Geometric implications of Poincaré inequalities in...
Transcript of Geometric implications of Poincaré inequalities in...
![Page 1: Geometric implications of Poincaré inequalities in …helper.ipam.ucla.edu/publications/iagws1/iagws1_11193.pdfGeometric implications of Poincare´ inequalities in metric measure](https://reader034.fdocuments.net/reader034/viewer/2022050118/5f4e7d3cb6f9633f2c3bdee1/html5/thumbnails/1.jpg)
Geometric implications of Poincareinequalities in metric measure spaces
Estibalitz Durand Cartagena
UNED (Spain)Dpto. de Matematica Aplicada
IPAM program “Interactions between Analysis and Geometry”Workshop I: Analysis on metric spaces
IPAM, UCLA
![Page 2: Geometric implications of Poincaré inequalities in …helper.ipam.ucla.edu/publications/iagws1/iagws1_11193.pdfGeometric implications of Poincare´ inequalities in metric measure](https://reader034.fdocuments.net/reader034/viewer/2022050118/5f4e7d3cb6f9633f2c3bdee1/html5/thumbnails/2.jpg)
Jesus A. Jaramillo (Universidad Complutense de Madrid)Nages Shanmugalingam (University of Cincinnati)Alex Williams (Texas Tech University)
![Page 3: Geometric implications of Poincaré inequalities in …helper.ipam.ucla.edu/publications/iagws1/iagws1_11193.pdfGeometric implications of Poincare´ inequalities in metric measure](https://reader034.fdocuments.net/reader034/viewer/2022050118/5f4e7d3cb6f9633f2c3bdee1/html5/thumbnails/3.jpg)
Analysis on metric spaces
Zero Order Calculus
Spaces of Homogeneous type: doubling
measures (Coifman-Weiss
70’s)
Lebesgue points
Vitali covering
Maximal operator
Doubling spaces with Poincaré
inequalities (Heinonen-Koskela
98)
Cheeger Differentiable
structure
First order Sobolev-type spaces
First order calculus
![Page 4: Geometric implications of Poincaré inequalities in …helper.ipam.ucla.edu/publications/iagws1/iagws1_11193.pdfGeometric implications of Poincare´ inequalities in metric measure](https://reader034.fdocuments.net/reader034/viewer/2022050118/5f4e7d3cb6f9633f2c3bdee1/html5/thumbnails/4.jpg)
Lipschitz function spaces
(X, d) metric space
DefinitionA function f : X −→ R is Lipschitz if there is a constant C > 0such that
|f (x)− f (y)| ≤ C d(x, y) ∀x, y ∈ X.
? LIP(X) = f : X −→ R : f is Lipschitz? LIP∞(X) = f : X −→ R : f is Lipschitz and bounded
‖f‖LIP∞ = ‖f‖∞ + LIP(f )
![Page 5: Geometric implications of Poincaré inequalities in …helper.ipam.ucla.edu/publications/iagws1/iagws1_11193.pdfGeometric implications of Poincare´ inequalities in metric measure](https://reader034.fdocuments.net/reader034/viewer/2022050118/5f4e7d3cb6f9633f2c3bdee1/html5/thumbnails/5.jpg)
Lipschitz function spaces
(X, d) metric space
DefinitionA function f : X −→ R is Lipschitz if there is a constant C > 0such that
|f (x)− f (y)| ≤ C d(x, y) ∀x, y ∈ X.
? LIP(X) = f : X −→ R : f is Lipschitz? LIP∞(X) = f : X −→ R : f is Lipschitz and bounded
‖f‖LIP∞ = ‖f‖∞ + LIP(f )
![Page 6: Geometric implications of Poincaré inequalities in …helper.ipam.ucla.edu/publications/iagws1/iagws1_11193.pdfGeometric implications of Poincare´ inequalities in metric measure](https://reader034.fdocuments.net/reader034/viewer/2022050118/5f4e7d3cb6f9633f2c3bdee1/html5/thumbnails/6.jpg)
Pointwise Lipschitz function spaces
DefinitionGiven a function f : X→ R the pointwise Lipschitz constant off at x ∈ X is defined as
Lip f (x) = lim supy→xy6=x
|f (x)− f (y)|d(x, y)
.
ExampleIf f ∈ C1(Ω), Ω
op⊂ Rn (or of a Riemannian manifold), then
Lip f (x) = |∇f (x)| ∀x ∈ Ω.
![Page 7: Geometric implications of Poincaré inequalities in …helper.ipam.ucla.edu/publications/iagws1/iagws1_11193.pdfGeometric implications of Poincare´ inequalities in metric measure](https://reader034.fdocuments.net/reader034/viewer/2022050118/5f4e7d3cb6f9633f2c3bdee1/html5/thumbnails/7.jpg)
Pointwise Lipschitz function spaces
DefinitionGiven a function f : X→ R the pointwise Lipschitz constant off at x ∈ X is defined as
Lip f (x) = lim supy→xy6=x
|f (x)− f (y)|d(x, y)
.
ExampleIf f ∈ C1(Ω), Ω
op⊂ Rn (or of a Riemannian manifold), then
Lip f (x) = |∇f (x)| ∀x ∈ Ω.
![Page 8: Geometric implications of Poincaré inequalities in …helper.ipam.ucla.edu/publications/iagws1/iagws1_11193.pdfGeometric implications of Poincare´ inequalities in metric measure](https://reader034.fdocuments.net/reader034/viewer/2022050118/5f4e7d3cb6f9633f2c3bdee1/html5/thumbnails/8.jpg)
Doubling measures
(X, d, µ) metric measure space, µ Borel regular measure
Definitionµ is doubling if ∃C > 0 constant such that
0 < µ(B(x, 2r)) ≤ Cµ(B(x, r)) <∞ ∀ x ∈ X, r > 0.
X complete + µ doubling =⇒ X proper
DefinitionA curve in X is a continuous mapping γ : [a, b]→ X.A rectifiable curve is a curve with finite length.
![Page 9: Geometric implications of Poincaré inequalities in …helper.ipam.ucla.edu/publications/iagws1/iagws1_11193.pdfGeometric implications of Poincare´ inequalities in metric measure](https://reader034.fdocuments.net/reader034/viewer/2022050118/5f4e7d3cb6f9633f2c3bdee1/html5/thumbnails/9.jpg)
Doubling measures
(X, d, µ) metric measure space, µ Borel regular measure
Definitionµ is doubling if ∃C > 0 constant such that
0 < µ(B(x, 2r)) ≤ Cµ(B(x, r)) <∞ ∀ x ∈ X, r > 0.
X complete + µ doubling =⇒ X proper
DefinitionA curve in X is a continuous mapping γ : [a, b]→ X.A rectifiable curve is a curve with finite length.
![Page 10: Geometric implications of Poincaré inequalities in …helper.ipam.ucla.edu/publications/iagws1/iagws1_11193.pdfGeometric implications of Poincare´ inequalities in metric measure](https://reader034.fdocuments.net/reader034/viewer/2022050118/5f4e7d3cb6f9633f2c3bdee1/html5/thumbnails/10.jpg)
Doubling measures
(X, d, µ) metric measure space, µ Borel regular measure
Definitionµ is doubling if ∃C > 0 constant such that
0 < µ(B(x, 2r)) ≤ Cµ(B(x, r)) <∞ ∀ x ∈ X, r > 0.
X complete + µ doubling =⇒ X proper
DefinitionA curve in X is a continuous mapping γ : [a, b]→ X.A rectifiable curve is a curve with finite length.
![Page 11: Geometric implications of Poincaré inequalities in …helper.ipam.ucla.edu/publications/iagws1/iagws1_11193.pdfGeometric implications of Poincare´ inequalities in metric measure](https://reader034.fdocuments.net/reader034/viewer/2022050118/5f4e7d3cb6f9633f2c3bdee1/html5/thumbnails/11.jpg)
Examples
(Rn, | · |,L n) C = 2n
(C, | · |,Hlog2log3 )
([0, 1], |x− y|1/2,H 2)
f (x) = x|f (x)− f (y)||x− y|1/2 = |x− y|1/2 y→x−→ 0
![Page 12: Geometric implications of Poincaré inequalities in …helper.ipam.ucla.edu/publications/iagws1/iagws1_11193.pdfGeometric implications of Poincare´ inequalities in metric measure](https://reader034.fdocuments.net/reader034/viewer/2022050118/5f4e7d3cb6f9633f2c3bdee1/html5/thumbnails/12.jpg)
Sierpinski carpet
Q0 = [0, 1]2
![Page 13: Geometric implications of Poincaré inequalities in …helper.ipam.ucla.edu/publications/iagws1/iagws1_11193.pdfGeometric implications of Poincare´ inequalities in metric measure](https://reader034.fdocuments.net/reader034/viewer/2022050118/5f4e7d3cb6f9633f2c3bdee1/html5/thumbnails/13.jpg)
Sierpinski carpetQ1
![Page 14: Geometric implications of Poincaré inequalities in …helper.ipam.ucla.edu/publications/iagws1/iagws1_11193.pdfGeometric implications of Poincare´ inequalities in metric measure](https://reader034.fdocuments.net/reader034/viewer/2022050118/5f4e7d3cb6f9633f2c3bdee1/html5/thumbnails/14.jpg)
Sierpinski carpetQ2
![Page 15: Geometric implications of Poincaré inequalities in …helper.ipam.ucla.edu/publications/iagws1/iagws1_11193.pdfGeometric implications of Poincare´ inequalities in metric measure](https://reader034.fdocuments.net/reader034/viewer/2022050118/5f4e7d3cb6f9633f2c3bdee1/html5/thumbnails/15.jpg)
Sierpinski carpetQ3
![Page 16: Geometric implications of Poincaré inequalities in …helper.ipam.ucla.edu/publications/iagws1/iagws1_11193.pdfGeometric implications of Poincare´ inequalities in metric measure](https://reader034.fdocuments.net/reader034/viewer/2022050118/5f4e7d3cb6f9633f2c3bdee1/html5/thumbnails/16.jpg)
Sierpinski carpetQ4
![Page 17: Geometric implications of Poincaré inequalities in …helper.ipam.ucla.edu/publications/iagws1/iagws1_11193.pdfGeometric implications of Poincare´ inequalities in metric measure](https://reader034.fdocuments.net/reader034/viewer/2022050118/5f4e7d3cb6f9633f2c3bdee1/html5/thumbnails/17.jpg)
Sierpinski carpet
…
![Page 18: Geometric implications of Poincaré inequalities in …helper.ipam.ucla.edu/publications/iagws1/iagws1_11193.pdfGeometric implications of Poincare´ inequalities in metric measure](https://reader034.fdocuments.net/reader034/viewer/2022050118/5f4e7d3cb6f9633f2c3bdee1/html5/thumbnails/18.jpg)
Sierpinski carpet: S3 = (X, d, µ)d = de|X
Equally distributing unit mass over Qn leads to a naturalprobability doubling measure µ on S3.( µ is comparable toHs, s =
log 8log 3 ).
![Page 19: Geometric implications of Poincaré inequalities in …helper.ipam.ucla.edu/publications/iagws1/iagws1_11193.pdfGeometric implications of Poincare´ inequalities in metric measure](https://reader034.fdocuments.net/reader034/viewer/2022050118/5f4e7d3cb6f9633f2c3bdee1/html5/thumbnails/19.jpg)
Classical Poincare inequalityOne way to view the Fundamental Theorem of Calculus is:
infinitesimal data local control
This principle can apply in very general situation in the form ofa Poincare inequality:
∃C = C(n) > 0: ∀B ≡ B(x, r) ⊂ Rn ∀f ∈W1,p(Rn)∫B|f − fB|dL n ≤ C(n) r
(∫B|∇f |pdL n
)1/p
Notation: ∫B
f dL n = fB =1
L n(B)
∫B
f dL n
![Page 20: Geometric implications of Poincaré inequalities in …helper.ipam.ucla.edu/publications/iagws1/iagws1_11193.pdfGeometric implications of Poincare´ inequalities in metric measure](https://reader034.fdocuments.net/reader034/viewer/2022050118/5f4e7d3cb6f9633f2c3bdee1/html5/thumbnails/20.jpg)
Classical Poincare inequalityOne way to view the Fundamental Theorem of Calculus is:
infinitesimal data local control
This principle can apply in very general situation in the form ofa Poincare inequality:
∃C = C(n) > 0: ∀B ≡ B(x, r) ⊂ Rn ∀f ∈W1,p(Rn)∫B|f − fB|dL n ≤ C(n) r
(∫B|∇f |pdL n
)1/p
Notation: ∫B
f dL n = fB =1
L n(B)
∫B
f dL n
![Page 21: Geometric implications of Poincaré inequalities in …helper.ipam.ucla.edu/publications/iagws1/iagws1_11193.pdfGeometric implications of Poincare´ inequalities in metric measure](https://reader034.fdocuments.net/reader034/viewer/2022050118/5f4e7d3cb6f9633f2c3bdee1/html5/thumbnails/21.jpg)
Classical Poincare inequalityOne way to view the Fundamental Theorem of Calculus is:
infinitesimal data local control
This principle can apply in very general situation in the form ofa Poincare inequality:
∃C = C(n) > 0: ∀B ≡ B(x, r) ⊂ Rn ∀f ∈W1,p(Rn)∫B|f − fB|dL n ≤ C(n) r
(∫B|∇f |pdL n
)1/p
Notation: ∫B
f dL n = fB =1
L n(B)
∫B
f dL n
![Page 22: Geometric implications of Poincaré inequalities in …helper.ipam.ucla.edu/publications/iagws1/iagws1_11193.pdfGeometric implications of Poincare´ inequalities in metric measure](https://reader034.fdocuments.net/reader034/viewer/2022050118/5f4e7d3cb6f9633f2c3bdee1/html5/thumbnails/22.jpg)
Poincare inequalities in metric measure spaces(X, d, µ) metric measure space
Definition (Heinonen-Koskela 98)A non-negative Borel function g on X is an upper gradient forf : X→ R ∪ ±∞ if
|f (x)− f (y)| ≤∫γ
g,
∀x, y ∈ X and every rectifiable curve γxy.
Examples
g ≡ ∞ is an upper gradient of every function on X.If there are no rectifiable curves in X then g ≡ 0 is an uppergradient of every function.If f ∈ LIP(X) then g ≡ LIP(f ) and g(x) = Lip f (x) are uppergradients for f .
![Page 23: Geometric implications of Poincaré inequalities in …helper.ipam.ucla.edu/publications/iagws1/iagws1_11193.pdfGeometric implications of Poincare´ inequalities in metric measure](https://reader034.fdocuments.net/reader034/viewer/2022050118/5f4e7d3cb6f9633f2c3bdee1/html5/thumbnails/23.jpg)
Poincare inequalities in metric measure spaces(X, d, µ) metric measure space
Definition (Heinonen-Koskela 98)A non-negative Borel function g on X is an upper gradient forf : X→ R ∪ ±∞ if
|f (x)− f (y)| ≤∫γ
g,
∀x, y ∈ X and every rectifiable curve γxy.
Examples
g ≡ ∞ is an upper gradient of every function on X.If there are no rectifiable curves in X then g ≡ 0 is an uppergradient of every function.If f ∈ LIP(X) then g ≡ LIP(f ) and g(x) = Lip f (x) are uppergradients for f .
![Page 24: Geometric implications of Poincaré inequalities in …helper.ipam.ucla.edu/publications/iagws1/iagws1_11193.pdfGeometric implications of Poincare´ inequalities in metric measure](https://reader034.fdocuments.net/reader034/viewer/2022050118/5f4e7d3cb6f9633f2c3bdee1/html5/thumbnails/24.jpg)
p-Poincare inequality
Definition (Heinonen-Koskela 98)Let 1 ≤ p <∞.We say that (X, d, µ) supports a weak p-Poincareinequality if there exist constants Cp > 0 and λ ≥ 1 such that forevery Borel measurable function f : X→ R and every uppergradient g : X→ [0,∞] of f , the pair (f , g) satisfies theinequality∫
B(x,r)|f − fB(x,r)| dµ ≤ Cp r
(∫B(x,λr)
gpdµ)1/p
∀B(x, r) ⊂ X.
Notation: ∫B
f dµ = fB =1
µ(B)
∫B
f dµ
![Page 25: Geometric implications of Poincaré inequalities in …helper.ipam.ucla.edu/publications/iagws1/iagws1_11193.pdfGeometric implications of Poincare´ inequalities in metric measure](https://reader034.fdocuments.net/reader034/viewer/2022050118/5f4e7d3cb6f9633f2c3bdee1/html5/thumbnails/25.jpg)
Examples
(Rn, | · |,L n)
Riemannian manifolds with non-negative Ricci curvatureHeisenberg group with its Carnot-Caratheodory metricand Haar measure Subriemannian geometryBoundaries of certain hyperbolic buildings: Bourdon-Pajotspaces Geometric group theoryLaakso spaces, . . .
![Page 26: Geometric implications of Poincaré inequalities in …helper.ipam.ucla.edu/publications/iagws1/iagws1_11193.pdfGeometric implications of Poincare´ inequalities in metric measure](https://reader034.fdocuments.net/reader034/viewer/2022050118/5f4e7d3cb6f9633f2c3bdee1/html5/thumbnails/26.jpg)
Geometric implications of p-Poincare inequalities
X is connected
Semmes 98 p <∞
X complete p-PI
µ doubling
=⇒ X is quasiconvex
DefinitionA metric space (X, d) is quasiconvex if there exists a constantC ≥ 1 such that for each pair of points x, y ∈ X, there exists acurve γ connecting x and y with
`(γ) ≤ Cd(x, y).
![Page 27: Geometric implications of Poincaré inequalities in …helper.ipam.ucla.edu/publications/iagws1/iagws1_11193.pdfGeometric implications of Poincare´ inequalities in metric measure](https://reader034.fdocuments.net/reader034/viewer/2022050118/5f4e7d3cb6f9633f2c3bdee1/html5/thumbnails/27.jpg)
Geometric implications of p-Poincare inequalities
X is connectedSemmes 98 p <∞
X complete p-PI
µ doubling
=⇒ X is quasiconvex
DefinitionA metric space (X, d) is quasiconvex if there exists a constantC ≥ 1 such that for each pair of points x, y ∈ X, there exists acurve γ connecting x and y with
`(γ) ≤ Cd(x, y).
![Page 28: Geometric implications of Poincaré inequalities in …helper.ipam.ucla.edu/publications/iagws1/iagws1_11193.pdfGeometric implications of Poincare´ inequalities in metric measure](https://reader034.fdocuments.net/reader034/viewer/2022050118/5f4e7d3cb6f9633f2c3bdee1/html5/thumbnails/28.jpg)
Geometric implications of p-Poincare inequalities
X is connectedSemmes 98 p <∞
X complete p-PI
µ doubling
=⇒ X is quasiconvex
DefinitionA metric space (X, d) is quasiconvex if there exists a constantC ≥ 1 such that for each pair of points x, y ∈ X, there exists acurve γ connecting x and y with
`(γ) ≤ Cd(x, y).
![Page 29: Geometric implications of Poincaré inequalities in …helper.ipam.ucla.edu/publications/iagws1/iagws1_11193.pdfGeometric implications of Poincare´ inequalities in metric measure](https://reader034.fdocuments.net/reader034/viewer/2022050118/5f4e7d3cb6f9633f2c3bdee1/html5/thumbnails/29.jpg)
Geometric implications of p-Poincare inequalities
X is connectedSemmes 98 p <∞
X complete p-PI
µ doubling
=⇒ X is quasiconvex
: (S3, d, µ) is quasiconvex but does not admit any p-PI
Heinonen-Koskela 98, Kinnunen-Latvala 02, Saloff-Coste02, Keith 03, Miranda 03, Korte 07, ....
![Page 30: Geometric implications of Poincaré inequalities in …helper.ipam.ucla.edu/publications/iagws1/iagws1_11193.pdfGeometric implications of Poincare´ inequalities in metric measure](https://reader034.fdocuments.net/reader034/viewer/2022050118/5f4e7d3cb6f9633f2c3bdee1/html5/thumbnails/30.jpg)
Geometric implications of p-Poincare inequalities
X is connectedSemmes 98 p <∞
X complete p-PI
µ doubling
=⇒ X is quasiconvex
: (S3, d, µ) is quasiconvex but does not admit any p-PI
Heinonen-Koskela 98, Kinnunen-Latvala 02, Saloff-Coste02, Keith 03, Miranda 03, Korte 07, ....
![Page 31: Geometric implications of Poincaré inequalities in …helper.ipam.ucla.edu/publications/iagws1/iagws1_11193.pdfGeometric implications of Poincare´ inequalities in metric measure](https://reader034.fdocuments.net/reader034/viewer/2022050118/5f4e7d3cb6f9633f2c3bdee1/html5/thumbnails/31.jpg)
Counterexample
(S3, d, µ) does not admit a 1-PI
Let Tn be the vertical strip of width 3−n.
![Page 32: Geometric implications of Poincaré inequalities in …helper.ipam.ucla.edu/publications/iagws1/iagws1_11193.pdfGeometric implications of Poincare´ inequalities in metric measure](https://reader034.fdocuments.net/reader034/viewer/2022050118/5f4e7d3cb6f9633f2c3bdee1/html5/thumbnails/32.jpg)
Counterexample
(S3, d, µ) does not admit a 1-PI
Let Tn be the vertical strip of width 3−n.
![Page 33: Geometric implications of Poincaré inequalities in …helper.ipam.ucla.edu/publications/iagws1/iagws1_11193.pdfGeometric implications of Poincare´ inequalities in metric measure](https://reader034.fdocuments.net/reader034/viewer/2022050118/5f4e7d3cb6f9633f2c3bdee1/html5/thumbnails/33.jpg)
Counterexample
T1
![Page 34: Geometric implications of Poincaré inequalities in …helper.ipam.ucla.edu/publications/iagws1/iagws1_11193.pdfGeometric implications of Poincare´ inequalities in metric measure](https://reader034.fdocuments.net/reader034/viewer/2022050118/5f4e7d3cb6f9633f2c3bdee1/html5/thumbnails/34.jpg)
Counterexample
T2
![Page 35: Geometric implications of Poincaré inequalities in …helper.ipam.ucla.edu/publications/iagws1/iagws1_11193.pdfGeometric implications of Poincare´ inequalities in metric measure](https://reader034.fdocuments.net/reader034/viewer/2022050118/5f4e7d3cb6f9633f2c3bdee1/html5/thumbnails/35.jpg)
Counterexample
T3
![Page 36: Geometric implications of Poincaré inequalities in …helper.ipam.ucla.edu/publications/iagws1/iagws1_11193.pdfGeometric implications of Poincare´ inequalities in metric measure](https://reader034.fdocuments.net/reader034/viewer/2022050118/5f4e7d3cb6f9633f2c3bdee1/html5/thumbnails/36.jpg)
Counterexample∫B(x,r)|f − fB(x,r)| dµ ≤ C r
(∫B(x,r)
gpdµ)1/p
Tn
Define fn ∈ LIP(S3) such that∫
S3
|fn − (fn)S3 |dµ > C but
∫S3
lip(fn)dµ = 3n · µ(Tn) = 3n · 2n
8n → 0 (n→∞)
![Page 37: Geometric implications of Poincaré inequalities in …helper.ipam.ucla.edu/publications/iagws1/iagws1_11193.pdfGeometric implications of Poincare´ inequalities in metric measure](https://reader034.fdocuments.net/reader034/viewer/2022050118/5f4e7d3cb6f9633f2c3bdee1/html5/thumbnails/37.jpg)
Counterexample
(S3, d, µ) does not admit any p-PI
Bourdon-Pajot 02 Let (X, d, µ) be a bounded metric measurespace with µ doubling and p-PI, and let f : X −→ I be asurjective Lipschitz function from X onto an interval I ⊂ R.Then, L 1
|I f#µ. Here f#µ denotes the push-forward measureof µ under f .
Proof.Let f be the projection on the horizontal axis. It can be checkedthat f#µ⊥L 1.
Question Higher dimensions?
![Page 38: Geometric implications of Poincaré inequalities in …helper.ipam.ucla.edu/publications/iagws1/iagws1_11193.pdfGeometric implications of Poincare´ inequalities in metric measure](https://reader034.fdocuments.net/reader034/viewer/2022050118/5f4e7d3cb6f9633f2c3bdee1/html5/thumbnails/38.jpg)
Generalized Sierpinski carpets: Sa
a = (a−11 , a−1
2 , . . .) ∈
13 ,
15 ,
17 , . . .
N
For a =(1
a,
1a,
1a, . . .
)a odd,
Sa does not admit any p−PI
Mackay, Tyson, Wildrick (To appear)(Sa, d, µ) supports a 1-PI if and only if a ∈ `1
(Sa, d, µ) supports a p-PI for some p > 1 if and only if a ∈ `2
![Page 39: Geometric implications of Poincaré inequalities in …helper.ipam.ucla.edu/publications/iagws1/iagws1_11193.pdfGeometric implications of Poincare´ inequalities in metric measure](https://reader034.fdocuments.net/reader034/viewer/2022050118/5f4e7d3cb6f9633f2c3bdee1/html5/thumbnails/39.jpg)
Generalized Sierpinski carpets: Sa
a = (a−11 , a−1
2 , . . .) ∈
13 ,
15 ,
17 , . . .
N
For a =(1
a,
1a,
1a, . . .
)a odd,
Sa does not admit any p−PI
Mackay, Tyson, Wildrick (To appear)(Sa, d, µ) supports a 1-PI if and only if a ∈ `1
(Sa, d, µ) supports a p-PI for some p > 1 if and only if a ∈ `2
![Page 40: Geometric implications of Poincaré inequalities in …helper.ipam.ucla.edu/publications/iagws1/iagws1_11193.pdfGeometric implications of Poincare´ inequalities in metric measure](https://reader034.fdocuments.net/reader034/viewer/2022050118/5f4e7d3cb6f9633f2c3bdee1/html5/thumbnails/40.jpg)
Generalized Sierpinski carpets: Sa
a = (a−11 , a−1
2 , . . .) ∈
13 ,
15 ,
17 , . . .
N
For a =(1
a,
1a,
1a, . . .
)a odd,
Sa does not admit any p−PI
Mackay, Tyson, Wildrick (To appear)(Sa, d, µ) supports a 1-PI if and only if a ∈ `1
(Sa, d, µ) supports a p-PI for some p > 1 if and only if a ∈ `2
![Page 41: Geometric implications of Poincaré inequalities in …helper.ipam.ucla.edu/publications/iagws1/iagws1_11193.pdfGeometric implications of Poincare´ inequalities in metric measure](https://reader034.fdocuments.net/reader034/viewer/2022050118/5f4e7d3cb6f9633f2c3bdee1/html5/thumbnails/41.jpg)
Which is the role of the exponent p?∫B(x,r)|f − fB(x,r)| dµ ≤ Cr
(∫B(x,λr)
gpdµ)1/p
Holder inequality: p-PI=⇒q-PI for q ≥ p
Federer-Fleming, Mazya 60 Miranda 03(Rn) p = 1 ⇐⇒ Isoperimetric inequality
Example Example
X := (x, y) ∈ R2 : x ≥ 0, 0 ≤ y ≤ xm
(X, | · |,L 2|X) X has p−PI ⇐⇒
p > m + 1
![Page 42: Geometric implications of Poincaré inequalities in …helper.ipam.ucla.edu/publications/iagws1/iagws1_11193.pdfGeometric implications of Poincare´ inequalities in metric measure](https://reader034.fdocuments.net/reader034/viewer/2022050118/5f4e7d3cb6f9633f2c3bdee1/html5/thumbnails/42.jpg)
Which is the role of the exponent p?∫B(x,r)|f − fB(x,r)| dµ ≤ Cr
(∫B(x,λr)
gpdµ)1/p
Holder inequality: p-PI=⇒q-PI for q ≥ p
Federer-Fleming, Mazya 60 Miranda 03(Rn) p = 1 ⇐⇒ Isoperimetric inequality
Example Example
X := (x, y) ∈ R2 : x ≥ 0, 0 ≤ y ≤ xm
(X, | · |,L 2|X) X has p−PI ⇐⇒
p > m + 1
![Page 43: Geometric implications of Poincaré inequalities in …helper.ipam.ucla.edu/publications/iagws1/iagws1_11193.pdfGeometric implications of Poincare´ inequalities in metric measure](https://reader034.fdocuments.net/reader034/viewer/2022050118/5f4e7d3cb6f9633f2c3bdee1/html5/thumbnails/43.jpg)
Which is the role of the exponent p?∫B(x,r)|f − fB(x,r)| dµ ≤ Cr
(∫B(x,λr)
gpdµ)1/p
Holder inequality: p-PI=⇒q-PI for q ≥ p
Federer-Fleming, Mazya 60 Miranda 03(Rn) p = 1 ⇐⇒ Isoperimetric inequality
Example Example
X := (x, y) ∈ R2 : x ≥ 0, 0 ≤ y ≤ xm
(X, | · |,L 2|X) X has p−PI ⇐⇒
p > m + 1
![Page 44: Geometric implications of Poincaré inequalities in …helper.ipam.ucla.edu/publications/iagws1/iagws1_11193.pdfGeometric implications of Poincare´ inequalities in metric measure](https://reader034.fdocuments.net/reader034/viewer/2022050118/5f4e7d3cb6f9633f2c3bdee1/html5/thumbnails/44.jpg)
Glueing spaces together
Heinonen-Koskela 98Suppose X and Y are locally compact Q−regular metricmeasure spaces and that A is a closed subset of X that has anisometric copy inside Y. Suppose there are numbersQ ≥ s > Q− p and C ≥ 1 so thatH∞s (A ∩ BR) ≥ C−1Rs for allballs BR either in X or in Y that are centered at A with radius0 < R < mindiamX,diamY.
If X and Y admit a p- PI =⇒ X∪AY admits a p- PI.
![Page 45: Geometric implications of Poincaré inequalities in …helper.ipam.ucla.edu/publications/iagws1/iagws1_11193.pdfGeometric implications of Poincare´ inequalities in metric measure](https://reader034.fdocuments.net/reader034/viewer/2022050118/5f4e7d3cb6f9633f2c3bdee1/html5/thumbnails/45.jpg)
What happens when p→∞?
∫B(x,r)|f − fB(x,r)| dµ ≤ Cr
(∫B(x,λr)
gpdµ)1/p
Holder inequality: p-PI=⇒q-PI for q ≥ p
Definition(X, d, µ) supports a weak∞-Poincare inequality if there existconstants C > 0 and λ ≥ 1 such that for every functionf : X→ R and every upper gradient g of f , the pair (f , g)satisfies ∫
B(x,r)|f − fB(x,r)| dµ ≤ Cr‖g‖L∞(B(x,λr))
∀B(x, r) ⊂ X.
![Page 46: Geometric implications of Poincaré inequalities in …helper.ipam.ucla.edu/publications/iagws1/iagws1_11193.pdfGeometric implications of Poincare´ inequalities in metric measure](https://reader034.fdocuments.net/reader034/viewer/2022050118/5f4e7d3cb6f9633f2c3bdee1/html5/thumbnails/46.jpg)
What happens when p→∞?
∫B(x,r)|f − fB(x,r)| dµ ≤ Cr
(∫B(x,λr)
gpdµ)1/p
Holder inequality: p-PI=⇒q-PI for q ≥ p
Definition(X, d, µ) supports a weak∞-Poincare inequality if there existconstants C > 0 and λ ≥ 1 such that for every functionf : X→ R and every upper gradient g of f , the pair (f , g)satisfies ∫
B(x,r)|f − fB(x,r)| dµ ≤ Cr‖g‖L∞(B(x,λr))
∀B(x, r) ⊂ X.
![Page 47: Geometric implications of Poincaré inequalities in …helper.ipam.ucla.edu/publications/iagws1/iagws1_11193.pdfGeometric implications of Poincare´ inequalities in metric measure](https://reader034.fdocuments.net/reader034/viewer/2022050118/5f4e7d3cb6f9633f2c3bdee1/html5/thumbnails/47.jpg)
X complete and∞-PI
µ doubling
=⇒X is quasiconvex
:
Sierpinski carpet
d = de|X µ = Hs, s =log 8log 3
(X, d) is quasiconvex(X, d, µ) does not admitany p-PI, 1 ≤ p ≤ ∞
![Page 48: Geometric implications of Poincaré inequalities in …helper.ipam.ucla.edu/publications/iagws1/iagws1_11193.pdfGeometric implications of Poincare´ inequalities in metric measure](https://reader034.fdocuments.net/reader034/viewer/2022050118/5f4e7d3cb6f9633f2c3bdee1/html5/thumbnails/48.jpg)
Modulus of a family of curvesDefinitionLet Γ ⊂ Υ = non constant rectifiable curves of X and1 ≤ p ≤ ∞. For Γ ⊂ Υ, let F(Γ) be the family of all Borelmeasurable functions ρ : X→ [0,∞] such that∫
γρ ≥ 1 for all γ ∈ Γ.
Modp(Γ) =
infρ∈F(Γ)
∫X ρ
p dµ, if p <∞
infρ∈F(Γ) ‖ρ‖L∞ , if p =∞
If some property holds for all curves γ ∈ Υ\Γ, whereModp Γ = 0, then we say that the property holds for p−a.e.curve.
RemarkModp is an outer measure
![Page 49: Geometric implications of Poincaré inequalities in …helper.ipam.ucla.edu/publications/iagws1/iagws1_11193.pdfGeometric implications of Poincare´ inequalities in metric measure](https://reader034.fdocuments.net/reader034/viewer/2022050118/5f4e7d3cb6f9633f2c3bdee1/html5/thumbnails/49.jpg)
LemmaLet Γ ⊂ Υ and 1 ≤ p ≤ ∞. The following conditions are equivalent:(a) Modp Γ = 0.(b) There exists a Borel function 0 ≤ ρ ∈ Lp(X) such that∫
γ ρ = +∞, for each γ ∈ Γ and ‖ρ‖L∞ = 0.
ExamplesRn,n ≥ 2
![Page 50: Geometric implications of Poincaré inequalities in …helper.ipam.ucla.edu/publications/iagws1/iagws1_11193.pdfGeometric implications of Poincare´ inequalities in metric measure](https://reader034.fdocuments.net/reader034/viewer/2022050118/5f4e7d3cb6f9633f2c3bdee1/html5/thumbnails/50.jpg)
p-“thick” quasiconvexity
Definition(X, d, µ) is a p-“thick” quasiconvex space if there exists C ≥ 1such that ∀ x, y ∈ X, 0 < ε < 1
4 d(x, y),
Modp(Γ(B(x, ε),B(y, ε),C)) > 0,
where Γ(B(x, ε),B(y, ε),C) denotes the set of curves γp,qconnecting p ∈ B(x, ε) and q ∈ B(y, ε) with `(γp,q) ≤ Cd(p, q).
![Page 51: Geometric implications of Poincaré inequalities in …helper.ipam.ucla.edu/publications/iagws1/iagws1_11193.pdfGeometric implications of Poincare´ inequalities in metric measure](https://reader034.fdocuments.net/reader034/viewer/2022050118/5f4e7d3cb6f9633f2c3bdee1/html5/thumbnails/51.jpg)
Geometric characterization: p =∞
D-C, Jaramillo, Shanmugalingam 11Let (X, d, µ) be a complete metric space with µ doubling. Then,
X is∞-“thick” quasi-convex ⇐⇒ X admits∞-PI
RemarkIf µ ∼ λ =⇒Mod∞(Γ, µ) = Mod∞(Γ, λ)=⇒ (X, d, µ)admits∞-PI if and only if (X, d, λ)admits∞-PI
![Page 52: Geometric implications of Poincaré inequalities in …helper.ipam.ucla.edu/publications/iagws1/iagws1_11193.pdfGeometric implications of Poincare´ inequalities in metric measure](https://reader034.fdocuments.net/reader034/viewer/2022050118/5f4e7d3cb6f9633f2c3bdee1/html5/thumbnails/52.jpg)
Analytic characterization
D-C, Jaramillo, Shanmugalingam 11 Let (X, d, µ) be aconnected complete metric space supporting a doubling Borelmeasure µ. Then
LIP∞(X) = N1,∞(X) with c.e.s. ⇐⇒ X admits∞-PI
![Page 53: Geometric implications of Poincaré inequalities in …helper.ipam.ucla.edu/publications/iagws1/iagws1_11193.pdfGeometric implications of Poincare´ inequalities in metric measure](https://reader034.fdocuments.net/reader034/viewer/2022050118/5f4e7d3cb6f9633f2c3bdee1/html5/thumbnails/53.jpg)
Geometric implications of p-PI
X complete and p-PI
µ doubling
=⇒X is p-“thick” quasiconvex
Remarks
p-“thick” quasiconvex =⇒ quasiconvexThe characterization is no longer true for p <∞ :
Question Are there∞-thick qc spaces which are not p-thick qcfor any p <∞?
![Page 54: Geometric implications of Poincaré inequalities in …helper.ipam.ucla.edu/publications/iagws1/iagws1_11193.pdfGeometric implications of Poincare´ inequalities in metric measure](https://reader034.fdocuments.net/reader034/viewer/2022050118/5f4e7d3cb6f9633f2c3bdee1/html5/thumbnails/54.jpg)
A counterexample
µ =∑
j χQj · µj doubling measure
X is p-thick quasi-convex 1 ≤ p ≤ ∞ =⇒∞-PIX admits an∞-PI but does not admit any p-PI(1 ≤ p <∞)
![Page 55: Geometric implications of Poincaré inequalities in …helper.ipam.ucla.edu/publications/iagws1/iagws1_11193.pdfGeometric implications of Poincare´ inequalities in metric measure](https://reader034.fdocuments.net/reader034/viewer/2022050118/5f4e7d3cb6f9633f2c3bdee1/html5/thumbnails/55.jpg)
Persistence of p-PI under GH-limits
Cheeger 99If Xn, dn, µnn with µn doubling measures supporting a p-PIp <∞ (with constants uniformly bounded), and
Xn, dn, µnnG−H−→ (X, d, µ), then (X, d, µ) has µ doubling and
supports a p-PI.
CorollaryThe∞-PI is non-stable under measured Gromov-Hausdorff limits.
![Page 56: Geometric implications of Poincaré inequalities in …helper.ipam.ucla.edu/publications/iagws1/iagws1_11193.pdfGeometric implications of Poincare´ inequalities in metric measure](https://reader034.fdocuments.net/reader034/viewer/2022050118/5f4e7d3cb6f9633f2c3bdee1/html5/thumbnails/56.jpg)
Not Self-improvement of∞-PI
Keith-Zhong 08 If X is a complete metric space equipped with adoubling measure satisfying a p-Poincare inequality for some1 < p <∞, then there exists ε > 0 such that X supports aq-Poincare inequality for all q > p− ε.
![Page 57: Geometric implications of Poincaré inequalities in …helper.ipam.ucla.edu/publications/iagws1/iagws1_11193.pdfGeometric implications of Poincare´ inequalities in metric measure](https://reader034.fdocuments.net/reader034/viewer/2022050118/5f4e7d3cb6f9633f2c3bdee1/html5/thumbnails/57.jpg)
∞-admissible weights
Definitionw ≥ 0, w ∈ L1
loc(Rn) is a p-admissible weight with p ≥ 1 if the
measure µ given by dµ = wdL n is doubling, and (Rn, | · |, µ)admits a weak p-Poincare inequality.
Muckenhoupt-Wheeden 741 ≤ p <∞ w ∈ Ap =⇒ w is p-admissible
RemarkA∞ weights are∞-admissible
A∞ =⋃p>1
Ap :
![Page 58: Geometric implications of Poincaré inequalities in …helper.ipam.ucla.edu/publications/iagws1/iagws1_11193.pdfGeometric implications of Poincare´ inequalities in metric measure](https://reader034.fdocuments.net/reader034/viewer/2022050118/5f4e7d3cb6f9633f2c3bdee1/html5/thumbnails/58.jpg)
A counterexample
(R, | · |, µ) admits an∞-PI but no p-PI
In R, the Riesz product
dν(x) =
∞∏k=1
(1 + a cos(3k · 2πx))dL 1(x) |a| < 1
is a doubling measure and ν⊥L 1.
Idea construct a sequence of weights wk, k ≥ 1 such that wkdL 1
“approximates” dν better as k→∞:
w(x) = w1(x)χ(−∞,2](x) +∞∑
k=2
wk(x− k)χ[k,k+1](x),
and dµ = w dL 1.
![Page 59: Geometric implications of Poincaré inequalities in …helper.ipam.ucla.edu/publications/iagws1/iagws1_11193.pdfGeometric implications of Poincare´ inequalities in metric measure](https://reader034.fdocuments.net/reader034/viewer/2022050118/5f4e7d3cb6f9633f2c3bdee1/html5/thumbnails/59.jpg)
Measured differentiable structures
X complete, µ doubling
Cheeger 99
X supports p-PI
1 ≤ p <∞
=⇒ X admits a “differentiable structure”
Keith 04
X satisfies Lip-lip =⇒ X admits a“differentiable structure”
![Page 60: Geometric implications of Poincaré inequalities in …helper.ipam.ucla.edu/publications/iagws1/iagws1_11193.pdfGeometric implications of Poincare´ inequalities in metric measure](https://reader034.fdocuments.net/reader034/viewer/2022050118/5f4e7d3cb6f9633f2c3bdee1/html5/thumbnails/60.jpg)
Lip-lip condition“The infinitesimal behaviour at a generic point is essentially independent ofthe scales used for the blow-up at that point”
Definition
X satisfies Lip-lip if ∃C > 0 such that ∀f ∈ LIP(X),
Lip f (x) ≤ C lip f (x) µ-a.e.x
Here,Lip f (x) := lim sup
r→0sup
0<d(y,x)<r
|f (y)− f (x)|r
,
andlip f (x) := lim inf
r→0sup
0<d(y,x)<r
|f (y)− f (x)|r
.
RemarkIf µ ∼ λ =⇒ (X, d, µ) has Lip-lip iff (X, d, λ) has Lip-lip
![Page 61: Geometric implications of Poincaré inequalities in …helper.ipam.ucla.edu/publications/iagws1/iagws1_11193.pdfGeometric implications of Poincare´ inequalities in metric measure](https://reader034.fdocuments.net/reader034/viewer/2022050118/5f4e7d3cb6f9633f2c3bdee1/html5/thumbnails/61.jpg)
Lip-lip conditionKeith 02
X complete and p-PI
µ doubling
=⇒ X has the Lip-lip condition
:
Proof.For µ-a.e.x,
1C
Lip f (x) ≤ lim supr→0
1r
∫B(x,r)|f − fB(x,r)| dµ
≤ L lim supr→0
(∫B(x,r)
lip f (x)pdµ) 1
p= L lip f (x)
![Page 62: Geometric implications of Poincaré inequalities in …helper.ipam.ucla.edu/publications/iagws1/iagws1_11193.pdfGeometric implications of Poincare´ inequalities in metric measure](https://reader034.fdocuments.net/reader034/viewer/2022050118/5f4e7d3cb6f9633f2c3bdee1/html5/thumbnails/62.jpg)
Lip-lip conditionKeith 02
X complete and p-PI
µ doubling
=⇒ X has the Lip-lip condition
:
Proof.For µ-a.e.x,
1C
Lip f (x) ≤ lim supr→0
1r
∫B(x,r)|f − fB(x,r)| dµ
≤ L lim supr→0
(∫B(x,r)
lip f (x)pdµ) 1
p= L lip f (x)
![Page 63: Geometric implications of Poincaré inequalities in …helper.ipam.ucla.edu/publications/iagws1/iagws1_11193.pdfGeometric implications of Poincare´ inequalities in metric measure](https://reader034.fdocuments.net/reader034/viewer/2022050118/5f4e7d3cb6f9633f2c3bdee1/html5/thumbnails/63.jpg)
Lip-lip conditionKeith 02
X complete and p-PI
µ doubling
=⇒ X has the Lip-lip condition
:Proof.For µ-a.e.x,
1C
Lip f (x) ≤ lim supr→0
1r
∫B(x,r)|f − fB(x,r)| dµ
≤ L lim supr→0
(∫B(x,r)
lip f (x)pdµ) 1
p= L lip f (x)
![Page 64: Geometric implications of Poincaré inequalities in …helper.ipam.ucla.edu/publications/iagws1/iagws1_11193.pdfGeometric implications of Poincare´ inequalities in metric measure](https://reader034.fdocuments.net/reader034/viewer/2022050118/5f4e7d3cb6f9633f2c3bdee1/html5/thumbnails/64.jpg)
Question
X complete and∞-PI
µ doubling
?
=⇒ X has the Lip-lip condition
Bate (Preprint 12), Gong (Preprint 12)
X satisfies σ−Lip-lip
µ pointwise doubling
⇐⇒ X admits a“differentiable structure”
![Page 65: Geometric implications of Poincaré inequalities in …helper.ipam.ucla.edu/publications/iagws1/iagws1_11193.pdfGeometric implications of Poincare´ inequalities in metric measure](https://reader034.fdocuments.net/reader034/viewer/2022050118/5f4e7d3cb6f9633f2c3bdee1/html5/thumbnails/65.jpg)
Thank you for your attention!