Besicovitch covering property on graded groups and applications to - - PowerPoint PPT Presentation

Besicovitch covering property on graded groups and applications to measure differentiation

Séverine Rigot

(Université Nice Sophia Antipolis)

based on a joint work with E. Le Donne (University of Jyväskylä)

Geometric Measure Theory, University of Warwick, July 10-14, 2017

Besicovitch covering property

A family B of closed balls in a metric space (X, d) is said to be a Besicovitch family of balls if for every B ∈ B, xB ∈

B′∈B\{B} B′

• B∈B B = ∅.

Definition (Besicovitch covering property) We say that the Besicovitch covering property (BCP) holds on (X, d) if there is C ≥ 1 such that Card B ≤ C for every Besicovitch family of balls B.

Besicovitch covering property

A family B of closed balls in a metric space (X, d) is said to be a Besicovitch family of balls if for every B ∈ B, xB ∈

B′∈B\{B} B′

• B∈B B = ∅.

Definition (Besicovitch covering property) We say that the Besicovitch covering property (BCP) holds on (X, d) if there is C ≥ 1 such that Card B ≤ C for every Besicovitch family of balls B. An application to differentiation of measures: Theorem (based on Besicovitch’s ideas) Let (X, d) be a separable metric space. Assume that BCP holds on (X, d). Then the differentiation theorem holds for every locally finite Borel regular measure over (X, d). The differentiation theorem holds for a measure µ over (X, d) if ∀f ∈ L1

loc(µ), lim r↓0+

1 µ(B(x, r))

• B(x,r)

f dµ = f(x) µ-a.e.

Besicovitch covering property

Examples: Finite dimensional normed vector spaces satisfy BCP . Infinite dimensional normed vector spaces do not satisfy BCP .

Besicovitch covering property

The validity or non validity of BCP is sensitive to perturbations of the metric.

Besicovitch covering property

The validity or non validity of BCP is sensitive to perturbations of the metric. Assume that there exists an accumulation point in (X, d). For every 0 < c < 1, there is a distance dc on X such that cd ≤ dc ≤ d for which BCP does not hold ([Le Donne, R] following a slightly different statement given by D. Preiss).

Besicovitch covering property

The validity or non validity of BCP is sensitive to perturbations of the metric. Assume that there exists an accumulation point in (X, d). For every 0 < c < 1, there is a distance dc on X such that cd ≤ dc ≤ d for which BCP does not hold ([Le Donne, R] following a slightly different statement given by D. Preiss). There are examples of metric spaces (X, d) on which BCP does not hold but are such that for every ε > 0, there is a distance dε such that d ≤ dε ≤ (1 + ε)d and for which BCP holds.

Besicovitch covering property

The validity or non validity of BCP is sensitive to perturbations of the metric. Assume that there exists an accumulation point in (X, d). For every 0 < c < 1, there is a distance dc on X such that cd ≤ dc ≤ d for which BCP does not hold ([Le Donne, R] following a slightly different statement given by D. Preiss). There are examples of metric spaces (X, d) on which BCP does not hold but are such that for every ε > 0, there is a distance dε such that d ≤ dε ≤ (1 + ε)d and for which BCP holds. In particular the validity or non validity of BCP is not preserved by a bi-Lipschitz change of distance.

Besicovitch covering property

The validity or non validity of BCP is sensitive to perturbations of the metric. Assume that there exists an accumulation point in (X, d). For every 0 < c < 1, there is a distance dc on X such that cd ≤ dc ≤ d for which BCP does not hold ([Le Donne, R] following a slightly different statement given by D. Preiss). There are examples of metric spaces (X, d) on which BCP does not hold but are such that for every ε > 0, there is a distance dε such that d ≤ dε ≤ (1 + ε)d and for which BCP holds. In particular the validity or non validity of BCP is not preserved by a bi-Lipschitz change of distance. On the other hand, if BCP holds on (X, d) then BCP holds on the snowflakes (X, dα) for every 0 < α < 1.

Graded Lie algebras and Lie groups

A Lie algebra g is positively graduable if it can be written as g = ⊕t>0Vt where [Vt, Vs] ⊂ Vt+s for all t, s > 0. A Lie algebra is said to be graded if it is positively graduable and endowed with a given positive grading. Associated dilations (δλ)λ>0 on a graded Lie algebra g = ⊕t>0Vt are defined as the unique linear maps δλ : g → g such that δλ(X) = λtX for X ∈ Vt. The family (δλ)λ>0 is a one-parameter group of Lie algebra automorphisms.

Graded Lie algebras and Lie groups

A Lie algebra g is positively graduable if it can be written as g = ⊕t>0Vt where [Vt, Vs] ⊂ Vt+s for all t, s > 0. A Lie algebra is said to be graded if it is positively graduable and endowed with a given positive grading. Associated dilations (δλ)λ>0 on a graded Lie algebra g = ⊕t>0Vt are defined as the unique linear maps δλ : g → g such that δλ(X) = λtX for X ∈ Vt. The family (δλ)λ>0 is a one-parameter group of Lie algebra automorphisms. A graded group G is a connected and simply connected Lie group whose Lie algebra g is graded. Associated dilations on a graded group G are defined as the unique Lie group automorphisms δλ : G → G such that δλ ◦ exp = exp ◦δλ. We may identify a graded group with its Lie algebra, the group law being given by the Baker-Campbell-Hausdorff formula, x · y = x + y + [x, y]/2 + · · · , and the exponential map from g to G being the identity.

Homogeneous distances and homogeneous groups

A distance d on a graded group G is called homogeneous if it is left-invariant and one-homogeneous w.r.t. the family of associated dilations, that is, d(x · y, x · y ′) = d(y, y ′) d(δλ(x), δλ(y)) = λ d(x, y) for every x, y, y ′ ∈ G and every λ > 0.

Homogeneous distances and homogeneous groups

A distance d on a graded group G is called homogeneous if it is left-invariant and one-homogeneous w.r.t. the family of associated dilations, that is, d(x · y, x · y ′) = d(y, y ′) d(δλ(x), δλ(y)) = λ d(x, y) for every x, y, y ′ ∈ G and every λ > 0. Homogeneous distances do exist on a graded group if and only if Vt = {0} for all 0 < t < 1. We call such groups homogeneous. On general graded groups, one can look at homogeneous quasi-distances. BCP makes sense for quasi-distances. For simplicity, we restrict ourselves in this talk to the study of the validity of BCP for homogeneous distances on homogeneous groups.

Homogeneous distances and homogeneous groups

A distance d on a graded group G is called homogeneous if it is left-invariant and one-homogeneous w.r.t. the family of associated dilations, that is, d(x · y, x · y ′) = d(y, y ′) d(δλ(x), δλ(y)) = λ d(x, y) for every x, y, y ′ ∈ G and every λ > 0. Homogeneous distances do exist on a graded group if and only if Vt = {0} for all 0 < t < 1. We call such groups homogeneous. On general graded groups, one can look at homogeneous quasi-distances. BCP makes sense for quasi-distances. For simplicity, we restrict ourselves in this talk to the study of the validity of BCP for homogeneous distances on homogeneous groups. A homogeneous distance on a homogeneous group induces the manifold topology on G.

Homogeneous distances and homogeneous groups

A distance d on a graded group G is called homogeneous if it is left-invariant and one-homogeneous w.r.t. the family of associated dilations, that is, d(x · y, x · y ′) = d(y, y ′) d(δλ(x), δλ(y)) = λ d(x, y) for every x, y, y ′ ∈ G and every λ > 0. Homogeneous distances do exist on a graded group if and only if Vt = {0} for all 0 < t < 1. We call such groups homogeneous. On general graded groups, one can look at homogeneous quasi-distances. BCP makes sense for quasi-distances. For simplicity, we restrict ourselves in this talk to the study of the validity of BCP for homogeneous distances on homogeneous groups. A homogeneous distance on a homogeneous group induces the manifold topology on G. Any two homogeneous distances on a homogeneous group are bi-Lipschitz equivalent.

Examples

Finite dimensional normed vector spaces correspond to Abelian homogeneous groups with Lie algebra endowed with the positive grading g = V1 and equipped with a homogeneous distance.

Examples

Finite dimensional normed vector spaces correspond to Abelian homogeneous groups with Lie algebra endowed with the positive grading g = V1 and equipped with a homogeneous distance. Snowflakes of finite dimensional normed vector spaces can be recovered as Abelian homogeneous groups with Lie algebra endowed with the positive grading g = Vα for some α > 1. Associated dilations are given by δλ(X) = λαX. Distances dα(x, y) = x − y1/α are left-invariant and one-homogeneous w.r.t. the associated dilations.

t-power of a positive grading

Let t > 0. The t-power of a positive grading g = ⊕s>0Vs is the positive grading g = ⊕s>0Ws where Wts = Vs. If (x1, . . . , xn) → (λs1x1, . . . , λsnxn) are the dilations associated to the initial grading, then dilations associated to its t-power are given by (x1, . . . , xn) → (λts1x1, . . . , λtsnxn).

t-power of a positive grading

Let t > 0. The t-power of a positive grading g = ⊕s>0Vs is the positive grading g = ⊕s>0Ws where Wts = Vs. If (x1, . . . , xn) → (λs1x1, . . . , λsnxn) are the dilations associated to the initial grading, then dilations associated to its t-power are given by (x1, . . . , xn) → (λts1x1, . . . , λtsnxn). If G is a graded group, the t-power of G is the group G considered as a graded group whose Lie algebra is endowed with the t-power of the initial grading. If d is a homogeneous distance on a homogeneous group and if t > 1, then d1/t is a homogeneous distance on its t-power.

t-power of a positive grading

Let t > 0. The t-power of a positive grading g = ⊕s>0Vs is the positive grading g = ⊕s>0Ws where Wts = Vs. If (x1, . . . , xn) → (λs1x1, . . . , λsnxn) are the dilations associated to the initial grading, then dilations associated to its t-power are given by (x1, . . . , xn) → (λts1x1, . . . , λtsnxn). If G is a graded group, the t-power of G is the group G considered as a graded group whose Lie algebra is endowed with the t-power of the initial grading. If d is a homogeneous distance on a homogeneous group and if t > 1, then d1/t is a homogeneous distance on its t-power. If d is a homogeneous distance on a homogeneous group that satisfies BCP , then, for every t > 1, d1/t is a homogeneous distance on its t-power that satisfies BCP .

Examples

The first Heisenberg Lie algebra is h =< X, Y, Z > where [X, Y] = Z is the

• nly non trivial bracket relation.

The first Heisenberg group can be identified with R3 equipped with the group law (x, y, z) · (x′, y ′, z′) = (x + x′, y + y ′, z + z′ + 1 2(xy ′ − x′y)).

Examples

The first Heisenberg Lie algebra is h =< X, Y, Z > where [X, Y] = Z is the

• nly non trivial bracket relation.

The first Heisenberg group can be identified with R3 equipped with the group law (x, y, z) · (x′, y ′, z′) = (x + x′, y + y ′, z + z′ + 1 2(xy ′ − x′y)). Stratified first Heisenberg group: h is stratifiable of step 2, namely, h = V1 ⊕ V2 where V1 =< X, Y >, V2 =< Z >, and hence [V1, V1] = V2. Associated dilations are given by (x, y, z) → (λx, λy, λ2z). The stratified first Heisenberg group is the graded group whose Lie algebra is the first Heisenberg Lie algebra endowed with the above stratification.

Examples

The Korányi distance dK(p, q) = p−1 · qK where (x, y, z)K = ((x2 + y 2)2 + 16z2)1/4 is a homogeneous distance on the stratified first Heisenberg group.

Examples

The Korányi distance dK(p, q) = p−1 · qK where (x, y, z)K = ((x2 + y 2)2 + 16z2)1/4 is a homogeneous distance on the stratified first Heisenberg group. It does not satisfy BCP ([Sawyer, Wheeden], [Korányi, Reimann]).

Examples

The Korányi distance dK(p, q) = p−1 · qK where (x, y, z)K = ((x2 + y 2)2 + 16z2)1/4 is a homogeneous distance on the stratified first Heisenberg group. It does not satisfy BCP ([Sawyer, Wheeden], [Korányi, Reimann]). For every ε > 0, dε(p, q) = p−1 · qε where (x, y, z)ε = (ε(x2 + y 2) + (x, y, z)2

K)1/2

defines a homogeneous distance on the stratified first Heisenberg group that satisfies BCP ([Le Donne, R] for ε = 1, [Nicolussi Golo, R] for every ε > 0 (on going work)).

Examples

Non-standard Heisenberg groups: The first Heisenberg Lie algebra admits positive gradings that are not

• stratifications. For α > 1,

h = W1 ⊕ Wα ⊕ W1+α where W1 =< X >, Wα =< Y >, and W1+α =< Z >, is called the non-standard grading of exponent α. Associated dilations are given by (x, y, z) → (λx, λαy, λ1+αz). The non-standard Heisenberg group of exponent α is the graded group whose Lie algebra is the first Heisenberg Lie algebra endowed with the non-standard grading of exponent α.

Examples

Non-standard Heisenberg groups: The first Heisenberg Lie algebra admits positive gradings that are not

• stratifications. For α > 1,

h = W1 ⊕ Wα ⊕ W1+α where W1 =< X >, Wα =< Y >, and W1+α =< Z >, is called the non-standard grading of exponent α. Associated dilations are given by (x, y, z) → (λx, λαy, λ1+αz). The non-standard Heisenberg group of exponent α is the graded group whose Lie algebra is the first Heisenberg Lie algebra endowed with the non-standard grading of exponent α. There do not exist homogeneous distances on the non-standard Heisenberg group of exponent α > 1 satisfying BCP .

Main theorem

We say that a graded group with positive grading of its Lie algebra given by g = ⊕t>0Vt has commuting different layers if [Vt, Vs] = {0} for every t = s.

Main theorem

We say that a graded group with positive grading of its Lie algebra given by g = ⊕t>0Vt has commuting different layers if [Vt, Vs] = {0} for every t = s. Theorem (Le Donne, R) Let G be a homogeneous group. There exist homogeneous distances on G for which BCP holds iff G has commuting different layers.

Main theorem

We say that a graded group with positive grading of its Lie algebra given by g = ⊕t>0Vt has commuting different layers if [Vt, Vs] = {0} for every t = s. Theorem (Le Donne, R) Let G be a homogeneous group. There exist homogeneous distances on G for which BCP holds iff G has commuting different layers. We say that a graded group is stratified of step s for some integer s ≥ 1 if the associated positive grading of its Lie algebra is of the form g = V1 ⊕ V2 ⊕ · · · ⊕ Vs where [V1, Vj] = Vj+1 for 1 ≤ j ≤ s with Vs+1 = {0}. A stratified group has commuting different layers iff it is of step ≤ 2. Corollary Let G be a stratified group. There exist homogeneous distances on G for which BCP holds iff G is of step 1 or 2.

Homogeneous distances with BCP on homogeneous groups with commuting different layers

Algebraic structure of homogeneous groups with commuting different layers: Proposition Let G be a homogeneous group with commuting different layers. Then G can be written as the direct product of ti-powers of stratified groups Gi of step ≤ 2 for some ti’s ≥ 1.

Homogeneous distances with BCP on homogeneous groups with commuting different layers

Algebraic structure of homogeneous groups with commuting different layers: Proposition Let G be a homogeneous group with commuting different layers. Then G can be written as the direct product of ti-powers of stratified groups Gi of step ≤ 2 for some ti’s ≥ 1. BCP for product of metric spaces: Let 1 ≤ p < ∞ and s > p. Then BCP does not hold on R2 equipped with the distance d((x, y), (x′, y ′)) = (|x′ − x|p + |y ′ − y|p/s)1/p which is the lp-mean

• f distances satisfying BCP on R.

Homogeneous distances with BCP on homogeneous groups with commuting different layers

Algebraic structure of homogeneous groups with commuting different layers: Proposition Let G be a homogeneous group with commuting different layers. Then G can be written as the direct product of ti-powers of stratified groups Gi of step ≤ 2 for some ti’s ≥ 1. BCP for product of metric spaces: Let 1 ≤ p < ∞ and s > p. Then BCP does not hold on R2 equipped with the distance d((x, y), (x′, y ′)) = (|x′ − x|p + |y ′ − y|p/s)1/p which is the lp-mean

• f distances satisfying BCP on R. However,

Proposition Let (X, dX) and (Y, dY) be metric spaces satisfying BCP . Then BCP holds on X × Y equipped with the distance dX×Y((x, y), (x′, y ′)) = max(dX(x, x′), dY(y, y ′)).

Stratified free nilptotent Lie groups of step 2

Stratified free nilptotent Lie groups of step 2: Theorem Homogeneous distances on stratified free nilpotent Lie groups of step 2 whose unit ball centered at the origin coincides with a Euclidean ball centered at the origin satisfy BCP .

Stratified free nilptotent Lie groups of step 2

Stratified free nilpotent Lie algebra fr2 of step 2 and rank r ≥ 2: fr2 = V1 ⊕ V2 with dim V1 = r and dim V2 = r(r − 1)/2.

Stratified free nilptotent Lie groups of step 2

Stratified free nilpotent Lie algebra fr2 of step 2 and rank r ≥ 2: fr2 = V1 ⊕ V2 with dim V1 = r and dim V2 = r(r − 1)/2. Fix a basis (X1, . . . , Xr) of V1. Then ([Xi, Xj])i<j is a basis of V2. Denote by · the Euclidean norm on fr2 making ((X1, . . . , Xr), ([Xi, Xj])i<j)

• rthonormal and by Fr2 the stratified free nilpotent Lie group of step 2

and rank r.

Stratified free nilptotent Lie groups of step 2

Stratified free nilpotent Lie algebra fr2 of step 2 and rank r ≥ 2: fr2 = V1 ⊕ V2 with dim V1 = r and dim V2 = r(r − 1)/2. Fix a basis (X1, . . . , Xr) of V1. Then ([Xi, Xj])i<j is a basis of V2. Denote by · the Euclidean norm on fr2 making ((X1, . . . , Xr), ([Xi, Xj])i<j)

• rthonormal and by Fr2 the stratified free nilpotent Lie group of step 2

and rank r. For R > 0, set AR = {p ∈ Fr2 : p ≤ R}.

Stratified free nilptotent Lie groups of step 2

Stratified free nilpotent Lie algebra fr2 of step 2 and rank r ≥ 2: fr2 = V1 ⊕ V2 with dim V1 = r and dim V2 = r(r − 1)/2. Fix a basis (X1, . . . , Xr) of V1. Then ([Xi, Xj])i<j is a basis of V2. Denote by · the Euclidean norm on fr2 making ((X1, . . . , Xr), ([Xi, Xj])i<j)

• rthonormal and by Fr2 the stratified free nilpotent Lie group of step 2

and rank r. For R > 0, set AR = {p ∈ Fr2 : p ≤ R}. ([Hebisch, Sikora]) There is R∗ > 0 such that, for every 0 < R ≤ R∗, dR(p, q) = inf{λ > 0 : δ1/λ(p−1 · q) ∈ AR} defines a homogeneous distance on Fr2. Its unit ball centered at the

• rigin is given by the set AR.

Stratified free nilptotent Lie groups of step 2

Stratified free nilpotent Lie algebra fr2 of step 2 and rank r ≥ 2: fr2 = V1 ⊕ V2 with dim V1 = r and dim V2 = r(r − 1)/2. Fix a basis (X1, . . . , Xr) of V1. Then ([Xi, Xj])i<j is a basis of V2. Denote by · the Euclidean norm on fr2 making ((X1, . . . , Xr), ([Xi, Xj])i<j)

• rthonormal and by Fr2 the stratified free nilpotent Lie group of step 2

and rank r. For R > 0, set AR = {p ∈ Fr2 : p ≤ R}. ([Hebisch, Sikora]) There is R∗ > 0 such that, for every 0 < R ≤ R∗, dR(p, q) = inf{λ > 0 : δ1/λ(p−1 · q) ∈ AR} defines a homogeneous distance on Fr2. Its unit ball centered at the

• rigin is given by the set AR.

If d is a homogeneous distance on Fr2 whose unit ball centered at the origin coincides with the set AR for some R > 0, then d satisfies BCP .

The example of the stratified first Heisenberg group

On the stratified first Heisenberg group F22, the homogeneous distance d(p, q) = inf{λ > 0 : δ1/λ(p−1 · q) ≤ 2} satisfies BCP .

The example of the stratified first Heisenberg group

On the stratified first Heisenberg group F22, the homogeneous distance d(p, q) = inf{λ > 0 : δ1/λ(p−1 · q) ≤ 2} satisfies BCP . The distance d and the Korányi distance dK are related via the formula 2 √ 2 d(0, p) = ((x2 + y 2) + dK(0, p)2)1/2.

The example of the stratified first Heisenberg group

On the stratified first Heisenberg group F22, the homogeneous distance d(p, q) = inf{λ > 0 : δ1/λ(p−1 · q) ≤ 2} satisfies BCP . The distance d and the Korányi distance dK are related via the formula 2 √ 2 d(0, p) = ((x2 + y 2) + dK(0, p)2)1/2. For every ε > 0, dε(0, p) = (ε(x2 + y 2) + dK(0, p)2)1/2 defines a homogeneous distance on the stratified first Heisenberg group that satisfies BCP .

Homogeneous distances with BCP on homogeneous groups with commuting different layers

From stratified free nilptotent Lie groups of step 2 to stratified groups of step 2: Proposition Let G be a stratified group of step 2 and rank r. There is a surjective Lie group homomorphism π : Fr2 → G such that if d is a homogeneous distance

• n Fr2, then dG(x, y) = d(π−1(x), π−1(y)) defines a homogeneous distance
• n G and π is a submetry from (Fr2, d) onto (G, dG).

Homogeneous distances with BCP on homogeneous groups with commuting different layers

From stratified free nilptotent Lie groups of step 2 to stratified groups of step 2: Proposition Let G be a stratified group of step 2 and rank r. There is a surjective Lie group homomorphism π : Fr2 → G such that if d is a homogeneous distance

• n Fr2, then dG(x, y) = d(π−1(x), π−1(y)) defines a homogeneous distance
• n G and π is a submetry from (Fr2, d) onto (G, dG).

A map π : (X, dX) → (Y, dY) is a submetry if it is surjective and such that π(BX(x, r)) = BY(π(x), r) for every x ∈ X and every r > 0.

Homogeneous distances with BCP on homogeneous groups with commuting different layers

From stratified free nilptotent Lie groups of step 2 to stratified groups of step 2: Proposition Let G be a stratified group of step 2 and rank r. There is a surjective Lie group homomorphism π : Fr2 → G such that if d is a homogeneous distance

• n Fr2, then dG(x, y) = d(π−1(x), π−1(y)) defines a homogeneous distance
• n G and π is a submetry from (Fr2, d) onto (G, dG).

A map π : (X, dX) → (Y, dY) is a submetry if it is surjective and such that π(BX(x, r)) = BY(π(x), r) for every x ∈ X and every r > 0. Proposition Let (X, dX) and (Y, dY) be metric spaces. Assume that BCP holds on (X, dX) and that there exists a submetry from (X, dX) onto (Y, dY). Then BCP holds

• n (Y, dY).

Non existence of homogeneous distances with BCP on homogeneous groups with two different layers not commuting

Non existence of homogeneous distances with BCP on homogeneous groups with two different layers not commuting

Non-standard Heisenberg groups: Theorem Let α > 1. There do not exist continuous homogeneous quasi-distances for which BCP holds on the non-standard Heisenberg group of exponent α.

Non existence of homogeneous distances with BCP on homogeneous groups with two different layers not commuting

From non-standard Heisenberg groups to homogeneous groups with two different layers not commuting: Proposition Let G be a homogeneous group with positive grading of its Lie algebra given by g = ⊕u≥1Vu. Assume that [Vt, Vs] = {0} for some t < s. There exist a homogeneous subgroup G and a surjective Lie group homomorphism ϕ : G → H where H is the t-power of the non-standard Heisenberg group of exponent s/t such that if d is a homogeneous distance

• n

G, then dH(p, q) = d(ϕ−1(p), ϕ−1(q)) defines a homogeneous distance

• n H and ϕ is a submetry from (

G, d) onto (H, dH).

Application to differentiation of measures

Let (X, d) be a metric space. We say that d is finite dimensional on a subset Y of X if there are C ≥ 1 and 0 < r ≤ ∞ such that Card B ≤ C for every Besicovitch family of balls centered on Y and with radii < r.

Application to differentiation of measures

Let (X, d) be a metric space. We say that d is finite dimensional on a subset Y of X if there are C ≥ 1 and 0 < r ≤ ∞ such that Card B ≤ C for every Besicovitch family of balls centered on Y and with radii < r. We say that d is σ-finite dimensional if X can be written as a countable union

• f subsets on which d is finite dimensional.

Application to differentiation of measures

Let (X, d) be a metric space. We say that d is finite dimensional on a subset Y of X if there are C ≥ 1 and 0 < r ≤ ∞ such that Card B ≤ C for every Besicovitch family of balls centered on Y and with radii < r. We say that d is σ-finite dimensional if X can be written as a countable union

• f subsets on which d is finite dimensional.

Theorem (Preiss) Let (X, d) be a complete separable metric space. The differentiation theorem holds for every locally finite Borel regular measure over (X, d) iff d is σ-finite dimensional.

Application to differentiation of measures

Let (X, d) be a metric space. We say that d is finite dimensional on a subset Y of X if there are C ≥ 1 and 0 < r ≤ ∞ such that Card B ≤ C for every Besicovitch family of balls centered on Y and with radii < r. We say that d is σ-finite dimensional if X can be written as a countable union

• f subsets on which d is finite dimensional.

Theorem (Preiss) Let (X, d) be a complete separable metric space. The differentiation theorem holds for every locally finite Borel regular measure over (X, d) iff d is σ-finite dimensional. Theorem (Le Donne, R) Let d be a homogeneous distance on a homogeneous group. Then d is σ-finite dimensional iff d satisfies BCP .

Application to differentiation of measures

Let (X, d) be a metric space. We say that d is finite dimensional on a subset Y of X if there are C ≥ 1 and 0 < r ≤ ∞ such that Card B ≤ C for every Besicovitch family of balls centered on Y and with radii < r. We say that d is σ-finite dimensional if X can be written as a countable union

• f subsets on which d is finite dimensional.

Theorem (Preiss) Let (X, d) be a complete separable metric space. The differentiation theorem holds for every locally finite Borel regular measure over (X, d) iff d is σ-finite dimensional. Theorem (Le Donne, R) Let d be a homogeneous distance on a homogeneous group. Then d is σ-finite dimensional iff d satisfies BCP . Corollary Let G be a homogeneous group. There exist homogeneous distances d on G such that the differentiation theorem holds for every locally finite Borel regular measure over (G, d) iff G has commuting different layers.

Sub-Riemannian geometry

Sub-Riemannian distances on stratified groups: Theorem ([R], [Le Donne, R]) Let G be a stratified group of step ≥ 2 and let d be a sub-Riemannian distance on G. Then BCP does not hold on (G, d).

Sub-Riemannian geometry

Sub-Riemannian distances on stratified groups: Theorem ([R], [Le Donne, R]) Let G be a stratified group of step ≥ 2 and let d be a sub-Riemannian distance on G. Then BCP does not hold on (G, d). Riemannian distances on Riemannian manifolds of class ≥ 2 are σ-finite dimensional.

Sub-Riemannian geometry

Sub-Riemannian distances on stratified groups: Theorem ([R], [Le Donne, R]) Let G be a stratified group of step ≥ 2 and let d be a sub-Riemannian distance on G. Then BCP does not hold on (G, d). Riemannian distances on Riemannian manifolds of class ≥ 2 are σ-finite

• dimensional. On the contrary,

Theorem (Le Donne, R) Let M be a sub-Riemannian manifold. Then its sub-Riemannian distance is not σ-finite dimensional.

Thanks for your attention!