Sub-Gaussian heat kernel bounds imply singularity of ...sesaki/JG2019/KAJINO.pdf · Sub-Gaussian...

Sub-Gaussian heat kernel boundsimply singularity of energy measures

Naotaka Kajino (Kobe University)

梶野直孝（神戸大学）Joint work with Mathav Murugan (UBC)

Japanese-GermanOpenConf. on Stochastic Analysis 2019

@Fukuoka University, Fukuoka, Japan

September 6, 2019

15:50–16:20β=2 (Gaussian)⇒Γ(u, u)≪µ.

β>2 (sub-Gaussian)⇒Γ(u, u)⊥µ!

pt(x, y)≍cµ(B(x, t

)−1exp(−c(d(x, y)β/t)

1β−1

1 Introduction: Str. local Dirich. sp.& energy meas.

▷(K, d, µ, E,F): strongly local reg. symmet. Dir. sp.

↔({Xt}t≥0,{Px}x∈K∂):µ-sym. diffusion, no killing

1β−1

)Fact. u(Xt)−u(X0)=M

[u]t +N

[u]t &M [u] satisfies∫

Kf d ∃1Γ(u, u)=limt↓0 t

−1Eµ

[∫ t

0f(Xs)d⟨M [u]⟩s

]=limt↓0

∫Kf(x)Ex

[|u(Xt) − u(X0)|2

]dµ(x)

=E(fu, u)−E(f, u2)/2 = “∫K

f · |∇u|2 dµ”.

•Assume Px[Xt∈dy]=∃pt(x, y) dµ(y) & ∃β>1,

1β−1

[u]t +N

−1Eµ

[∫ t

0f(Xs)d⟨M [u]⟩s

]=limt↓0

∫Kf(x)Ex

[|u(Xt) − u(X0)|2

]dµ(x)

=E(fu, u)−E(f, u2)/2 = “∫K

f · |∇u|2 dµ”.

•Assume Px[Xt∈dy]=∃pt(x, y) dµ(y) & ∃β≥2,

1β−1

[u]t +N

−1Eµ

[∫ t

0f(Xs)d⟨M [u]⟩s

]=limt↓0

∫Kf(x)Ex

[|u(Xt) − u(X0)|2

]dµ(x)

=E(fu, u)−E(f, u2)/2 = “∫K

f · |∇u|2 dµ”.

1β−1

)Q.How similar is {u(Xt)}t to 1-dim. BM? (u∈F)

Fact. u(Xt)−u(X0)=M[u]t +N

−1Eµ

[∫ t

0f(Xs)d⟨M [u]⟩s

]=limt↓0

∫Kf(x)Ex

[|u(Xt) − u(X0)|2

]dµ(x)

=E(fu, u)−E(f, u2)/2 = “∫K

f · |∇u|2 dµ”.

1β−1

)Q.How similar is {u(Xt)}t to 1-dim. BM? (u∈F)

−1Eµ

[∫ t

0f(Xs)d⟨M [u]⟩s

]=limt↓0

∫Kf(x)Ex

[|u(Xt) − u(X0)|2

]dµ(x)

=E(fu, u)−E(f, u2)/2 = “∫K

f · |∇u|2 dµ”.

1β−1

)Q′. How similar is {M [u]

t }t to 1-dim. BM? (u∈F)

−1Eµ

[∫ t

0f(Xs)d⟨M [u]⟩s

]=limt↓0

∫Kf(x)Ex

[|u(Xt) − u(X0)|2

]dµ(x)

=E(fu, u)−E(f, u2)/2 = “∫K

f · |∇u|2 dµ”.

1β−1

−1Eµ

[∫ t

0f(Xs)d⟨M [u]⟩s

]=limt↓0

∫Kf(x)Ex

[|u(Xt) − u(X0)|2

]dµ(x)

=E(fu, u)−E(f, u2)/2 = “∫K

f · |∇u|2 dµ”.

1β−1

−1Eµ

[∫ t

0f(Xs)d⟨M [u]⟩s

]=limt↓0

∫Kf(x)Ex

[|u(Xt) − u(X0)|2

]dµ(x)

=E(fu, u)−E(f, u2)/2 = “∫K

f · |∇u|2 dµ”.

1β−1

−1Eµ

[∫ t

0f(Xs)d⟨M [u]⟩s

]=limt↓0

∫Kf(x)Ex

[|u(Xt) − u(X0)|2

]dµ(x)

=E(fu, u)−E(f, u2)/2 = “∫K

f · |∇u|2 dµ”.

2/9▷(K, d, µ, E,F): strongly local reg. symmet. Dir. sp.

1β−1

[u]t +N

−1Eµ

[∫ t

0f(Xs)d⟨M [u]⟩s

]=E(fu, u)−E(f, u2)/2 = “

f · |∇u|2 dµ”.

Q′. How similar is {M [u]t }t to 1-dim. BM? (u∈F)

Q′′. d⟨M [u]⟩t ⊥ dt?

Thm(K.–Murugan). Assume (K, d) is complete. Then

under pt(x, y)≍β · · · , • β> 2⇒ ∀u ∈ F , Γ(u, u)⊥µ.• β= 2⇒ ∀u ∈ F , Γ(u, u)≪µ.

1β−1

[u]t +N

−1Eµ

[∫ t

0f(Xs)d⟨M [u]⟩s

]=E(fu, u)−E(f, u2)/2 = “

f · |∇u|2 dµ”.

Q′′. d⟨M [u]⟩t ⊥ dt?

1β−1

[u]t +N

−1Eµ

[∫ t

0f(Xs)d⟨M [u]⟩s

]=E(fu, u)−E(f, u2)/2 = “

f · |∇u|2 dµ”.

Q′′. d⟨M [u]⟩t ⊥ dt? Q′′′. Γ(u, u) ⊥ µ?

1β−1

[u]t +N

−1Eµ

[∫ t

0f(Xs)d⟨M [u]⟩s

]=E(fu, u)−E(f, u2)/2 = “

f · |∇u|2 dµ”.

Q′′. d⟨M [u]⟩t ⊥ dt? Q′′′. Γ(u, u) ⊥ µ?

1β−1

[u]t +N

−1Eµ

[∫ t

0f(Xs)d⟨M [u]⟩s

]=E(fu, u)−E(f, u2)/2 = “

f · |∇u|2 dµ”.

Q′′. d⟨M [u]⟩t ⊥ dt? Q′′′. Γ(u, u) ⊥ µ?

1β−1

)▷∃1Γ(u, u):

∫Kf dΓ(u, u)=E(fu, u)−E(f, u2)/2

Preceding ⊥ results (ONLY for self-similar sets!)

•(Kusuoka ’89,’93) p.c.f. s.-s. sets⊃ d-dimSG (d≥2)

•(Ben-Bassat–Strichartz–Teplyaev ’99) Similar results, simpler prf

•(Hino ’05) General s.-s. sets⊃∀GSCs!(cf.Barlow–Bass ’92, ’99, Kusuoka–Zhou ’92, Barlow–Bass–Kumagai ’06)•(Hino–Nakahara ’06) p.c.f., topol. criteria excluding≪case

1β−1

)▷∃1Γ(u, u):

1β−1

)▷∃1Γ(u, u):

1β−1

)▷∃1Γ(u, u):

1β−1

)▷∃1Γ(u, u):

2 General setting & conditions for HK estimates

▷Ψ: homeo. on [0,∞) with 1<∃β1≤∃β2,∃c≥1,

0<∀r ≤∀R, c−1(Rr

≤Ψ(R)

Ψ(r)≤c

▷Φ(R, t) :=ΦΨ(R, t) := supr>0

(R/r − t/Ψ(r)

HKE(Ψ): ∃pt(x, y),∀t > 0, (µ-almost)∀x, y ∈ K,

pt(x, y)≍cµ(B(x,Ψ−1(t))

)−1exp

(−cΦ(cd(x, y), t)

VD: ∃cD>0, 0<µ(B(x, 2r))≤cDµ(B(x, r))<∞.

PI(Ψ): ∃cP>0, ∃A≥1, ∀x ∈ K, ∀r > 0, ∀u ∈ F ,∫B(x,r)

∣∣u−uB(x,r)∣∣2dµ≤cPΨ(r)

∫B(x,Ar)

dΓ(u, u).

0<∀r ≤∀R, c−1(Rr

≤Ψ(R)

Ψ(r)≤c

▷Φ(R, t) :=ΦΨ(R, t) := supr>0

(R/r − t/Ψ(r)

)−1exp

VD: ∃cD>0, 0<µ(B(x, 2r))≤cDµ(B(x, r))<∞.

PI(Ψ): ∃cP>0, ∃A≥1, ∀x ∈ K, ∀r > 0, ∀u ∈ F ,∫B(x,r)

∫B(x,Ar)

dΓ(u, u).

0<∀r ≤∀R, c−1(Rr

≤Ψ(R)

Ψ(r)≤c

▷Φ(R, t) :=ΦΨ(R, t) := supr>0

(R/r − t/Ψ(r)

)−1exp

VD: ∃cD>0, 0<µ(B(x, 2r))≤cDµ(B(x, r))<∞.

PI(Ψ): ∃cP>0, ∃A≥1, ∀x ∈ K, ∀r > 0, ∀u ∈ F ,∫B(x,r)

∫B(x,Ar)

dΓ(u, u).

0<∀r ≤∀R, c−1(Rr

≤Ψ(R)

Ψ(r)≤c

▷Φ(R, t) :=ΦΨ(R, t) := supr>0

(R/r − t/Ψ(r)

)−1exp

VD: ∃cD>0, 0<µ(B(x, 2r))≤cDµ(B(x, r))<∞.

PI(Ψ): ∃cP>0, ∃A≥1, ∀x ∈ K, ∀r > 0, ∀u ∈ F ,∫B(x,r)

∫B(x,Ar)

dΓ(u, u).

0<∀r ≤∀R, c−1(Rr

≤Ψ(R)

Ψ(r)≤c

▷Φ(R, t) :=ΦΨ(R, t) := supr>0

(R/r − t/Ψ(r)

)−1exp

VD: ∃cD>0, 0<µ(B(x, 2r))≤cDµ(B(x, r))<∞.

PI(Ψ): ∃cP>0, ∃A≥1, ∀x ∈ K, ∀r > 0, ∀u ∈ F ,∫B(x,r)

∫B(x,Ar)

dΓ(u, u).

5/9HKE(Ψ): ∃pt(x, y),∀t > 0, (µ-almost)∀x, y ∈ K,

)−1exp

VD: ∃cD>0, 0<µ(B(x, 2r))≤cDµ(B(x, r))<∞.

PI(Ψ): ∃cP>0, ∃A≥1, ∀x ∈ K, ∀r > 0, ∀u ∈ F ,∫B(x,r)

∫B(x,Ar)

dΓ(u, u).

CS(Ψ): ∃cS>0, ∀x ∈ K, ∀R, r > 0, ∃φx,R,r ∈ F ,

1B(x,R) ≤ φ := φx,R,r ≤ 1B(x,R+r) q.e., ∀u ∈ F ,∫u2dΓ(φ)≤

∫BR+r(x)\BR(x)

18φ2dΓ(u)+ cS

Ψ(r)u2dµ.

Thm(Grigor’yan–Hu–Lau ’15, Andres–Barlow ’15,⇒qG:Mu., cf.BBK’06Lierl ’15).

Assume (K, d) is complete. Then HKE(Ψ)⇔HKEa.e.(Ψ)

⇔VD+PI(Ψ)+CS(Ψ)+quasiGeodesic(d)←crucial for us!

)−1exp

VD: ∃cD>0, 0<µ(B(x, 2r))≤cDµ(B(x, r))<∞.

PI(Ψ): ∃cP>0, ∃A≥1, ∀x ∈ K, ∀r > 0, ∀u ∈ F ,∫B(x,r)

∫B(x,Ar)

dΓ(u, u).

CS(Ψ): ∃cS>0, ∀x ∈ K, ∀R, r > 0, ∃φx,R,r ∈ F ,

∫BR+r(x)\BR(x)

18φ2dΓ(u)+ cS

Ψ(r)u2dµ.

)−1exp

VD: ∃cD>0, 0<µ(B(x, 2r))≤cDµ(B(x, r))<∞.

PI(Ψ): ∃cP>0, ∃A≥1, ∀x ∈ K, ∀r > 0, ∀u ∈ F ,∫B(x,r)

∫B(x,Ar)

dΓ(u, u).

CS(Ψ): ∃cS>0, ∀x ∈ K, ∀R, r > 0, ∃φx,R,r ∈ F ,

∫BR+r(x)\BR(x)

18φ2dΓ(u)+ cS

Ψ(r)u2dµ.

3 Main result: Almost dichotomy Γ⊥µ v.s.Γ∼µ

)−1exp

Thm 1 (K.–Murugan). Assume (K, d) is complete andthat VD+PI(Ψ)+CS(Ψ)+quasiGeodesic(d) hold. Then:

(a) lim infλ→∞

lim infr↓0

λ2Ψ(r/λ)

Ψ(r)=0⇒ ∀u ∈ F , Γ(u, u)⊥µ.

(b) lim supr↓0 r−2Ψ(r)>0⇒ ∀u ∈ F , Γ(u, u)≪µ, &µ≪

∑n∈N Γ(un, un) for any dense {un}n∈N ⊂ F .

Rmk.∃in-between case (e.g.Ψ(r)=r2/log(e+r

−1)). Situation UNclear.

)−1exp

(a) lim infλ→∞

lim infr↓0

λ2Ψ(r/λ)

Ψ(r)=0⇒ ∀u ∈ F , Γ(u, u)⊥µ.

)−1exp

(a) lim infλ→∞

lim infr↓0

λ2Ψ(r/λ)

Ψ(r)=0⇒ ∀u ∈ F , Γ(u, u)⊥µ.

)−1exp

(a) lim infλ→∞

lim infr↓0

λ2Ψ(r/λ)

Ψ(r)=0⇒ ∀u ∈ F , Γ(u, u)⊥µ.

)−1exp

(a) lim infλ→∞

lim infr↓0

λ2Ψ(r/λ)

Ψ(r)=0⇒ ∀u ∈ F , Γ(u, u)⊥µ.

7/9Thm 1 (K.–Murugan). Assume (K, d) is complete andthat VD+PI(Ψ)+CS(Ψ)+quasiGeodesic(d) hold. Then:

(a) lim infλ→∞

lim infr↓0

λ2Ψ(r/λ)

Ψ(r)=0⇒ ∀u ∈ F , Γ(u, u)⊥µ.

Example: Scale irregular SGs (Hambly ’92, Barlow–Hambly ’97)

Thm(Barlow–Hambly ’97).∀(ℓn)n≥1 bounded,

(K(ℓn), dgeo, µunif , {XBMt }t) satisfies

HKE(Ψ(ℓn)) with Ψ(ℓn)(ℓ−11 · · · ℓ−1

k ) := t−1ℓ1

· · · t−1ℓk

Prop. 0<∀r ≤∀R ≤1,Ψ(ℓn)(R)

Ψ(ℓn)(r)≥ c

)βmin(ℓn)>2 by Hino–

Nakahara ’06

(a) lim infλ→∞

lim infr↓0

λ2Ψ(r/λ)

Ψ(r)=0⇒ ∀u ∈ F , Γ(u, u)⊥µ.

ℓ = 2 ℓ = 3 ℓ = 4

· · ·⇝Thm(Barlow–Hambly ’97).

∀(ℓn)n≥1 bounded,

k ) := t−1ℓ1

· · · t−1ℓk

Prop. 0<∀r ≤∀R ≤1,Ψ(ℓn)(R)

Ψ(ℓn)(r)≥ c

Nakahara ’06

(a) lim infλ→∞

lim infr↓0

λ2Ψ(r/λ)

Ψ(r)=0⇒ ∀u ∈ F , Γ(u, u)⊥µ.

ℓ = 2 ℓ = 3 ℓ = 4

· · ·⇝K(2,3,4,ℓ4,ℓ5,...)

k ) := t−1ℓ1

· · · t−1ℓk

Prop. 0<∀r ≤∀R ≤1,Ψ(ℓn)(R)

Ψ(ℓn)(r)≥ c

Nakahara ’06

(a) lim infλ→∞

lim infr↓0

λ2Ψ(r/λ)

Ψ(r)=0⇒ ∀u ∈ F , Γ(u, u)⊥µ.

ℓ = 2 ℓ = 3 ℓ = 4

k ) := t−1ℓ1

· · · t−1ℓk

Prop. 0<∀r ≤∀R ≤1,Ψ(ℓn)(R)

Ψ(ℓn)(r)≥ c

Nakahara ’06

(a) lim infλ→∞

lim infr↓0

λ2Ψ(r/λ)

Ψ(r)=0⇒ ∀u ∈ F , Γ(u, u)⊥µ.

ℓ = 2 ℓ = 3 ℓ = 4

k ) := t−1ℓ1

· · · t−1ℓk

Prop. 0<∀r ≤∀R ≤1,Ψ(ℓn)(R)

Ψ(ℓn)(r)≥ c

Nakahara ’06

(a) lim infλ→∞

lim infr↓0

λ2Ψ(r/λ)

Ψ(r)=0⇒ ∀u ∈ F , Γ(u, u)⊥µ.

ℓ = 2 ℓ = 3 ℓ = 4

k ) := t−1ℓ1

· · · t−1ℓk

Prop. 0<∀r ≤∀R ≤1,Ψ(ℓn)(R)

Ψ(ℓn)(r)≥ c

Nakahara ’06

(a) lim infλ→∞

lim infr↓0

λ2Ψ(r/λ)

Ψ(r)=0⇒ ∀u ∈ F , Γ(u, u)⊥µ.

ℓ = 2 ℓ = 3 ℓ = 4

k ) := t−1ℓ1

· · · t−1ℓk

Prop. 0<∀r ≤∀R ≤1,Ψ(ℓn)(R)

Ψ(ℓn)(r)≥ c

Nakahara ’06

(a) lim infλ→∞

lim infr↓0

λ2Ψ(r/λ)

Ψ(r)=0⇒ ∀u ∈ F , Γ(u, u)⊥µ.

ℓ = 2 ℓ = 3 ℓ = 4

k ) := t−1ℓ1

· · · t−1ℓk

Prop. 0<∀r ≤∀R ≤1,Ψ(ℓn)(R)

Ψ(ℓn)(r)≥ c

Nakahara ’06

(a) lim infλ→∞

lim infr↓0

λ2Ψ(r/λ)

Ψ(r)=0⇒ ∀u ∈ F , Γ(u, u)⊥µ.

ℓ = 2 ℓ = 3 ℓ = 4

k ) := t−1ℓ1

· · · t−1ℓk

Prop. 0<∀r ≤∀R ≤1,Ψ(ℓn)(R)

Ψ(ℓn)(r)≥ c

Nakahara ’06

(a) lim infλ→∞

lim infr↓0

λ2Ψ(r/λ)

Ψ(r)=0⇒ ∀u ∈ F , Γ(u, u)⊥µ.

ℓ = 2 ℓ = 3 ℓ = 4

k ) := t−1ℓ1

· · · t−1ℓk

Prop. 0<∀r ≤∀R ≤1,Ψ(ℓn)(R)

Ψ(ℓn)(r)≥ c

Nakahara ’06

(a) lim infλ→∞

lim infr↓0

λ2Ψ(r/λ)

Ψ(r)=0⇒ ∀u ∈ F , Γ(u, u)⊥µ.

ℓ = 2 ℓ = 3 ℓ = 4

· · ·⇝K(2,3,4,ℓ4,ℓ5,...)

k ) := t−1ℓ1

· · · t−1ℓk

Prop. 0<∀r ≤∀R ≤1,Ψ(ℓn)(R)

Ψ(ℓn)(r)≥ c

Nakahara ’06

(a) lim infλ→∞

lim infr↓0

λ2Ψ(r/λ)

Ψ(r)=0⇒ ∀u ∈ F , Γ(u, u)⊥µ.

ℓ = 2 ℓ = 3 ℓ = 4

· · ·⇝K(2,3,4,ℓ4,ℓ5,...)

k ) := t−1ℓ1

· · · t−1ℓk

Prop. 0<∀r ≤∀R ≤1,Ψ(ℓn)(R)

Ψ(ℓn)(r)≥ c

Nakahara ’06

(a) lim infλ→∞

lim infr↓0

λ2Ψ(r/λ)

Ψ(r)=0⇒ ∀u ∈ F , Γ(u, u)⊥µ.

ℓ = 2 ℓ = 3 ℓ = 4

· · ·⇝K(2,3,4,ℓ4,ℓ5,...)

k ) := t−1ℓ1

· · · t−1ℓk

Prop. 0<∀r ≤∀R ≤1,Ψ(ℓn)(R)

Ψ(ℓn)(r)≥ c

Nakahara ’06

(a) lim infλ→∞

lim infr↓0

λ2Ψ(r/λ)

Ψ(r)=0⇒ ∀u ∈ F , Γ(u, u)⊥µ.

ℓ = 2 ℓ = 3 ℓ = 4

· · ·⇝K(2,3,4,ℓ4,ℓ5,...)

k ) := t−1ℓ1

· · · t−1ℓk

Prop. 0<∀r ≤∀R ≤1,Ψ(ℓn)(R)

Ψ(ℓn)(r)≥ c

Nakahara ’06

(a) lim infλ→∞

lim infr↓0

λ2Ψ(r/λ)

Ψ(r)=0⇒ ∀u ∈ F , Γ(u, u)⊥µ.

•Suffices to show Γ(u, u)|V ⊥µ|V whenu|V is harmonic:

Lem. Let u∈F ∩ Cc(K), u ≥ 0, Fn :=u−1(2−nZ) and

Then Γ(un)(Fn)=0 & ∥u−un∥qsup+E(u − un)→0.

Prop(Reverse PI).If h∈F ∩ L∞&h|B(x,2r) is harmonic,

CS(Ψ) ⇒∫B(x,r)

dΓ(h, h)≤8cSΨ(r)−1∫B(x,2r)

h2 dµ.

cf.PI(Ψ): ∃cP>0, ∃A≥1, ∀x∈K, ∀r>0, ∀u∈F ,∫B(x,r)

∫B(x,Ar)

dΓ(u, u).

(a) lim infλ→∞

lim infr↓0

λ2Ψ(r/λ)

Ψ(r)=0⇒ ∀u ∈ F , Γ(u, u)⊥µ.

un :=“HFn(u|Fn) :=E(·)[u(XσFn)]”(harmonic extension).

CS(Ψ) ⇒∫B(x,r)

h2 dµ.

∫B(x,Ar)

dΓ(u, u).

(a) lim infλ→∞

lim infr↓0

λ2Ψ(r/λ)

Ψ(r)=0⇒ ∀u ∈ F , Γ(u, u)⊥µ.

un:=∑∞

k=0H{k2−n<u<(k+1)2−n}c((u−k2−n)+∧2−n).

CS(Ψ) ⇒∫B(x,r)

h2 dµ.

∫B(x,Ar)

dΓ(u, u).

(a) lim infλ→∞

lim infr↓0

λ2Ψ(r/λ)

Ψ(r)=0⇒ ∀u ∈ F , Γ(u, u)⊥µ.

CS(Ψ) ⇒∫B(x,r)

h2 dµ.

∫B(x,Ar)

dΓ(u, u).

(a) lim infλ→∞

lim infr↓0

λ2Ψ(r/λ)

Ψ(r)=0⇒ ∀u ∈ F , Γ(u, u)⊥µ.

CS(Ψ) ⇒∫B(x,r)

h2 dµ.

∫B(x,Ar)

dΓ(u, u).

(a) lim infλ→∞

lim infr↓0

λ2Ψ(r/λ)

Ψ(r)=0⇒ ∀u ∈ F , Γ(u, u)⊥µ.

CS(Ψ) ⇒∫B(x,r)

h2 dµ.

∫B(x,Ar)

dΓ(u, u).

(a) lim infλ→∞

lim infr↓0

λ2Ψ(r/λ)

Ψ(r)=0⇒ ∀u ∈ F , Γ(u, u)⊥µ.

CS(Ψ) ⇒∫B(x,r)

h2 dµ.

∫B(x,Ar)

dΓ(u, u).

(a) lim infλ→∞

lim infr↓0

λ2Ψ(r/λ)

Ψ(r)=0⇒ ∀u ∈ F , Γ(u, u)⊥µ.

•Suffices to show Γ(u, u)|V ⊥µ|V whenu|V is harmonic.

CS(Ψ) ⇒∫B(x,r)

h2 dµ.

Proof of Thm 1-(a) for u|V harmonic. By contradiction.

If f :=dΓ(u)|V, ac

dµ|V≡0, ∃Lebesgue pt x∈V of f , f(x)>0.

Forλ−1, r>0 small, Γ(u)(B(y, r/λ))≍ µ(B(y, r/λ)).

•SumupPI along qGeod shows VarB(x,r)u≲λ2Ψ(r/λ),(use VD!)•whereas Reverse PI yields VarB(x,r)u ≳ Ψ(r).

⇝Ψ(r)≲λ2Ψ(r/λ) for small r depending onλ. Contrdct!

(a) lim infλ→∞

lim infr↓0

λ2Ψ(r/λ)

Ψ(r)=0⇒ ∀u ∈ F , Γ(u, u)⊥µ.

CS(Ψ) ⇒∫B(x,r)

h2 dµ.

If f :=dΓ(u)|V, ac

(a) lim infλ→∞

lim infr↓0

λ2Ψ(r/λ)

Ψ(r)=0⇒ ∀u ∈ F , Γ(u, u)⊥µ.

CS(Ψ) ⇒∫B(x,r)

h2 dµ.

If f :=dΓ(u)|V, ac

(a) lim infλ→∞

lim infr↓0

λ2Ψ(r/λ)

Ψ(r)=0⇒ ∀u ∈ F , Γ(u, u)⊥µ.

CS(Ψ) ⇒∫B(x,r)

h2 dµ.

If f :=dΓ(u)|V, ac

(a) lim infλ→∞

lim infr↓0

λ2Ψ(r/λ)

Ψ(r)=0⇒ ∀u ∈ F , Γ(u, u)⊥µ.

CS(Ψ) ⇒∫B(x,r)

h2 dµ.

If f :=dΓ(u)|V, ac

(a) lim infλ→∞

lim infr↓0

λ2Ψ(r/λ)

Ψ(r)=0⇒ ∀u ∈ F , Γ(u, u)⊥µ.

CS(Ψ) ⇒∫B(x,r)

h2 dµ.

If f :=dΓ(u)|V, ac

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