(A∩A)∪A whe e Ais
he se o densi y poin s o Ain he di ec ion o Rn−p×{0}((x, y)∈
A⇒lim
ε→0
m((Bε(x)×{y})∩A)
m(Bε(x)) = 1, whe e mis he (n−p)-dimensional
measu e). The measu e o A =A−Ais ze o, since i is ze o in each
plane Rn−p×{y}.
Fo (x0,y
0)∈A ake B((x0,y
0),ε(x0,y
0)) a ball con ained in Uand
ψ=Id|B((x0,y0),ε(x0,y0)). We ha e lim sup
(x,y0)→(x0,y0)
(x, y0)
|x−x0|1+α<+∞,
since (x, y0)= (x, y0)− (x0,y
0)=d ( x0+(1− )x)(x−x0), ∈
(0,1) ⇒| (x, y0)|≤Kx0|x−x0|1+α. Fo (x0,y
0)∈A,
lim
δ→0
1
ol(Sn−p−1)Sn−p−11
δδ
0
χA(x0+ , y0)d d =1,
so ∀ε>0∃δ0>0 s. . |x−x0|<δ
0⇒∃ ∈Sn−p−1wi h
 −x−x0
|x−x0|<ε
and 
1
|x−x0||x−x0|
0
χA(x0+ , y0)d −1<ε,
152 C. G. T. de A. Mo ei a
so, i ˜x=x0+|x−x0| ,
| (x, y0)− (x0,y
0)|≤| (x, y0)− (˜x, y0)|+| (˜x, y0)− (x0,y
0)|,
bu
(x, y0)− (˜x, y0)=d (θ +(1−θ)˜x, y0)·(x−˜x),θ∈(0,1)
⇒| (x, y0)− (˜x, y0)|≤Kx0|x−x0|α·ε|x−x0|=εKx0|x−x0|1+α
and
(˜x, y0)− (x0,y
0)
=|˜x−x0|
0
d (x0+ , y0)· d
≤Kx0|x−x0|α·m ∈[0,|˜x−x0|]|∂
∂x(x0+ , y0)=0

≤Kx0|x−x0|α·ε|x−x0|=εKx0|x−x0|1+α.
So
| (x, y0)|=| (x, y0)− (x0,y
0)|≤2εKx0|x−x0|1+α
⇒lim
(x,y0)→(x0,y0)
(x, y0)
|x−x0|1+α=0.
We can ake a coun able subco e ing o Aby he B((x0,y
0),ε(x0,y
0))
o finish he p oo in his case.
Conside now he case k≥1, na bi a y. We ha e A=A∗∪A∗∗
whe e A∗={x∈A|∃g:UCk
−→ R,g|A≡0, ∃ ∈Rn−p×{0},
dg(x)· =0}.A∗∗ =A A∗.I (x0,y
0)∈A∗ he e is gas abo e, so
he e is ε>0 such ha g−1(0) ∩Bε(x0,y
0) is con ained in he image
o ψ:B×VCk
−→ Uwhe e Bis a ball in Rn−p−1, as in he s a emen ,
and A⊂g−1(0). Taking a coun able subco e ing o A∗by hese balls
we educe he p oo in his case o a case wi h smalle n.I k= 1, he
esul was ye p o ed o A∗∗.I k>1 , and assuming by induc ion he
esul o k−1, we ha e
A∗∗ =∞

i=1
A∗∗
i,A
∗∗
i=(A∗∗
i)∪(A∗∗
i),A
∗∗
i⊂ψi(Bi×Vi),ψ
i∈C1,
Hausdo Measu es and he Mo se-Sa d Theo em 153
ψi(x0,y
0)∈A∗∗
i⇒lim sup
x→x0
||d (ψi(x, y0))|Rn−p×{0}||
|x−x0|k−1+α<+∞
⇒lim sup
x→x0
| (ψi(x, y0))|
|x−x0|k+α<+∞
and
ψi(x0,y
0)∈(A∗∗
i)⇒lim
x→x0
||d (ψi(x, y0))|Rn−p×{0}||
|x−x0|k−1+α=0
⇒lim
x→x0
(ψi(x, y0))
|x−x0|k+α=0,
bo h by he mean alue heo em, and he p oo is finished by induc-
ion.
Co olla y 2.2. Le k≥1,α∈[0,1],n>pand A⊂U⊂Rn, whe e
Uis an open se . Then he e a e se s A1,A
2... ⊂Asuch ha A=
∞
i=1 Ai, whe e o each i=1,2,... he e is a unc ion ψi:Bi×Vi
C1
−→ U
whe e Biis a ball in some R i, i≥0and Viis a ball in Rpsuch ha
ψi(x, y)=(
ψi(x, y),y), and |ψi(x1,y
1)−ψi(x2,y
2)|≥|(x1,y
1)−(x2,y
2)|,
∀(x1,y
1),(x2,y
2)∈Bi×Viand Ai⊂ψi(Bi×Vi), wi h he ollowing
p ope y: We can w i e Ai=A
i∪A
iso ha ψ−1
i(A
i)has measu e ze o
in Bi×Vi, and i :U→Ris Ck+(α)a Aand Dx ≡0in Awe ha e:
•lim sup
(x,y0)→(x0,y0)
| (ψi(x, y0)) − (ψi(x0,y
0))|
|x−x0|k+α<+∞,∀(x0,y
0)∈Bi×
Visuch ha ψi(x0,y
0)∈Ai,
•lim
(x,y0)→(x0,y0)| (ψi(x, y0)) − (ψi(x0,y
0))|
|x−x0|k+α=0,∀(x0,y
0)∈Bi×Vi
such ha ψi(x0,y
0)) ∈A
i.
P oo : I k≥2 his is an immedia e consequence o Theo em 2.1 applied
o Dx and o he mean alue heo em. I k= 1 his can be p o ed
exac ly as he case k= 1 o he Theo em 2.1.
Co olla y 2.3. In he s a emen s o Theo em 2.1 and Co olla y 2.2,
o any x∈Bis. . ψi(x)∈Ai he e a e εx>0,Kx>0such ha
|y−x|<ε
x⇒| (ψi(y)) − (ψi(x))|≤Kx|y−x|k+α, and o any ε>0
he e is a δ>0so ha λ(ψ−1
i(Ai)∩B (x))
λ(B (x)) >1−δ⇒| (ψi(y))− (ψi(x))|≤
εKx k+α,i ≤εxand |y−x|≤ (δdepends only on εand n, bu no
on o on x).

154 C. G. T. de A. Mo ei a
P oo : This is only a mo e p ecise o mula ion o he esul s p o ed in
he demons a ion o he heo em.
Rema k 2.1.Fo k= 0 we ha e he same esul s, excep he s a e-
men lim
y→x
(ψi(y))
|y−x|k+α= 0, o each x∈Bisuch ha ψi(x)∈A
i.
3. The main esul s
Lemma 3.1. Le A⊂Rmwi h λ(A)<∞and le Ube a amily o
balls B (x),x∈Asuch ha o each x∈A he e is an εx>0such ha
≤εx⇒B (x)∈U. Then o each ε>0 he e a e xn∈A, n>0
wi h B n(xn)∈Uand A⊂∞
n=1 B n(xn)such ha ∞
n=1 λ(B n(xn)) <
λ(A)+ε.
P oo : This lemma is essen ially he Vi ali co e ing heo em om mea-
su e heo y. Take U⊃Aan open se wi h λ(U)<λ(A)+ε
2.I we
choosed B˜ 1(x1),... ,B˜ n(xn), define sn= sup{ >0|∃x∈As. . <
εx
5,B (x)⊂Uand B (x)∩(B˜ 1(x1)∪···∪B˜ n(xn)) = ∅}. Choose
B˜ n+1 (xn+1) such ha ˜ n+1 >sn
2,˜ n+1 <εxn+1
5,B˜ n+1 (xn+1)⊂Uand
B˜ n+1 (xn+1)∩(B˜ 1(x1)∪···∪B˜ n(xn)) = ∅. Since he B˜ i(xi) a e disjoin
and con ained in Uwe ha e ∞
i=1 λ(B˜ i(xi)) <λ(A)+ε
2, and so he e is
an0∈Nsuch ha ∞
i=n0λ(B5˜ i(xi)) <ε
2. We ake B i(xi)=B˜ i(xi),
i<n
0and B i(xi)=B5˜ i(xi), i≥n0.
Clea ly we ha e ∞
i=1 λ(B i(xi)) <λ(A)+ε. To p o e ha A⊂
∞
n=1B n
(xn), ake x∈Aand =min{˜ n0,ε
x/5,d(x, Uc∪i<n0B i
(xi))}.
I >0, ake n≥n0such ha sn< ≤sn−1(we ha e ≤˜ n0≤sn0−1),
and no e ha sn< ⇒B (x)∩(B˜ 1(x1)∪···∪B˜ n(xn)) =∅⇒∃i≤n
such ha B (x)∩B˜ i(xi)=∅. We ha e n≥n0since ≤d(x, B˜ i(xi)),
and ˜ i>sn−1
2≥
2, since i≤n. The e o e, we ha e x∈B5˜ i(xi). I =0
hen x∈B i(xi) o some i<n
0. This p o es ha A⊂∞
n=1 B n(xn).
Taking ˜ n=( λ(A)+ε
∞
i=1 λ(B i(xi)) )1/2m· n, we ha e A⊂∞
n=1 B˜ n(xn), wi h
∞
n=1 λ(B˜ n(xn)) = (λ(A)+ε)1/2(∞
i=1 λ(B i(xi)))1/2<λ(A)+ε.
Rema k 3.1.In he Lemma 3.1 we can eplace a amily o balls B (x)by
a amily o cubes C (x)=m
i=1[xi− , xi+ ], whe e x=(x1,... ,x
m),
using he same p oo .
Lemma 3.2. Le F:U⊂Rn→Rmbe a unc ion, A⊂Uand d>0
such ha o any x∈A he e a e εx>0,Kx>0such ha md(F(Bε(x)∩
A)) ≤Kx.λ(Bε(x)),∀ε<ε
x, whe e mdis he Hausdo ff measu e
o dimension d, and he e is A⊂Asuch ha λ(A A)=0and
lim
ε→0
md(F(Bε(x)∩A))
λ(Bε(x)) =0,∀x∈A. Then md(F(A)) = 0.
Hausdo Measu es and he Mo se-Sa d Theo em 155
Rema k 3.2.The same esul is ue i we eplace Bε(x)byCε(x).
Rema k 3.3.We can eplace he condi ion
“md(F(Bε(x)∩A)) ≤Kxλ(Bε(x)),∀ε<ε
x”
by
“F(Bε(x)∩A) can be co e ed by balls Bδi(yi),i∈N,
wi h ∞

i=1
δd
i≤Kxλ(Bε(x)),∀ε<ε
x”,
and he condi ion
“ lim
ε→0
md(F(Bε(x)∩A)
λ(Bε(x)) =0,∀x∈A”
by
“F(Bε(x)∩A) can be co e ed by balls Bδ(ε)
i
(yi),i∈N
wi h lim
ε→0∞
i=1(δ(ε)
i)d
λ(Bε(x)) =0,∀x∈A”.
The p oo emains essen ially he same, and Rema k 3.2 is s ill alid.
Rema k 3.4.I we eplace he condi ions o his lemma by “F(Bε(x)∩A)
can be co e ed by balls Bδi(yi), i∈N, wi h ∞
i=1 δd
i≤kλ(Bε(x)),
∀ε<ε
x(no e ha he e kdoes no depend on x), and λ(A)<∞”, hen
we can conclude, using he same p oo , ha md(F(A)) ≤kλ(A).
P oo : We may suppose ha Ahas fini e Lebesgue measu e, since Ais a
coun able union o se s wi h fini e measu e, and a coun able union o se s
wi h Hausdo ff d-measu e ze o has Hausdo ff d-measu e ze o. Mo eo e ,
since A=∞
k=1 Ak, whe e Ak={x∈A|Kx≤k}, we may suppose
Kx≤K,∀x∈A. Le Cbe he Lebesgue measu e o A.
Le ε>0. Fo each x∈A ake δx>0 such ha Bδx(x)⊂Uand
≤δx⇒md(F(B (x)∩A))
λ(B (x)) ≤ε
2(C+1) . By he Lemma 3.1 we can co e A
by ∞
n=1 B n(xn) wi h
∞

n=1
λ(B n(xn)) <C+1
156 C. G. T. de A. Mo ei a
and
n≤δxn⇒∞

n=1
md(F(B n(xn)∩A))
≤ε
2(C+1)·(C+1)= ε
2⇒md(F(A)) ≤ε
2.
By Lemma 3.1 we can co e A Aby ∞
n=1 B˜ n(xn) such ha B˜ n(xn)⊂
Uand ˜ n<ε
xn,∀n∈N, wi h
∞

n=1
λ(B˜ n(xn)) <ε
2K⇒∞

n=1
md(F(B˜ n(xn)))
≤ε
2K·K=ε
2⇒md(F(A A))
≤ε
2⇒md(F(A)) ≤ε
2+ε
2=ε.
Since ε>0 is a bi a y we ha e md(F(A)) = 0.
We fi s use Lemma 3.2 o p o e he ollowing s ong e sion o Con-
s an in’s esul ([Co]), ha does no suppose con inui y o he de i a-
i es. He e we do no suppose diffe en iabili y in e e y poin , bu only
in he se o c i ical poin s unde conside a ion.
Theo em 3.3. Le F:X⊂Rn→Rnbe a unc ion, and le A={x∈
X|DF(x)exis s and is no su jec i e}. Then λ(F(A)) = 0.
P oo : I is a simple consequence o Lemma 3.2, since i x∈A hen
lim
→0
λ(F(B (x)))
λ(B (x)) = 0. Indeed, x∈A⇒F(x+h)=F(x)+DF(x).h +
(h), whe e lim
h→0
(h)
|h|= 0. Le K=||DF(x)||, and le ε∈(0,1). Le
δ>0 such ha |h|≤δ⇒| (h)|
|h|<ε
2(K+1)n−1. Then, i |h|≤δ,
F(x+h)−F(x) belongs o an ε.|h|
2(K+1)n−1neighbou hood o a ball o
adius K|h|in a subspace o Rno dimension n−1 (a fixed subspace
o Rno dimension n−1 which con ains he image o DF), and hus
belongs o he o hogonal p oduc o a ball o adius (K+1)|h|in his
subspace by an in e al o adius ε|h|
2(K+1)n−1. The e o e, λ(F(B (x)) ≤
ε. . n−1(K+1)n−1
(K+1)n−1 n−1=ε n n−1, whe e n−1is he olume o he uni a y
ball in Rn−1, and, since ε>0 is a bi a y, lim
→0
λ(F(B (x)))
λ(B (x)) =0.
Theo em 3.4. Le F:U⊂RnCk
−→ Rmbe a unc ion o class Ck+(α)(α∈
(0,1]) a Cp(F):={x∈U| ank(DF(x)) ≤p}. Then he Hausdo ff
measu e o dimension d=p+n−p
k+αo F(Cp(F)) is ze o, ∀p<min{m, n}.
Hausdo Measu es and he Mo se-Sa d Theo em 157
P oo : Since Cp(F)=p
=0{x∈U| ank(DF(x)) = }, and +n−
k+α≤
p+n−p
k+α o 0 ≤ ≤p, we may es ic ou a en ion o 
Cp(F)={x∈
U| ank(DF(x)) = p}.I x0∈Cp(F), we ha e, a e a change o
coo dina es o class Ck,F(z,y)=(z,G(z,y)), wi h (z,y)∈Rp×Rn−p
and G(z,y)∈Rm−p, in a neighbou hood Vo x0=(z0,y
0). We shall
es ic ou a en ion o his neighbou hood. We ha e x=(z,y)∈
Cp(F)⇔DyG(z,y) = 0. We can apply he esul s o he Sec ion 2
(Theo em 2.1, Co olla y 2.3 and Rema k 2.1) o he unc ion DyG,
and ob ain he decomposi ion A=∞
i=1 Ai,Ai⊂ψi(Vi×Bi), whe e
A={(z,y)∈V|DyG(z,y)=0}. Le us fix such an Ai.
Since ψ−1
i(Ai)=m∈N{x∈ψ−1
i(Ai)|εx≥1
m,K
x≤m},wemay
suppose εx≥1
M,Kx≤M,∀x∈ψ−1
i(Ai), o some fixed Mand also
ha Vhas fini e Lebesgue measu e λ(V).
Wi h hese assump ions, we shall p o e ha he e is a cons an K0
such ha o any X⊂V,ν>0, we can co e F(Ai∩X) by balls Bδi(pi)
so ha ∞
i=1 δd
i≤K0(λ(X)+ν). Fo his, gi en a poin x∈Ai∩X
and an ε< 1
2√nM , we can di ide he cube Cε(x)=Cε(z)×Cε(y)
in o ([ε1−(k+α)]+1)
pboxes Cδ(zi)×Cε(y), δ<ε
k+α. I he e is some
poin (zi,y
i)in(Cδ(zi)×Cε(y)) ∩(Ai∩X), hen o any poin (z
i,y
i)in
(Cδ(z1)×Cε(y)) ∩(Ai∩X), we ha e |F(z
i,y)−F(zi,y
i)|≤|F(z
i,y)−
F(zi,y)|+|F(zi,y)−F(zi,y
i)|≤Kδ+|F(zi,y
i)−F(zi,y
i)|(whe e K
is √p imes a Lipschi z cons an o F|Vwhich we may suppose o exis )
≤Kδ+|G(zi,y
i)−G(zi,y
i)|.
Obse e now ha (zi,y
i)=(zi,
ψi(p1)) and (z
i,y
i)=(zi,
ψi(p2)), o
some p1,p2in {zi}×Biwi h |p1−p2|≤|yi−y
i|≤2ε√n. Le γ:[0,1] →
Vi×Bibe a s aigh pa h joining p1and p2. Then G(zi,y
i)−G(zi,y
i)=
1
0
∂G
∂y (γ( )) ·γ( )d , whe e γ:= ψi◦γ. We ha e ∂G
∂y (γ(0)) = 0, so




∂G
∂y (γ( ))


=



∂G
∂y (γ( )) −∂G
∂y (γ(0))



≤M|p1−p2|k+α−1
≤M(2ε√n)k+α−1⇒



∂G
∂y (γ( ))


|γ( )|
≤Kεk+α,
o some cons an K. Indeed, |γ( )|is limi ed by a cons an mul iple
o |p1−p2|≤2√nε.So
|G(z1,y
i)−G(zi,y
i)|≤1
0
∂G
∂y (γ( )) ◦γ( )d ≤Kεk+α