Quantitative rectifiability in the Heisenberg group Katrin F assler - - PowerPoint PPT Presentation

Quantitative rectifiability in the Heisenberg group

Katrin F¨ assler University of Fribourg, Switzerland

Joint work with Vasileios Chousionis and Tuomas Orponen

Quantitative rectifiability in the Heisenberg group

(of codimension 1 sets) Katrin F¨ assler University of Fribourg, Switzerland

Joint work with Vasileios Chousionis and Tuomas Orponen

The Heisenberg group

H1 = (R3, ∗) d(p, q) = q−1 ∗ p, where (x, y, t) = max{|(x, y)|, |t|1/2} The resulting space (H1, d) has Hausdorff dimension 4.

Rectifiability in H1

Rectifiability in H1

• L. Ambrosio and B. Kirchheim, 2000

Let k ≥ 2. If f : A ⊆ Rk → H1 is Lipschitz, then Hk(f (A)) = 0.

Rectifiability in H1

• L. Ambrosio and B. Kirchheim, 2000

Let k ≥ 2. If f : A ⊆ Rk → H1 is Lipschitz, then Hk(f (A)) = 0.

• B. Franchi, R. Serapioni, and F. Serra Cassano, 2001–. . .

In H1, consider an alternative to Federer’s definition of 3-rectifiability, for instance using intrinsic Lipschitz graphs.

Rectifiability in H1

• L. Ambrosio and B. Kirchheim, 2000

Let k ≥ 2. If f : A ⊆ Rk → H1 is Lipschitz, then Hk(f (A)) = 0.

• B. Franchi, R. Serapioni, and F. Serra Cassano, 2001–. . .

In H1, consider an alternative to Federer’s definition of 3-rectifiability, for instance using intrinsic Lipschitz graphs. Goal: study quantitative rectifiability in H1 (for codimension 1) (low dimensional sets: Franchi-Ferrari-Pajot, Juillet, Li-Schul, Hahlomaa) Motivation: uniform rectifiability in Rn (G. David and S. Semmes).

Towards the definition of intrinsic Lipschitz graphs

Write H1 = W ∗ V, where W - vertical 2-d subspace in R3, V - perpendicular 1-d horizontal subspace.

Towards the definition of intrinsic Lipschitz graphs

Write H1 = W ∗ V, where W - vertical 2-d subspace in R3, V - perpendicular 1-d horizontal subspace.

Heisenberg projections

Vertical projections: πW : H1 → W, p = pW ∗ pV → pW. Horizontal projections: πV : H1 → V, p = pW ∗ pV → pV.

Towards the definition of intrinsic Lipschitz graphs

Write H1 = W ∗ V, where W - vertical 2-d subspace in R3, V - perpendicular 1-d horizontal subspace.

Heisenberg projections

Vertical projections: πW : H1 → W, p = pW ∗ pV → pW. Horizontal projections: πV : H1 → V, p = pW ∗ pV → pV.

Heisenberg cone

CW(α) := {p ∈ H1 : πW(p) ≤ απV(p)}.

Intrinsic Lipschitz functions

Intrinsic L-Lipschitz graph

Γ ⊂ H1 such that (p ∗ CW( 1

L)) ∩ Γ = {p},

for all p ∈ Γ.

Intrinsic Lipschitz functions

Intrinsic L-Lipschitz graph

Γ ⊂ H1 such that (p ∗ CW( 1

L)) ∩ Γ = {p},

for all p ∈ Γ.

Intrinsic L-Lipschitz function

φ : A ⊆ W → V such that Γφ := {w ∗ φ(w) : w ∈ A} is an intrinsic L-Lipschitz graph. (φ is in general not metrically Lipschitz!)

The main result

Theorem

The following are equivalent for an Ahlfors 3-regular E ⊂ H1:

1 big pieces of intrinsic Lipschitz graphs 2 big vertical projections + weak geometric lemma for vertical β’s.

The main result

Theorem

The following are equivalent for an Ahlfors 3-regular E ⊂ H1:

1 big pieces of intrinsic Lipschitz graphs 2 big vertical projections + weak geometric lemma for vertical β’s.

big pieces of intrinsic Lipschitz graphs: ∃L ≥ 1, θ > 0: ∀x ∈ E, 0 < R ≤ diam(E) ∃Γ intrinsic L-Lipschitz: H3(E ∩ Γ ∩ B(x, R)) ≥ θR3.

The main result

Theorem

The following are equivalent for an Ahlfors 3-regular E ⊂ H1:

1 big pieces of intrinsic Lipschitz graphs 2 big vertical projections + weak geometric lemma for vertical β’s.

big pieces of intrinsic Lipschitz graphs: ∃L ≥ 1, θ > 0: ∀x ∈ E, 0 < R ≤ diam(E) ∃Γ intrinsic L-Lipschitz: H3(E ∩ Γ ∩ B(x, R)) ≥ θR3. big vertical projections: ∃δ > 0: ∀x ∈ E, 0 < R ≤ diamE ∃W vertical: L2(πW(E ∩ B(x, R))) ≥ δR3

The main result

Theorem

The following are equivalent for an Ahlfors 3-regular E ⊂ H1:

1 big pieces of intrinsic Lipschitz graphs 2 big vertical projections + weak geometric lemma for vertical β’s.

big pieces of intrinsic Lipschitz graphs: ∃L ≥ 1, θ > 0: ∀x ∈ E, 0 < R ≤ diam(E) ∃Γ intrinsic L-Lipschitz: H3(E ∩ Γ ∩ B(x, R)) ≥ θR3. big vertical projections: ∃δ > 0: ∀x ∈ E, 0 < R ≤ diamE ∃W vertical: L2(πW(E ∩ B(x, R))) ≥ δR3 vertical β-numbers: β(B) := infW,z supy∈B∩E

dist(y,z∗W) rad(B)

.

The main result

Theorem

The following are equivalent for an Ahlfors 3-regular E ⊂ H1:

1 big pieces of intrinsic Lipschitz graphs 2 big vertical projections + weak geometric lemma for vertical β’s.

big pieces of intrinsic Lipschitz graphs: ∃L ≥ 1, θ > 0: ∀x ∈ E, 0 < R ≤ diam(E) ∃Γ intrinsic L-Lipschitz: H3(E ∩ Γ ∩ B(x, R)) ≥ θR3. big vertical projections: ∃δ > 0: ∀x ∈ E, 0 < R ≤ diamE ∃W vertical: L2(πW(E ∩ B(x, R))) ≥ δR3 vertical β-numbers: β(B) := infW,z supy∈B∩E

dist(y,z∗W) rad(B)

. weak geometric lemma: ∀ε > 0, x ∈ E, R > 0 R

• E∩B(x,R)

χ{(y,s)∈E×R+: β(B(y,s))≥ε}(y, s)dH3(y)ds s ε R3.

Properties of intrinsic Lipschitz functions

• B. Franchi, R. Serapioni, F. Serra Cassano (2011); A. Naor, R. Young (2017) :

Properties of intrinsic L-Lipschitz φ : A ⊆ W → V

Properties of intrinsic Lipschitz functions

• B. Franchi, R. Serapioni, F. Serra Cassano (2011); A. Naor, R. Young (2017) :

Properties of intrinsic L-Lipschitz φ : A ⊆ W → V

p ∗ Γφ is intrinsic L-Lipschitz ∀p ∈ H1

Properties of intrinsic Lipschitz functions

• B. Franchi, R. Serapioni, F. Serra Cassano (2011); A. Naor, R. Young (2017) :

Properties of intrinsic L-Lipschitz φ : A ⊆ W → V

p ∗ Γφ is intrinsic L-Lipschitz ∀p ∈ H1 φ can be extended to an L′(L)-intrinsic Lipschitz function W → V

Properties of intrinsic Lipschitz functions

• B. Franchi, R. Serapioni, F. Serra Cassano (2011); A. Naor, R. Young (2017) :

Properties of intrinsic L-Lipschitz φ : A ⊆ W → V

p ∗ Γφ is intrinsic L-Lipschitz ∀p ∈ H1 φ can be extended to an L′(L)-intrinsic Lipschitz function W → V Γφ is Ahlfors-3-regular with constant depending on L (for A = W)

Properties of intrinsic Lipschitz functions

• B. Franchi, R. Serapioni, F. Serra Cassano (2011); A. Naor, R. Young (2017) :

Properties of intrinsic L-Lipschitz φ : A ⊆ W → V

p ∗ Γφ is intrinsic L-Lipschitz ∀p ∈ H1 φ can be extended to an L′(L)-intrinsic Lipschitz function W → V Γφ is Ahlfors-3-regular with constant depending on L (for A = W) φ is intrinsically differentiable in L2 a.e. w ∈ A, with intrinsic differential dφw : W → V, dφw(y, t) = ∇φφ(w)y, Intrinsically differentiable: (for φ(0) = 0) ∃dφ0 : W → V, intrinsic linear: |φ(y, t) − dφ0(y, t)| = o((y, t)); the general definition is by left-translation of the graph

Properties of intrinsic Lipschitz functions

• B. Franchi, R. Serapioni, F. Serra Cassano (2011); A. Naor, R. Young (2017) :

Properties of intrinsic L-Lipschitz φ : A ⊆ W → V

p ∗ Γφ is intrinsic L-Lipschitz ∀p ∈ H1 φ can be extended to an L′(L)-intrinsic Lipschitz function W → V Γφ is Ahlfors-3-regular with constant depending on L (for A = W) φ is intrinsically differentiable in L2 a.e. w ∈ A, with intrinsic differential dφw : W → V, dφw(y, t) = ∇φφ(w)y, φ has bounded intrinsic gradient ∇φφ∞ ≤ L

Properties of intrinsic Lipschitz functions

• B. Franchi, R. Serapioni, F. Serra Cassano (2011); A. Naor, R. Young (2017) :

Properties of intrinsic L-Lipschitz φ : A ⊆ W → V

p ∗ Γφ is intrinsic L-Lipschitz ∀p ∈ H1 φ can be extended to an L′(L)-intrinsic Lipschitz function W → V Γφ is Ahlfors-3-regular with constant depending on L (for A = W) φ is intrinsically differentiable in L2 a.e. w ∈ A, with intrinsic differential dφw : W → V, dφw(y, t) = ∇φφ(w)y, φ has bounded intrinsic gradient ∇φφ∞ ≤ L

Proof: Intrinsic Lipschitz graph ⇒ WGL

Let Γ = Γφ be the graph of an intrinsic Lipschitz function φ over W.

Proof: Intrinsic Lipschitz graph ⇒ WGL

Let Γ = Γφ be the graph of an intrinsic Lipschitz function φ over W.

First attempt to prove WGL for Γ

Γ ∩ B(y, s) is far from a vertical plane

?

⇒ gradient of φ fluctuates significantly in Γ ∩ B(y, s) ⇒ (with L∞ bound for gradient) WGL.

Proof: Intrinsic Lipschitz graph ⇒ WGL

Let Γ = Γφ be the graph of an intrinsic Lipschitz function φ over W.

First attempt to prove WGL for Γ

Γ ∩ B(y, s) is far from a vertical plane

?

⇒ gradient of φ fluctuates significantly in Γ ∩ B(y, s) ⇒ (with L∞ bound for gradient) WGL. This argument does not work for the intrinsic gradient!

Proof: Intrinsic Lipschitz graph ⇒ WGL

Let Γ = Γφ be the graph of an intrinsic Lipschitz function φ over W.

First attempt to prove WGL for Γ

Γ ∩ B(y, s) is far from a vertical plane

?

⇒ gradient of φ fluctuates significantly in Γ ∩ B(y, s) ⇒ (with L∞ bound for gradient) WGL. This argument does not work for the intrinsic gradient!

Example with ∇φφ(y, t) = ∂yφ(y, t) + φ(y, t)∂tφ(y, t) ≡ 0

Proof: Intrinsic Lipschitz graph ⇒ WGL

1 Property that forces ∇φφ to fluctuate a lot:

Γ being far from a intrinsic Lip constant gradient (CG) graph, (quantified by βCG-numbers)

Proof: Intrinsic Lipschitz graph ⇒ WGL

1 Property that forces ∇φφ to fluctuate a lot:

Γ being far from a intrinsic Lip constant gradient (CG) graph, (quantified by βCG-numbers)

2 ⇒ WGL for βCG (as before)

Proof: Intrinsic Lipschitz graph ⇒ WGL

1 Property that forces ∇φφ to fluctuate a lot:

Γ being far from a intrinsic Lip constant gradient (CG) graph, (quantified by βCG-numbers)

2 ⇒ WGL for βCG (as before) 3 globally defined intrinsic Lip CG graphs are (translated) vertical planes

Proof: Intrinsic Lipschitz graph ⇒ WGL

1 Property that forces ∇φφ to fluctuate a lot:

Γ being far from a intrinsic Lip constant gradient (CG) graph, (quantified by βCG-numbers)

2 ⇒ WGL for βCG (as before) 3 globally defined intrinsic Lip CG graphs are (translated) vertical planes 4 ⇒ (compactness argument)

CG graph defined in B(x, r) is almost flat in all sufficiently small sub-balls of B(x, r).

Proof: WGL and BVP ⇒ BPiLG

Proof: WGL and BVP ⇒ BPiLG

1 △ - David cubes on E. Define Q ∈ △ to be good if

β(Q) ≤ ε and L2(πW(Q)) ≥ cH3(Q).

Proof: WGL and BVP ⇒ BPiLG

1 △ - David cubes on E. Define Q ∈ △ to be good if

β(Q) ≤ ε and L2(πW(Q)) ≥ cH3(Q).

2 Geometric part:

(c, ε, W)-good Q ∈ △ is ‘almost’ intrinsic Lipschitz graph over W

Proof: WGL and BVP ⇒ BPiLG

1 △ - David cubes on E. Define Q ∈ △ to be good if

β(Q) ≤ ε and L2(πW(Q)) ≥ cH3(Q).

2 Geometric part:

(c, ε, W)-good Q ∈ △ is ‘almost’ intrinsic Lipschitz graph over W

3 Abstract part: modeled on David-Semmes “Quantitative rectifiability

and Lipschitz mappings” and Jones “Lipschitz and bi-Lipschitz functions”, but πW is not Lipschitz! However:

Proof: WGL and BVP ⇒ BPiLG

1 △ - David cubes on E. Define Q ∈ △ to be good if

β(Q) ≤ ε and L2(πW(Q)) ≥ cH3(Q).

2 Geometric part:

(c, ε, W)-good Q ∈ △ is ‘almost’ intrinsic Lipschitz graph over W

3 Abstract part: modeled on David-Semmes “Quantitative rectifiability

and Lipschitz mappings” and Jones “Lipschitz and bi-Lipschitz functions”, but πW is not Lipschitz! However:

Projection lemma

∃ 0 < C < ∞ such that for A ⊆ H1, one has L2(πW(A)) ≤ CH3(A).

Proof: WGL and BVP ⇒ BPiLG

1 △ - David cubes on E. Define Q ∈ △ to be good if

β(Q) ≤ ε and L2(πW(Q)) ≥ cH3(Q).

2 Geometric part:

(c, ε, W)-good Q ∈ △ is ‘almost’ intrinsic Lipschitz graph over W

3 Abstract part: modeled on David-Semmes “Quantitative rectifiability

and Lipschitz mappings” and Jones “Lipschitz and bi-Lipschitz functions”, but πW is not Lipschitz! However:

Projection lemma

∃ 0 < C < ∞ such that for A ⊆ H1, one has L2(πW(A)) ≤ CH3(A).

1

WGL, geometric part, and projection lemma ⇒ ∀Q0 ∈ △, W: ∃Fj ⊂ Q0 intrinsic L-Lipschitz: L2(πW(Q0 \ Fj)) ≤ bH3(Q0)

Proof: WGL and BVP ⇒ BPiLG

1 △ - David cubes on E. Define Q ∈ △ to be good if

β(Q) ≤ ε and L2(πW(Q)) ≥ cH3(Q).

2 Geometric part:

(c, ε, W)-good Q ∈ △ is ‘almost’ intrinsic Lipschitz graph over W

3 Abstract part: modeled on David-Semmes “Quantitative rectifiability

and Lipschitz mappings” and Jones “Lipschitz and bi-Lipschitz functions”, but πW is not Lipschitz! However:

Projection lemma

∃ 0 < C < ∞ such that for A ⊆ H1, one has L2(πW(A)) ≤ CH3(A).

1

WGL, geometric part, and projection lemma ⇒ ∀Q0 ∈ △, W: ∃Fj ⊂ Q0 intrinsic L-Lipschitz: L2(πW(Q0 \ Fj)) ≤ bH3(Q0)

2

BVP ⇒ ∃W, Q0 such that L2(πW(Q0)) ≥ 2bH3(Q0)

Proof: WGL and BVP ⇒ BPiLG

1 △ - David cubes on E. Define Q ∈ △ to be good if

β(Q) ≤ ε and L2(πW(Q)) ≥ cH3(Q).

2 Geometric part:

(c, ε, W)-good Q ∈ △ is ‘almost’ intrinsic Lipschitz graph over W

3 Abstract part: modeled on David-Semmes “Quantitative rectifiability

and Lipschitz mappings” and Jones “Lipschitz and bi-Lipschitz functions”, but πW is not Lipschitz! However:

Projection lemma

∃ 0 < C < ∞ such that for A ⊆ H1, one has L2(πW(A)) ≤ CH3(A).

1

WGL, geometric part, and projection lemma ⇒ ∀Q0 ∈ △, W: ∃Fj ⊂ Q0 intrinsic L-Lipschitz: L2(πW(Q0 \ Fj)) ≤ bH3(Q0)

2

BVP ⇒ ∃W, Q0 such that L2(πW(Q0)) ≥ 2bH3(Q0)

3

⇒ BPiLG.

Thank you for your attention!