Maximal Directional Derivatives and Universal Differentiability Sets - - PowerPoint PPT Presentation

Maximal Directional Derivatives and Universal Differentiability Sets in Carnot Groups

Andrea Pinamonti and Gareth Speight

University of Trento and University of Cincinnati

Warwick GMT 2017

• A. Pinamonti and G. Speight

UDS in Carnot Groups Warwick GMT 2017 1 / 35

Rademacher’s Theorem

Theorem (Rademacher)

Every Lipschitz function f : Rn → Rm is differentiable Lebesgue almost everywhere.

• A. Pinamonti and G. Speight

UDS in Carnot Groups Warwick GMT 2017 2 / 35

Rademacher’s Theorem

Theorem (Rademacher)

Every Lipschitz function f : Rn → Rm is differentiable Lebesgue almost everywhere. Equivalently: If a Lipschitz function f : Rn → Rm is differentiable at no point of N ⊂ Rn, then N is Lebesgue null.

• A. Pinamonti and G. Speight

UDS in Carnot Groups Warwick GMT 2017 2 / 35

Converse to Rademacher’s Theorem

Question

If N ⊂ Rn is Lebesgue null, does there exist a Lipschitz function f : Rn → Rm which is differentiable at no point of N?

• A. Pinamonti and G. Speight

UDS in Carnot Groups Warwick GMT 2017 3 / 35

m ≥ n

YES It is enough to prove the result for f : Rn → Rn: (Zahorski and Fowler and Preiss) The case n = 1.

• A. Pinamonti and G. Speight

UDS in Carnot Groups Warwick GMT 2017 4 / 35

m ≥ n

YES It is enough to prove the result for f : Rn → Rn: (Zahorski and Fowler and Preiss) The case n = 1. (Alberti, Cs¨

• rnyei and Preiss) The case n = 2.
• A. Pinamonti and G. Speight

UDS in Carnot Groups Warwick GMT 2017 4 / 35

m ≥ n

YES It is enough to prove the result for f : Rn → Rn: (Zahorski and Fowler and Preiss) The case n = 1. (Alberti, Cs¨

• rnyei and Preiss) The case n = 2.

(Alberti, Cs¨

• rnyei and Preiss) + (Cs¨
• rnyei and Jones) The case

n > 2. Recent results by De Philippis and Rindler.

• A. Pinamonti and G. Speight

UDS in Carnot Groups Warwick GMT 2017 4 / 35

m < n

NO (Preiss) Let n > 1. There exists a Lebesgue null set N ⊂ Rn such that every Lipschitz function f : Rn → R is differentiable at a point of N.

• A. Pinamonti and G. Speight

UDS in Carnot Groups Warwick GMT 2017 5 / 35

m < n

NO (Preiss) Let n > 1. There exists a Lebesgue null set N ⊂ Rn such that every Lipschitz function f : Rn → R is differentiable at a point of N. (Preiss, Speight) There is a Lebesgue null set N ⊂ Rn such that every Lipschitz function f : Rn → Rn−1 is differentiable at a point of N.

• A. Pinamonti and G. Speight

UDS in Carnot Groups Warwick GMT 2017 5 / 35

m < n

NO (Preiss) Let n > 1. There exists a Lebesgue null set N ⊂ Rn such that every Lipschitz function f : Rn → R is differentiable at a point of N. (Preiss, Speight) There is a Lebesgue null set N ⊂ Rn such that every Lipschitz function f : Rn → Rn−1 is differentiable at a point of N.

Dor´ e-Maleva, Dymond-Maleva

The universal differentiability set N constructed in Preiss’s paper can be made compact and of Hausdorff (even upper Minkowski) dimension one.

• A. Pinamonti and G. Speight

UDS in Carnot Groups Warwick GMT 2017 5 / 35

Maximal Directional Derivatives

‘Maximal’ directional derivatives are key to differentiability in small sets.

• A. Pinamonti and G. Speight

UDS in Carnot Groups Warwick GMT 2017 6 / 35

Maximal Directional Derivatives

‘Maximal’ directional derivatives are key to differentiability in small sets.

Theorem (Fitzpatrick)

Let E be a Banach space and f : E → R be Lipschitz. Suppose there exist x ∈ E and e ∈ E with e = 1 such that f ′(x, e) = Lip(f ). If the norm of E is Fr´ echet differentiable at e with derivative e∗, then f is Fr´ echet differentiable at x and f ′(x) = Lip(f )e∗.

• A. Pinamonti and G. Speight

UDS in Carnot Groups Warwick GMT 2017 6 / 35

Almost Maximal Directional Derivatives

Let Df := {(x, e) ∈ E × E : e = 1, f ′(x, e) exists}.

• A. Pinamonti and G. Speight

UDS in Carnot Groups Warwick GMT 2017 7 / 35

Almost Maximal Directional Derivatives

Let Df := {(x, e) ∈ E × E : e = 1, f ′(x, e) exists}.

Theorem (Preiss)

Suppose f : E → R is Lipschitz and (x0, e0) ∈ Df .

• A. Pinamonti and G. Speight

UDS in Carnot Groups Warwick GMT 2017 7 / 35

Almost Maximal Directional Derivatives

Let Df := {(x, e) ∈ E × E : e = 1, f ′(x, e) exists}.

Theorem (Preiss)

Suppose f : E → R is Lipschitz and (x0, e0) ∈ Df . Let M denote the set of all pairs (x, e) ∈ Df such that f ′(x, e) ≥ f ′(x0, e0) and |(f (x + te0) − f (x)) − (f (x0 + te0) − f (x0))| ≤ 6|t|

• (f ′(x, e) − f ′(x0, e0))Lip(f )

for every t ∈ R.

• A. Pinamonti and G. Speight

UDS in Carnot Groups Warwick GMT 2017 7 / 35

Almost Maximal Directional Derivatives

Let Df := {(x, e) ∈ E × E : e = 1, f ′(x, e) exists}.

Theorem (Preiss)

Suppose f : E → R is Lipschitz and (x0, e0) ∈ Df . Let M denote the set of all pairs (x, e) ∈ Df such that f ′(x, e) ≥ f ′(x0, e0) and |(f (x + te0) − f (x)) − (f (x0 + te0) − f (x0))| ≤ 6|t|

• (f ′(x, e) − f ′(x0, e0))Lip(f )

for every t ∈ R. If the norm is Fr´ echet differentiable at e0 and lim

δ↓0 sup{f ′(x, e): (x, e) ∈ M and x − x0 ≤ δ} ≤ f ′(x0, e0),

then f is Fr´ echet differentiable at x0.

• A. Pinamonti and G. Speight

UDS in Carnot Groups Warwick GMT 2017 7 / 35

Existence of Almost Maximal Directional Derivatives

Preiss showed that any Lipschitz map f : E → R has an ‘almost maximal’ directional derivative (after linear perturbation), hence a point of Fr´ echet differentiability (if the norm is differentiable).

• A. Pinamonti and G. Speight

UDS in Carnot Groups Warwick GMT 2017 8 / 35

Existence of Almost Maximal Directional Derivatives

Preiss showed that any Lipschitz map f : E → R has an ‘almost maximal’ directional derivative (after linear perturbation), hence a point of Fr´ echet differentiability (if the norm is differentiable).

Remark

The same argument works inside any Lebesgue null Gδ subset of Rn containing all lines which join pairs of points in Qn. Hence there exists a measure zero universal differentiability set.

• A. Pinamonti and G. Speight

UDS in Carnot Groups Warwick GMT 2017 8 / 35

Question

What happens in other spaces?

• A. Pinamonti and G. Speight

UDS in Carnot Groups Warwick GMT 2017 9 / 35

Carnot Groups

Definition

A Carnot group G is a simply connected Lie group whose Lie algebra G (space of left invariant vector fields) admits a decomposition of the form G = V1 ⊕ V2 ⊕ . . . ⊕ Vs, such that Vi = [V1, Vi−1] for any i = 2, . . . , s, and [V1, Vs] = 0.

• A. Pinamonti and G. Speight

UDS in Carnot Groups Warwick GMT 2017 10 / 35

Carnot Groups

Definition

A Carnot group G is a simply connected Lie group whose Lie algebra G (space of left invariant vector fields) admits a decomposition of the form G = V1 ⊕ V2 ⊕ . . . ⊕ Vs, such that Vi = [V1, Vi−1] for any i = 2, . . . , s, and [V1, Vs] = 0. Members of V1 are called horizontal directions.

• A. Pinamonti and G. Speight

UDS in Carnot Groups Warwick GMT 2017 10 / 35

Carnot Groups

Definition

A Carnot group G is a simply connected Lie group whose Lie algebra G (space of left invariant vector fields) admits a decomposition of the form G = V1 ⊕ V2 ⊕ . . . ⊕ Vs, such that Vi = [V1, Vi−1] for any i = 2, . . . , s, and [V1, Vs] = 0. Members of V1 are called horizontal directions. The number s is called the step of G.

• A. Pinamonti and G. Speight

UDS in Carnot Groups Warwick GMT 2017 10 / 35

Carnot Groups

Definition

A Carnot group G is a simply connected Lie group whose Lie algebra G (space of left invariant vector fields) admits a decomposition of the form G = V1 ⊕ V2 ⊕ . . . ⊕ Vs, such that Vi = [V1, Vi−1] for any i = 2, . . . , s, and [V1, Vs] = 0. Members of V1 are called horizontal directions. The number s is called the step of G. Using exponential coordinates, every Carnot group can be identified with some Rn equipped with a group operation and left-invariant vector fields.

• A. Pinamonti and G. Speight

UDS in Carnot Groups Warwick GMT 2017 10 / 35

Horizontal Curves

Fix a basis X1, . . . , Xm of V1.

• A. Pinamonti and G. Speight

UDS in Carnot Groups Warwick GMT 2017 11 / 35

Horizontal Curves

Fix a basis X1, . . . , Xm of V1.

Definition

An absolutely continuous curve γ : [a, b] → G is horizontal if there exists h: [a, b] → Rm such that for almost every t: γ′(t) =

m

• i=1

hiXi(γ(t)).

• A. Pinamonti and G. Speight

UDS in Carnot Groups Warwick GMT 2017 11 / 35

Horizontal Curves

Fix a basis X1, . . . , Xm of V1.

Definition

An absolutely continuous curve γ : [a, b] → G is horizontal if there exists h: [a, b] → Rm such that for almost every t: γ′(t) =

m

• i=1

hiXi(γ(t)). The horizontal length of such a curve is L(γ) =

b

a |h|.

• A. Pinamonti and G. Speight

UDS in Carnot Groups Warwick GMT 2017 11 / 35

Horizontal Curves

Fix a basis X1, . . . , Xm of V1.

Definition

An absolutely continuous curve γ : [a, b] → G is horizontal if there exists h: [a, b] → Rm such that for almost every t: γ′(t) =

m

• i=1

hiXi(γ(t)). The horizontal length of such a curve is L(γ) =

b

a |h|.

Definition

Define the Carnot-Caratheodory (CC) distance d on G by: d(x, y) = inf{L(γ): γ horizontal and joins x to y}. For convenience we write d(x) instead of d(x, 0).

• A. Pinamonti and G. Speight

UDS in Carnot Groups Warwick GMT 2017 11 / 35

Dilations and Haar Measure

Dilations δr : G → G are defined for every r > 0. For every x, y ∈ G, these satisfy: δr(xy) = δr(x)δr(y), d(δr(x), δr(y)) = rd(x, y).

• A. Pinamonti and G. Speight

UDS in Carnot Groups Warwick GMT 2017 12 / 35

Dilations and Haar Measure

Dilations δr : G → G are defined for every r > 0. For every x, y ∈ G, these satisfy: δr(xy) = δr(x)δr(y), d(δr(x), δr(y)) = rd(x, y). Carnot groups admit a Haar measure µ. This is a non-trivial Borel measure satisfying µ(gA) = µ(A) for every Borel set A ⊂ G.

• A. Pinamonti and G. Speight

UDS in Carnot Groups Warwick GMT 2017 12 / 35

Heisenberg Group

Example

The Heisenberg group Hn (a step 2 Carnot group) is R2n+1 equipped with group law: (x, y, t)(x′, y′, t′) = (x + x′, y + y′, t + t′ − 2(x, y′ − y, x′)).

• A. Pinamonti and G. Speight

UDS in Carnot Groups Warwick GMT 2017 13 / 35

Heisenberg Group

Example

The Heisenberg group Hn (a step 2 Carnot group) is R2n+1 equipped with group law: (x, y, t)(x′, y′, t′) = (x + x′, y + y′, t + t′ − 2(x, y′ − y, x′)). Left-invariant horizontal vector fields on Hn are given by: Xi(x, y, t) = ∂xi + 2yi∂t, Yi(x, y, t) = ∂yi − 2xi∂t, 1 ≤ i ≤ n.

• A. Pinamonti and G. Speight

UDS in Carnot Groups Warwick GMT 2017 13 / 35

Heisenberg Group

Example

The Heisenberg group Hn (a step 2 Carnot group) is R2n+1 equipped with group law: (x, y, t)(x′, y′, t′) = (x + x′, y + y′, t + t′ − 2(x, y′ − y, x′)). Left-invariant horizontal vector fields on Hn are given by: Xi(x, y, t) = ∂xi + 2yi∂t, Yi(x, y, t) = ∂yi − 2xi∂t, 1 ≤ i ≤ n. Dilations are given by δr(x, y, t) = (rx, ry, r 2t).

• A. Pinamonti and G. Speight

UDS in Carnot Groups Warwick GMT 2017 13 / 35

Heisenberg Group

Example

The Heisenberg group Hn (a step 2 Carnot group) is R2n+1 equipped with group law: (x, y, t)(x′, y′, t′) = (x + x′, y + y′, t + t′ − 2(x, y′ − y, x′)). Left-invariant horizontal vector fields on Hn are given by: Xi(x, y, t) = ∂xi + 2yi∂t, Yi(x, y, t) = ∂yi − 2xi∂t, 1 ≤ i ≤ n. Dilations are given by δr(x, y, t) = (rx, ry, r 2t). Haar measure on Hn is L2n+1.

• A. Pinamonti and G. Speight

UDS in Carnot Groups Warwick GMT 2017 13 / 35

Horizontal Curves in the Heisenberg Group

x y t

• A. Pinamonti and G. Speight

UDS in Carnot Groups Warwick GMT 2017 14 / 35

Pansu Differentiability

Definition

A map L: G1 → G2 is group linear if for every x, y ∈ G1 and r > 0: L(xy) = L(x)L(y) and L(δr(x)) = δr(L(x)).

• A. Pinamonti and G. Speight

UDS in Carnot Groups Warwick GMT 2017 15 / 35

Pansu Differentiability

Definition

A map L: G1 → G2 is group linear if for every x, y ∈ G1 and r > 0: L(xy) = L(x)L(y) and L(δr(x)) = δr(L(x)). A map f : G1 → G2 is differentiable at x ∈ G1 if there exists a group linear map L: G1 → G2 such that lim

y→x

d(f (x)−1f (y), L(x−1y)) d(x, y) = 0.

• A. Pinamonti and G. Speight

UDS in Carnot Groups Warwick GMT 2017 15 / 35

Pansu Differentiability

Definition

A map L: G1 → G2 is group linear if for every x, y ∈ G1 and r > 0: L(xy) = L(x)L(y) and L(δr(x)) = δr(L(x)). A map f : G1 → G2 is differentiable at x ∈ G1 if there exists a group linear map L: G1 → G2 such that lim

y→x

d(f (x)−1f (y), L(x−1y)) d(x, y) = 0.

Theorem (Pansu)

Lipschitz maps between Carnot groups are differentiable almost everywhere.

• A. Pinamonti and G. Speight

UDS in Carnot Groups Warwick GMT 2017 15 / 35

Directional Derivatives

Definition

Suppose f : G → R is Lipschitz, x ∈ G and E ∈ V1. We define the directional derivative Ef (x) by: Ef (x) = lim

t→0

f (x exp tE) − f (x) t if the limit exists.

• A. Pinamonti and G. Speight

UDS in Carnot Groups Warwick GMT 2017 16 / 35

Directional Derivatives

Definition

Suppose f : G → R is Lipschitz, x ∈ G and E ∈ V1. We define the directional derivative Ef (x) by: Ef (x) = lim

t→0

f (x exp tE) − f (x) t if the limit exists. If f is differentiable at x, the derivative is df (x)(v) = (p(v), ∇Hf (x)). Here p is the horizontal projection and ∇Hf (x) = (X1f (x), . . . , Xmf (x)).

• A. Pinamonti and G. Speight

UDS in Carnot Groups Warwick GMT 2017 16 / 35

Directional Derivatives

Definition

Suppose f : G → R is Lipschitz, x ∈ G and E ∈ V1. We define the directional derivative Ef (x) by: Ef (x) = lim

t→0

f (x exp tE) − f (x) t if the limit exists. If f is differentiable at x, the derivative is df (x)(v) = (p(v), ∇Hf (x)). Here p is the horizontal projection and ∇Hf (x) = (X1f (x), . . . , Xmf (x)). Questions: Does existence of a maximal directional derivative suffice for differentiability in Carnot groups?

• A. Pinamonti and G. Speight

UDS in Carnot Groups Warwick GMT 2017 16 / 35

Directional Derivatives

Definition

Suppose f : G → R is Lipschitz, x ∈ G and E ∈ V1. We define the directional derivative Ef (x) by: Ef (x) = lim

t→0

f (x exp tE) − f (x) t if the limit exists. If f is differentiable at x, the derivative is df (x)(v) = (p(v), ∇Hf (x)). Here p is the horizontal projection and ∇Hf (x) = (X1f (x), . . . , Xmf (x)). Questions: Does existence of a maximal directional derivative suffice for differentiability in Carnot groups? Do measure zero universal differentiability sets exist in Carnot groups?

• A. Pinamonti and G. Speight

UDS in Carnot Groups Warwick GMT 2017 16 / 35

Maximal Directional Derivatives in Carnot Groups

Fix a Carnot group G with CC distance from a basis X1, . . . , Xr of V1.

• A. Pinamonti and G. Speight

UDS in Carnot Groups Warwick GMT 2017 17 / 35

Maximal Directional Derivatives in Carnot Groups

Fix a Carnot group G with CC distance from a basis X1, . . . , Xr of V1. Let ω be an inner product norm on V1 making X1, . . . , Xr orthonormal.

• A. Pinamonti and G. Speight

UDS in Carnot Groups Warwick GMT 2017 17 / 35

Maximal Directional Derivatives in Carnot Groups

Fix a Carnot group G with CC distance from a basis X1, . . . , Xr of V1. Let ω be an inner product norm on V1 making X1, . . . , Xr orthonormal.

Proposition

Let f : G → R be a Lipschitz map. Then: Lip(f ) = sup{Ef (x): x ∈ G, E ∈ V1, ω(E) = 1, Ef (x) exists}.

• A. Pinamonti and G. Speight

UDS in Carnot Groups Warwick GMT 2017 17 / 35

Maximal Directional Derivatives in Carnot Groups

Fix a Carnot group G with CC distance from a basis X1, . . . , Xr of V1. Let ω be an inner product norm on V1 making X1, . . . , Xr orthonormal.

Proposition

Let f : G → R be a Lipschitz map. Then: Lip(f ) = sup{Ef (x): x ∈ G, E ∈ V1, ω(E) = 1, Ef (x) exists}.

Definition

Let x ∈ G and E ∈ V1 with ω(E) = 1. We say the directional derivative Ef (x) is maximal if Ef (x) = Lip(f ).

• A. Pinamonti and G. Speight

UDS in Carnot Groups Warwick GMT 2017 17 / 35

Maximality and Differentiability of the CC Distance

Definition

We say that the CC distance d in G is differentiable in horizontal directions if it is differentiable at any point u of the form u = exp(E) with E ∈ V1 \ {0}.

• A. Pinamonti and G. Speight

UDS in Carnot Groups Warwick GMT 2017 18 / 35

Maximality and Differentiability of the CC Distance

Definition

We say that the CC distance d in G is differentiable in horizontal directions if it is differentiable at any point u of the form u = exp(E) with E ∈ V1 \ {0}.

Theorem (Le Donne, Pinamonti, Speight)

The CC distance in G is differentiable in horizontal directions if and only if whenever Ef (x) is a maximal directional derivative of a Lipschitz function f : G → R, then necessarily f must be differentiable at x.

• A. Pinamonti and G. Speight

UDS in Carnot Groups Warwick GMT 2017 18 / 35

Maximality and Differentiability of the CC Distance

Definition

We say that the CC distance d in G is differentiable in horizontal directions if it is differentiable at any point u of the form u = exp(E) with E ∈ V1 \ {0}.

Theorem (Le Donne, Pinamonti, Speight)

The CC distance in G is differentiable in horizontal directions if and only if whenever Ef (x) is a maximal directional derivative of a Lipschitz function f : G → R, then necessarily f must be differentiable at x. If is easy: if E ∈ V1 with ω(E) = 1 then Ed(exp(E)) = 1 = Lip(d), so d must be differentiable at exp(E).

• A. Pinamonti and G. Speight

UDS in Carnot Groups Warwick GMT 2017 18 / 35

Maximality and Differentiability of the CC Distance

Definition

We say that the CC distance d in G is differentiable in horizontal directions if it is differentiable at any point u of the form u = exp(E) with E ∈ V1 \ {0}.

Theorem (Le Donne, Pinamonti, Speight)

The CC distance in G is differentiable in horizontal directions if and only if whenever Ef (x) is a maximal directional derivative of a Lipschitz function f : G → R, then necessarily f must be differentiable at x. If is easy: if E ∈ V1 with ω(E) = 1 then Ed(exp(E)) = 1 = Lip(d), so d must be differentiable at exp(E). Only if adapts the proof from Banach spaces with a differentiable norm.

• A. Pinamonti and G. Speight

UDS in Carnot Groups Warwick GMT 2017 18 / 35

Maximality and Differentiability in Step 2

Theorem (Le Donne, Pinamonti, Speight)

The CC distance is differentiable in horizontal directions (equivalently, maximality implies differentiability) in any step 2 Carnot group.

• A. Pinamonti and G. Speight

UDS in Carnot Groups Warwick GMT 2017 19 / 35

Maximality and Differentiability in Step 2

Theorem (Le Donne, Pinamonti, Speight)

The CC distance is differentiable in horizontal directions (equivalently, maximality implies differentiability) in any step 2 Carnot group. Idea:

1 First work in a free Carnot group of step 2. Let u = exp(E) for some

E ∈ V1 \ {0}.

• A. Pinamonti and G. Speight

UDS in Carnot Groups Warwick GMT 2017 19 / 35

Maximality and Differentiability in Step 2

Theorem (Le Donne, Pinamonti, Speight)

The CC distance is differentiable in horizontal directions (equivalently, maximality implies differentiability) in any step 2 Carnot group. Idea:

1 First work in a free Carnot group of step 2. Let u = exp(E) for some

E ∈ V1 \ {0}.

2 Construct not-too-long horizontal curves joining 0 to points y = uz

close to u. They should stay close to the curve t → exp(tE).

• A. Pinamonti and G. Speight

UDS in Carnot Groups Warwick GMT 2017 19 / 35

Maximality and Differentiability in Step 2

Theorem (Le Donne, Pinamonti, Speight)

The CC distance is differentiable in horizontal directions (equivalently, maximality implies differentiability) in any step 2 Carnot group. Idea:

1 First work in a free Carnot group of step 2. Let u = exp(E) for some

E ∈ V1 \ {0}.

2 Construct not-too-long horizontal curves joining 0 to points y = uz

close to u. They should stay close to the curve t → exp(tE).

3 Use the curves constructed to get a good upper bound for

d(uz) − d(u). Projection arguments give a lower bound in any Carnot group, hence differentiability of the CC distance in step 2 free groups.

• A. Pinamonti and G. Speight

UDS in Carnot Groups Warwick GMT 2017 19 / 35

Maximality and Differentiability in Step 2

Theorem (Le Donne, Pinamonti, Speight)

The CC distance is differentiable in horizontal directions (equivalently, maximality implies differentiability) in any step 2 Carnot group. Idea:

1 First work in a free Carnot group of step 2. Let u = exp(E) for some

E ∈ V1 \ {0}.

2 Construct not-too-long horizontal curves joining 0 to points y = uz

close to u. They should stay close to the curve t → exp(tE).

3 Use the curves constructed to get a good upper bound for

d(uz) − d(u). Projection arguments give a lower bound in any Carnot group, hence differentiability of the CC distance in step 2 free groups.

4 Using homomorphism properties of free Lie algebras, a quotient

argument gives the result for general step 2 Carnot groups.

• A. Pinamonti and G. Speight

UDS in Carnot Groups Warwick GMT 2017 19 / 35

Free Carnot Groups of Step 2

Given integer r ≥ 2, let n = r + r(r − 1)/2. Denote coordinates in Rn by xi, 1 ≤ i ≤ r, and xij, 1 ≤ j < i ≤ r.

• A. Pinamonti and G. Speight

UDS in Carnot Groups Warwick GMT 2017 20 / 35

Free Carnot Groups of Step 2

Given integer r ≥ 2, let n = r + r(r − 1)/2. Denote coordinates in Rn by xi, 1 ≤ i ≤ r, and xij, 1 ≤ j < i ≤ r.

Definition

The free Carnot group of step 2 and r generators Gr is Rn with product: (x · y)i = xi + yi, (x · y)ij = xij + yij + 1 2(xiyj − yixj).

• A. Pinamonti and G. Speight

UDS in Carnot Groups Warwick GMT 2017 20 / 35

Free Carnot Groups of Step 2

Given integer r ≥ 2, let n = r + r(r − 1)/2. Denote coordinates in Rn by xi, 1 ≤ i ≤ r, and xij, 1 ≤ j < i ≤ r.

Definition

The free Carnot group of step 2 and r generators Gr is Rn with product: (x · y)i = xi + yi, (x · y)ij = xij + yij + 1 2(xiyj − yixj). We have V1 = Span{Xi : 1 ≤ i ≤ r} and V2 = Span{Xij : 1 ≤ j < i ≤ r}, where Xi := ∂i +

• j>i

xj 2 ∂ji −

• j<i

xj 2 ∂ij Xij := ∂ij.

• A. Pinamonti and G. Speight

UDS in Carnot Groups Warwick GMT 2017 20 / 35

Free Carnot Groups of Step 2

Given integer r ≥ 2, let n = r + r(r − 1)/2. Denote coordinates in Rn by xi, 1 ≤ i ≤ r, and xij, 1 ≤ j < i ≤ r.

Definition

The free Carnot group of step 2 and r generators Gr is Rn with product: (x · y)i = xi + yi, (x · y)ij = xij + yij + 1 2(xiyj − yixj). We have V1 = Span{Xi : 1 ≤ i ≤ r} and V2 = Span{Xij : 1 ≤ j < i ≤ r}, where Xi := ∂i +

• j>i

xj 2 ∂ji −

• j<i

xj 2 ∂ij Xij := ∂ij. Note [Xi, Xj] = Xij and G2 = H1.

• A. Pinamonti and G. Speight

UDS in Carnot Groups Warwick GMT 2017 20 / 35

Free Carnot Groups of Step 2

Given integer r ≥ 2, let n = r + r(r − 1)/2. Denote coordinates in Rn by xi, 1 ≤ i ≤ r, and xij, 1 ≤ j < i ≤ r.

Definition

The free Carnot group of step 2 and r generators Gr is Rn with product: (x · y)i = xi + yi, (x · y)ij = xij + yij + 1 2(xiyj − yixj). We have V1 = Span{Xi : 1 ≤ i ≤ r} and V2 = Span{Xij : 1 ≤ j < i ≤ r}, where Xi := ∂i +

• j>i

xj 2 ∂ji −

• j<i

xj 2 ∂ij Xij := ∂ij. Note [Xi, Xj] = Xij and G2 = H1. Horizontal curves in Gr are lifts of curves in Rr, with ij-coordinate given by areas swept out in the ij-plane.

• A. Pinamonti and G. Speight

UDS in Carnot Groups Warwick GMT 2017 20 / 35

Abstract View of Free Groups

Definition

Let r ≥ 2 and s ≥ 1 be integers. We say that Fr,s is the free-nilpotent Lie algebra with r generators x1, . . . , xr of step s if:

• A. Pinamonti and G. Speight

UDS in Carnot Groups Warwick GMT 2017 21 / 35

Abstract View of Free Groups

Definition

Let r ≥ 2 and s ≥ 1 be integers. We say that Fr,s is the free-nilpotent Lie algebra with r generators x1, . . . , xr of step s if:

1 Fr,s is a Lie algebra generated by elements x1, . . . , xr,

• A. Pinamonti and G. Speight

UDS in Carnot Groups Warwick GMT 2017 21 / 35

Abstract View of Free Groups

Definition

Let r ≥ 2 and s ≥ 1 be integers. We say that Fr,s is the free-nilpotent Lie algebra with r generators x1, . . . , xr of step s if:

1 Fr,s is a Lie algebra generated by elements x1, . . . , xr, 2 Fr,s is nilpotent of step s (nested Lie brackets of length s + 1 are 0),

• A. Pinamonti and G. Speight

UDS in Carnot Groups Warwick GMT 2017 21 / 35

Abstract View of Free Groups

Definition

Let r ≥ 2 and s ≥ 1 be integers. We say that Fr,s is the free-nilpotent Lie algebra with r generators x1, . . . , xr of step s if:

1 Fr,s is a Lie algebra generated by elements x1, . . . , xr, 2 Fr,s is nilpotent of step s (nested Lie brackets of length s + 1 are 0), 3 for every Lie algebra g nilpotent of step s and every map

Φ: {x1, . . . , xr} → g, there is a unique homomorphism of Lie algebras ˜ Φ: Fr,s → g that extends Φ.

• A. Pinamonti and G. Speight

UDS in Carnot Groups Warwick GMT 2017 21 / 35

Abstract View of Free Groups

Definition

Let r ≥ 2 and s ≥ 1 be integers. We say that Fr,s is the free-nilpotent Lie algebra with r generators x1, . . . , xr of step s if:

1 Fr,s is a Lie algebra generated by elements x1, . . . , xr, 2 Fr,s is nilpotent of step s (nested Lie brackets of length s + 1 are 0), 3 for every Lie algebra g nilpotent of step s and every map

Φ: {x1, . . . , xr} → g, there is a unique homomorphism of Lie algebras ˜ Φ: Fr,s → g that extends Φ. Free Carnot groups (of any step) are Carnot groups whose Lie algebra is isomorphic to a free-nilpotent Lie algebra Fr,s for some r ≥ 2 and s ≥ 1.

• A. Pinamonti and G. Speight

UDS in Carnot Groups Warwick GMT 2017 21 / 35

Rotations in Free Groups

Lemma

Suppose y ∈ Gr with L = |p(y)| = 0. Then there exists a group isometric isomorphism F : Gr → Gr such that F1(y) = L and Fi(y) = 0 for i > 1.

• A. Pinamonti and G. Speight

UDS in Carnot Groups Warwick GMT 2017 22 / 35

Rotations in Free Groups

Lemma

Suppose y ∈ Gr with L = |p(y)| = 0. Then there exists a group isometric isomorphism F : Gr → Gr such that F1(y) = L and Fi(y) = 0 for i > 1. Such a map can be chosen of the form F(x, y) = (A(x), B(y)), where A: Rr → Rr is a linear isometry and B is linear.

• A. Pinamonti and G. Speight

UDS in Carnot Groups Warwick GMT 2017 22 / 35

Rotations in Free Groups

Lemma

Suppose y ∈ Gr with L = |p(y)| = 0. Then there exists a group isometric isomorphism F : Gr → Gr such that F1(y) = L and Fi(y) = 0 for i > 1. Such a map can be chosen of the form F(x, y) = (A(x), B(y)), where A: Rr → Rr is a linear isometry and B is linear. Hence, to prove differentiability of the CC distance in horizontal directions, it suffices to construct horizontal curves joining 0 to points y ∈ Gr with y1 > 0 and yi = 0 for i > 1.

• A. Pinamonti and G. Speight

UDS in Carnot Groups Warwick GMT 2017 22 / 35

Useful Horizontal Curve in Gr

Lemma (Le Donne, Pinamonti, Speight)

Fix y ∈ Gr with y1 > 0 and yi = 0 for i > 1.

• A. Pinamonti and G. Speight

UDS in Carnot Groups Warwick GMT 2017 23 / 35

Useful Horizontal Curve in Gr

Lemma (Le Donne, Pinamonti, Speight)

Fix y ∈ Gr with y1 > 0 and yi = 0 for i > 1. Let A = max2≤i≤r |yi1| and B = maxi>j>1 |yij|.

• A. Pinamonti and G. Speight

UDS in Carnot Groups Warwick GMT 2017 23 / 35

Useful Horizontal Curve in Gr

Lemma (Le Donne, Pinamonti, Speight)

Fix y ∈ Gr with y1 > 0 and yi = 0 for i > 1. Let A = max2≤i≤r |yi1| and B = maxi>j>1 |yij|. Then there exists a Lipschitz horizontal curve γ : [0, 1] → Gr which is a concatenation of horizontal lines such that:

• A. Pinamonti and G. Speight

UDS in Carnot Groups Warwick GMT 2017 23 / 35

Useful Horizontal Curve in Gr

Lemma (Le Donne, Pinamonti, Speight)

Fix y ∈ Gr with y1 > 0 and yi = 0 for i > 1. Let A = max2≤i≤r |yi1| and B = maxi>j>1 |yij|. Then there exists a Lipschitz horizontal curve γ : [0, 1] → Gr which is a concatenation of horizontal lines such that:

1

γ(0) = 0 and γ(1) = y,

• A. Pinamonti and G. Speight

UDS in Carnot Groups Warwick GMT 2017 23 / 35

Useful Horizontal Curve in Gr

Lemma (Le Donne, Pinamonti, Speight)

Fix y ∈ Gr with y1 > 0 and yi = 0 for i > 1. Let A = max2≤i≤r |yi1| and B = maxi>j>1 |yij|. Then there exists a Lipschitz horizontal curve γ : [0, 1] → Gr which is a concatenation of horizontal lines such that:

1

γ(0) = 0 and γ(1) = y,

2

The Lipschitz constant of γ satisfies Lip(γ) ≤ y1 max

• 1 + CA2

y 4

1

1/2 ,

• 1 + CB

y 2

1

1/2 ,

• A. Pinamonti and G. Speight

UDS in Carnot Groups Warwick GMT 2017 23 / 35

Useful Horizontal Curve in Gr

Lemma (Le Donne, Pinamonti, Speight)

Fix y ∈ Gr with y1 > 0 and yi = 0 for i > 1. Let A = max2≤i≤r |yi1| and B = maxi>j>1 |yij|. Then there exists a Lipschitz horizontal curve γ : [0, 1] → Gr which is a concatenation of horizontal lines such that:

1

γ(0) = 0 and γ(1) = y,

2

The Lipschitz constant of γ satisfies Lip(γ) ≤ y1 max

• 1 + CA2

y 4

1

1/2 ,

• 1 + CB

y 2

1

1/2 ,

3

γ′(t) exists for all t ∈ [0, 1] except finitely many points and satisfies |(p ◦ γ)′(t) − p(y)| ≤ C max A y1 , √ B

• .
• A. Pinamonti and G. Speight

UDS in Carnot Groups Warwick GMT 2017 23 / 35

Useful Horizontal Curve in Gr

(y1, 0, . . . , 0) X1 y γ(0) = 0 In each subinterval we fix some vertical coordinate ij and leave the rest unchanged γ(1) = y = (y1, 0, . . . , 0, y21, y31, . . .)

• A. Pinamonti and G. Speight

UDS in Carnot Groups Warwick GMT 2017 24 / 35

The Engel Group

Definition

The Engel group E (a Carnot group of step 3) is R4 with group law x · y =

• x1 + y1, x2 + y2, x3 + y3 − x1y2, x4 + y4 − x1y3 + 1

2x2

1y2

• .
• A. Pinamonti and G. Speight

UDS in Carnot Groups Warwick GMT 2017 25 / 35

The Engel Group

Definition

The Engel group E (a Carnot group of step 3) is R4 with group law x · y =

• x1 + y1, x2 + y2, x3 + y3 − x1y2, x4 + y4 − x1y3 + 1

2x2

1y2

• .

We have V1 = Span{X1, X2}, V2 = Span{X3} and V3 = Span{X4}, where X1 = ∂1, X2 = ∂2 − x1∂3 + x2

1

2 ∂4, X3 = ∂3 − x1∂4, X4 = ∂4.

• A. Pinamonti and G. Speight

UDS in Carnot Groups Warwick GMT 2017 25 / 35

The Engel Group

Definition

The Engel group E (a Carnot group of step 3) is R4 with group law x · y =

• x1 + y1, x2 + y2, x3 + y3 − x1y2, x4 + y4 − x1y3 + 1

2x2

1y2

• .

We have V1 = Span{X1, X2}, V2 = Span{X3} and V3 = Span{X4}, where X1 = ∂1, X2 = ∂2 − x1∂3 + x2

1

2 ∂4, X3 = ∂3 − x1∂4, X4 = ∂4. Non-trivial bracket relations are X3 = [X2, X1] and X4 = [X3, X1].

• A. Pinamonti and G. Speight

UDS in Carnot Groups Warwick GMT 2017 25 / 35

The Engel Group

Definition

The Engel group E (a Carnot group of step 3) is R4 with group law x · y =

• x1 + y1, x2 + y2, x3 + y3 − x1y2, x4 + y4 − x1y3 + 1

2x2

1y2

• .

We have V1 = Span{X1, X2}, V2 = Span{X3} and V3 = Span{X4}, where X1 = ∂1, X2 = ∂2 − x1∂3 + x2

1

2 ∂4, X3 = ∂3 − x1∂4, X4 = ∂4. Non-trivial bracket relations are X3 = [X2, X1] and X4 = [X3, X1]. Note (0, 1, 0, 0) · (0, 0, 0, ε) = (0, 1, 0, ε).

• A. Pinamonti and G. Speight

UDS in Carnot Groups Warwick GMT 2017 25 / 35

Non-Differentiability in the Engel Group

Lemma

In the Engel group, there exists C > 0 such that d((0, 0, 0, 0), (0, 1, 0, ε)) ≥ 1 + C|ε|1/3 for ε ∈ (−1, 0).

• A. Pinamonti and G. Speight

UDS in Carnot Groups Warwick GMT 2017 26 / 35

Non-Differentiability in the Engel Group

Lemma

In the Engel group, there exists C > 0 such that d((0, 0, 0, 0), (0, 1, 0, ε)) ≥ 1 + C|ε|1/3 for ε ∈ (−1, 0).

Theorem (Le Donne, Pinamonti, Speight)

d is not differentiable at ¯ p = (0, 1, 0, 0) = exp(X2), but X2d(¯ p) = 1.

• A. Pinamonti and G. Speight

UDS in Carnot Groups Warwick GMT 2017 26 / 35

Non-Differentiability in the Engel Group

Lemma

In the Engel group, there exists C > 0 such that d((0, 0, 0, 0), (0, 1, 0, ε)) ≥ 1 + C|ε|1/3 for ε ∈ (−1, 0).

Theorem (Le Donne, Pinamonti, Speight)

d is not differentiable at ¯ p = (0, 1, 0, 0) = exp(X2), but X2d(¯ p) = 1.

Proof.

Suppose d has differential L at ¯

• p. Then L(h) = X1d(¯

p), X2d(¯ p) · h1, h2, so L((0, 0, 0, ε)) = 0 for ε ∈ R. Hence for ε ∈ (−1, 0): d(¯ p · (0, 0, 0, ε)) − d(¯ p) − L((0, 0, 0, ε)) = d((0, 1, 0, ε)) − 1 ≥ C|ε|1/3 ≥ Cd((0, 0, 0, ε)), which contradicts differentiability at ¯ p.

• A. Pinamonti and G. Speight

UDS in Carnot Groups Warwick GMT 2017 26 / 35

Universal Differentiability Sets in Step 2

Theorem (Le Donne, Pinamonti, Speight)

There is a Hausdorff dimension one (in particular, measure zero) universal differentiability set in any Carnot group G of step 2.

• A. Pinamonti and G. Speight

UDS in Carnot Groups Warwick GMT 2017 27 / 35

Universal Differentiability Sets in Step 2

Theorem (Le Donne, Pinamonti, Speight)

There is a Hausdorff dimension one (in particular, measure zero) universal differentiability set in any Carnot group G of step 2. (every Lipschitz map f : G → R is differentiable at some point of N)

• A. Pinamonti and G. Speight

UDS in Carnot Groups Warwick GMT 2017 27 / 35

Universal Differentiability Sets in Step 2

Theorem (Le Donne, Pinamonti, Speight)

There is a Hausdorff dimension one (in particular, measure zero) universal differentiability set in any Carnot group G of step 2. (every Lipschitz map f : G → R is differentiable at some point of N) Idea:

1 First work in a free Carnot group Gr of step 2 and r generators. Fix a

Gδ Hausdorff dimension one set N containing a special countable family of horizontal curves (to be used in step 3 below).

• A. Pinamonti and G. Speight

UDS in Carnot Groups Warwick GMT 2017 27 / 35

Universal Differentiability Sets in Step 2

Theorem (Le Donne, Pinamonti, Speight)

There is a Hausdorff dimension one (in particular, measure zero) universal differentiability set in any Carnot group G of step 2. (every Lipschitz map f : G → R is differentiable at some point of N) Idea:

1 First work in a free Carnot group Gr of step 2 and r generators. Fix a

Gδ Hausdorff dimension one set N containing a special countable family of horizontal curves (to be used in step 3 below).

2 Find an ‘almost maximal’ directional derivative Ef (x), where we

consider x ∈ N and E ∈ V1 with ω(E) = 1.

• A. Pinamonti and G. Speight

UDS in Carnot Groups Warwick GMT 2017 27 / 35

Universal Differentiability Sets in Step 2

Theorem (Le Donne, Pinamonti, Speight)

There is a Hausdorff dimension one (in particular, measure zero) universal differentiability set in any Carnot group G of step 2. (every Lipschitz map f : G → R is differentiable at some point of N) Idea:

1 First work in a free Carnot group Gr of step 2 and r generators. Fix a

Gδ Hausdorff dimension one set N containing a special countable family of horizontal curves (to be used in step 3 below).

2 Find an ‘almost maximal’ directional derivative Ef (x), where we

consider x ∈ N and E ∈ V1 with ω(E) = 1.

3 Using carefully constructed horizontal curves, show that if x ∈ N and

Ef (x) is ‘almost maximal’, then f is differentiable at x.

• A. Pinamonti and G. Speight

UDS in Carnot Groups Warwick GMT 2017 27 / 35

Universal Differentiability Sets in Step 2

Theorem (Le Donne, Pinamonti, Speight)

There is a Hausdorff dimension one (in particular, measure zero) universal differentiability set in any Carnot group G of step 2. (every Lipschitz map f : G → R is differentiable at some point of N) Idea:

1 First work in a free Carnot group Gr of step 2 and r generators. Fix a

Gδ Hausdorff dimension one set N containing a special countable family of horizontal curves (to be used in step 3 below).

2 Find an ‘almost maximal’ directional derivative Ef (x), where we

consider x ∈ N and E ∈ V1 with ω(E) = 1.

3 Using carefully constructed horizontal curves, show that if x ∈ N and

Ef (x) is ‘almost maximal’, then f is differentiable at x.

4 Using homomorphism properties of free Lie algebras, a quotient

argument gives the result for general step 2 Carnot groups.

• A. Pinamonti and G. Speight

UDS in Carnot Groups Warwick GMT 2017 27 / 35

Horizontal Curves for Almost Maximality

Lemma (Le Donne, Pinamonti, Speight)

Given η > 0, the following holds for sufficiently small ∆.

• A. Pinamonti and G. Speight

UDS in Carnot Groups Warwick GMT 2017 28 / 35

Horizontal Curves for Almost Maximality

Lemma (Le Donne, Pinamonti, Speight)

Given η > 0, the following holds for sufficiently small ∆. Suppose: x, u ∈ Gr with d(u) ≤ 1,

• A. Pinamonti and G. Speight

UDS in Carnot Groups Warwick GMT 2017 28 / 35

Horizontal Curves for Almost Maximality

Lemma (Le Donne, Pinamonti, Speight)

Given η > 0, the following holds for sufficiently small ∆. Suppose: x, u ∈ Gr with d(u) ≤ 1, E ∈ V1 with ω(E) = 1,

• A. Pinamonti and G. Speight

UDS in Carnot Groups Warwick GMT 2017 28 / 35

Horizontal Curves for Almost Maximality

Lemma (Le Donne, Pinamonti, Speight)

Given η > 0, the following holds for sufficiently small ∆. Suppose: x, u ∈ Gr with d(u) ≤ 1, E ∈ V1 with ω(E) = 1, 0 < r < ∆ and s := r/∆.

• A. Pinamonti and G. Speight

UDS in Carnot Groups Warwick GMT 2017 28 / 35

Horizontal Curves for Almost Maximality

Lemma (Le Donne, Pinamonti, Speight)

Given η > 0, the following holds for sufficiently small ∆. Suppose: x, u ∈ Gr with d(u) ≤ 1, E ∈ V1 with ω(E) = 1, 0 < r < ∆ and s := r/∆. Then there is a Lipschitz horizontal curve g : R → Gr, which is a concatenation of horizontal lines, such that:

• A. Pinamonti and G. Speight

UDS in Carnot Groups Warwick GMT 2017 28 / 35

Horizontal Curves for Almost Maximality

Lemma (Le Donne, Pinamonti, Speight)

Given η > 0, the following holds for sufficiently small ∆. Suppose: x, u ∈ Gr with d(u) ≤ 1, E ∈ V1 with ω(E) = 1, 0 < r < ∆ and s := r/∆. Then there is a Lipschitz horizontal curve g : R → Gr, which is a concatenation of horizontal lines, such that:

1 g(t) = x + tE(x) for |t| ≥ s,

• A. Pinamonti and G. Speight

UDS in Carnot Groups Warwick GMT 2017 28 / 35

Horizontal Curves for Almost Maximality

Lemma (Le Donne, Pinamonti, Speight)

Given η > 0, the following holds for sufficiently small ∆. Suppose: x, u ∈ Gr with d(u) ≤ 1, E ∈ V1 with ω(E) = 1, 0 < r < ∆ and s := r/∆. Then there is a Lipschitz horizontal curve g : R → Gr, which is a concatenation of horizontal lines, such that:

1 g(t) = x + tE(x) for |t| ≥ s, 2 g(ζ) = xδr(u), where ζ := δr(u), E(0),

• A. Pinamonti and G. Speight

UDS in Carnot Groups Warwick GMT 2017 28 / 35

Horizontal Curves for Almost Maximality

Lemma (Le Donne, Pinamonti, Speight)

Given η > 0, the following holds for sufficiently small ∆. Suppose: x, u ∈ Gr with d(u) ≤ 1, E ∈ V1 with ω(E) = 1, 0 < r < ∆ and s := r/∆. Then there is a Lipschitz horizontal curve g : R → Gr, which is a concatenation of horizontal lines, such that:

1 g(t) = x + tE(x) for |t| ≥ s, 2 g(ζ) = xδr(u), where ζ := δr(u), E(0), 3 Lip(g) ≤ 1 + η∆,

• A. Pinamonti and G. Speight

UDS in Carnot Groups Warwick GMT 2017 28 / 35

Horizontal Curves for Almost Maximality

Lemma (Le Donne, Pinamonti, Speight)

Given η > 0, the following holds for sufficiently small ∆. Suppose: x, u ∈ Gr with d(u) ≤ 1, E ∈ V1 with ω(E) = 1, 0 < r < ∆ and s := r/∆. Then there is a Lipschitz horizontal curve g : R → Gr, which is a concatenation of horizontal lines, such that:

1 g(t) = x + tE(x) for |t| ≥ s, 2 g(ζ) = xδr(u), where ζ := δr(u), E(0), 3 Lip(g) ≤ 1 + η∆, 4 |(p ◦ g)′(t) − p(E)| ≤ C∆ for t ∈ R outside a finite set.

• A. Pinamonti and G. Speight

UDS in Carnot Groups Warwick GMT 2017 28 / 35

Horizontal Curves for Almost Maximality

E x xδr(u) x + sE(x) x − sE(x) g(ζ) = xδr(u), where ζ := δr(u), E(0) Lip(g) ≤ 1 + η∆ |(p ◦ g)′(t) − p(E)| ≤ C∆

• A. Pinamonti and G. Speight

UDS in Carnot Groups Warwick GMT 2017 29 / 35

Almost Maximality implies Differentiability

Let Df := {(x, E) ∈ N × V1 : ω(E) = 1, Ef (x) exists}.

• A. Pinamonti and G. Speight

UDS in Carnot Groups Warwick GMT 2017 30 / 35

Almost Maximality implies Differentiability

Let Df := {(x, E) ∈ N × V1 : ω(E) = 1, Ef (x) exists}.

Theorem (Le Donne, Pinamonti, Speight)

Let f : Gr → R be Lipschitz and (x0, E0) ∈ Df .

• A. Pinamonti and G. Speight

UDS in Carnot Groups Warwick GMT 2017 30 / 35

Almost Maximality implies Differentiability

Let Df := {(x, E) ∈ N × V1 : ω(E) = 1, Ef (x) exists}.

Theorem (Le Donne, Pinamonti, Speight)

Let f : Gr → R be Lipschitz and (x0, E0) ∈ Df . Let M denote the set of pairs (x, E) ∈ Df such that Ef (x) ≥ E0f (x0) and |(f (x + tE0(x)) − f (x)) − (f (x0 + tE0(x0)) − f (x0))| ≤ 6|t|((Ef (x) − E0f (x0))Lip(f ))

1 4

for every t ∈ (−1, 1).

• A. Pinamonti and G. Speight

UDS in Carnot Groups Warwick GMT 2017 30 / 35

Almost Maximality implies Differentiability

Let Df := {(x, E) ∈ N × V1 : ω(E) = 1, Ef (x) exists}.

Theorem (Le Donne, Pinamonti, Speight)

Let f : Gr → R be Lipschitz and (x0, E0) ∈ Df . Let M denote the set of pairs (x, E) ∈ Df such that Ef (x) ≥ E0f (x0) and |(f (x + tE0(x)) − f (x)) − (f (x0 + tE0(x0)) − f (x0))| ≤ 6|t|((Ef (x) − E0f (x0))Lip(f ))

1 4

for every t ∈ (−1, 1). If lim

δ↓0 sup{Ef (x): (x, E) ∈ M and d(x, x0) ≤ δ} ≤ E0f (x0),

then f is differentiable at x0 with derivative x → E0f (x0)x, E0(0).

• A. Pinamonti and G. Speight

UDS in Carnot Groups Warwick GMT 2017 30 / 35

Free Step 2 to General Step 2

Let G and H be Carnot groups with horizontal layers V and W .

• A. Pinamonti and G. Speight

UDS in Carnot Groups Warwick GMT 2017 31 / 35

Free Step 2 to General Step 2

Let G and H be Carnot groups with horizontal layers V and W . Suppose V and W admit bases X1, . . . , Xr and Y1, . . . , Yr, together with a Lie group homomorphism F : G → H such that F∗(Xi) = Yi for 1 ≤ i ≤ r.

• A. Pinamonti and G. Speight

UDS in Carnot Groups Warwick GMT 2017 31 / 35

Free Step 2 to General Step 2

Let G and H be Carnot groups with horizontal layers V and W . Suppose V and W admit bases X1, . . . , Xr and Y1, . . . , Yr, together with a Lie group homomorphism F : G → H such that F∗(Xi) = Yi for 1 ≤ i ≤ r. Equip G and H with CC metrics dG and dH induced by the given bases. These make F Lipschitz with Lip(F) = 1.

• A. Pinamonti and G. Speight

UDS in Carnot Groups Warwick GMT 2017 31 / 35

Free Step 2 to General Step 2

Let G and H be Carnot groups with horizontal layers V and W . Suppose V and W admit bases X1, . . . , Xr and Y1, . . . , Yr, together with a Lie group homomorphism F : G → H such that F∗(Xi) = Yi for 1 ≤ i ≤ r. Equip G and H with CC metrics dG and dH induced by the given bases. These make F Lipschitz with Lip(F) = 1.

Lemma

Let x ∈ G and ˜ x = F(x) ∈ H. Then for any ˜ y ∈ H, there is y ∈ G with F(y) = ˜ y and dG(x, y) = dH(˜ x, ˜ y).

• A. Pinamonti and G. Speight

UDS in Carnot Groups Warwick GMT 2017 31 / 35

Free Step 2 to General Step 2

Proposition (Le Donne, Pinamonti, Speight)

If the CC distance in G is differentiable in horizontal directions then the CC distance in H is differentiable in horizontal directions.

• A. Pinamonti and G. Speight

UDS in Carnot Groups Warwick GMT 2017 32 / 35

Free Step 2 to General Step 2

Proposition (Le Donne, Pinamonti, Speight)

If the CC distance in G is differentiable in horizontal directions then the CC distance in H is differentiable in horizontal directions.

Proof.

Let ˜ u = (uh, 0) ∈ H for some uh ∈ Rr. Let u = (uh, 0) ∈ G, which satisfies F(u) = ˜ u and dG(u) = dH(˜ u). Given ˜ z ∈ H, choose z ∈ G such that F(z) = ˜ z and dG(z) = dH(˜ z). Then: dH(˜ u˜ z) ≤ dG(uz) ≤ dG(u) + uh/dG(u), pG(z) + o(dG(z)) = dH(˜ u) + uh/dH(˜ u), pH(˜ z) + o(dH(˜ z)).

• A. Pinamonti and G. Speight

UDS in Carnot Groups Warwick GMT 2017 32 / 35

Free Step 2 to General Step 2

Proposition (Le Donne, Pinamonti, Speight)

If the CC distance in G is differentiable in horizontal directions then the CC distance in H is differentiable in horizontal directions.

Proof.

Let ˜ u = (uh, 0) ∈ H for some uh ∈ Rr. Let u = (uh, 0) ∈ G, which satisfies F(u) = ˜ u and dG(u) = dH(˜ u). Given ˜ z ∈ H, choose z ∈ G such that F(z) = ˜ z and dG(z) = dH(˜ z). Then: dH(˜ u˜ z) ≤ dG(uz) ≤ dG(u) + uh/dG(u), pG(z) + o(dG(z)) = dH(˜ u) + uh/dH(˜ u), pH(˜ z) + o(dH(˜ z)).

Proposition (Le Donne, Pinamonti, Speight)

If N is a (Hausdorff dimension one) universal differentiability set N in G, then F(N) is a (Hausdorff dimension one) universal differentiability set in H.

• A. Pinamonti and G. Speight

UDS in Carnot Groups Warwick GMT 2017 32 / 35

Questions

Does existence of a maximal directional derivative imply differentiability in any Carnot groups of step greater than two?

• A. Pinamonti and G. Speight

UDS in Carnot Groups Warwick GMT 2017 33 / 35

Questions

Does existence of a maximal directional derivative imply differentiability in any Carnot groups of step greater than two? Do measure zero universal differentiability sets exist in the Engel group or other Carnot groups of step greater than two?

• A. Pinamonti and G. Speight

UDS in Carnot Groups Warwick GMT 2017 33 / 35

Questions

Does existence of a maximal directional derivative imply differentiability in any Carnot groups of step greater than two? Do measure zero universal differentiability sets exist in the Engel group or other Carnot groups of step greater than two? In groups where measure zero universal differentiability sets exist, can

• ne always make them compact and of small Hausdorff or Minkowski

dimension?

• A. Pinamonti and G. Speight

UDS in Carnot Groups Warwick GMT 2017 33 / 35

Conclusion

A converse to Rademacher’s theorem for Lipschitz functions Rn → Rm holds if and only if n ≤ m.

• A. Pinamonti and G. Speight

UDS in Carnot Groups Warwick GMT 2017 34 / 35

Conclusion

A converse to Rademacher’s theorem for Lipschitz functions Rn → Rm holds if and only if n ≤ m. Existence of a maximal directional derivative implies differentiability for real-valued Lipschitz maps on Euclidean spaces.

• A. Pinamonti and G. Speight

UDS in Carnot Groups Warwick GMT 2017 34 / 35

Conclusion

A converse to Rademacher’s theorem for Lipschitz functions Rn → Rm holds if and only if n ≤ m. Existence of a maximal directional derivative implies differentiability for real-valued Lipschitz maps on Euclidean spaces. In Carnot groups, the implication ‘maximality implies differentiability’ is equivalent to differentiability of the Carnot-Caratheodory distance at horizontal points.

• A. Pinamonti and G. Speight

UDS in Carnot Groups Warwick GMT 2017 34 / 35

Conclusion

A converse to Rademacher’s theorem for Lipschitz functions Rn → Rm holds if and only if n ≤ m. Existence of a maximal directional derivative implies differentiability for real-valued Lipschitz maps on Euclidean spaces. In Carnot groups, the implication ‘maximality implies differentiability’ is equivalent to differentiability of the Carnot-Caratheodory distance at horizontal points. In all step 2 Carnot groups, maximality implies differentiability and measure zero universal differentiability sets exist.

• A. Pinamonti and G. Speight

UDS in Carnot Groups Warwick GMT 2017 34 / 35

Conclusion

A converse to Rademacher’s theorem for Lipschitz functions Rn → Rm holds if and only if n ≤ m. Existence of a maximal directional derivative implies differentiability for real-valued Lipschitz maps on Euclidean spaces. In Carnot groups, the implication ‘maximality implies differentiability’ is equivalent to differentiability of the Carnot-Caratheodory distance at horizontal points. In all step 2 Carnot groups, maximality implies differentiability and measure zero universal differentiability sets exist. In the step 3 Engel group, maximality need not imply differentiability.

• A. Pinamonti and G. Speight

UDS in Carnot Groups Warwick GMT 2017 34 / 35

Conclusion

A converse to Rademacher’s theorem for Lipschitz functions Rn → Rm holds if and only if n ≤ m. Existence of a maximal directional derivative implies differentiability for real-valued Lipschitz maps on Euclidean spaces. In Carnot groups, the implication ‘maximality implies differentiability’ is equivalent to differentiability of the Carnot-Caratheodory distance at horizontal points. In all step 2 Carnot groups, maximality implies differentiability and measure zero universal differentiability sets exist. In the step 3 Engel group, maximality need not imply differentiability.

• A. Pinamonti and G. Speight

UDS in Carnot Groups Warwick GMT 2017 34 / 35

• A. Pinamonti and G. Speight

UDS in Carnot Groups Warwick GMT 2017 35 / 35