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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

  1. Horizontal Curves Fix a basis X 1 , . . . , X m of V 1 . Definition An absolutely continuous curve γ : [ a , b ] → G is horizontal if there exists h : [ a , b ] → R m such that for almost every t : m γ ′ ( t ) = � h i X i ( γ ( t )) . i =1 � b The horizontal length of such a curve is L ( γ ) = 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

  2. Dilations and Haar Measure Dilations δ r : G → G are defined for every r > 0. For every x , y ∈ G , these satisfy: δ r ( xy ) = δ r ( x ) δ r ( y ) , d ( δ r ( x ) , δ r ( y )) = rd ( x , y ) . A. Pinamonti and G. Speight UDS in Carnot Groups Warwick GMT 2017 12 / 35

  3. Dilations and Haar Measure Dilations δ r : G → G are defined for every r > 0. For every x , y ∈ G , these satisfy: δ r ( xy ) = δ r ( x ) δ r ( y ) , d ( δ r ( x ) , δ r ( 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

  4. Heisenberg Group Example The Heisenberg group H n (a step 2 Carnot group) is R 2 n +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

  5. Heisenberg Group Example The Heisenberg group H n (a step 2 Carnot group) is R 2 n +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 H n are given by: X i ( x , y , t ) = ∂ x i + 2 y i ∂ t , Y i ( x , y , t ) = ∂ y i − 2 x i ∂ t , 1 ≤ i ≤ n . A. Pinamonti and G. Speight UDS in Carnot Groups Warwick GMT 2017 13 / 35

  6. Heisenberg Group Example The Heisenberg group H n (a step 2 Carnot group) is R 2 n +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 H n are given by: X i ( x , y , t ) = ∂ x i + 2 y i ∂ t , Y i ( x , y , t ) = ∂ y i − 2 x i ∂ t , 1 ≤ i ≤ n . Dilations are given by δ r ( x , y , t ) = ( rx , ry , r 2 t ). A. Pinamonti and G. Speight UDS in Carnot Groups Warwick GMT 2017 13 / 35

  7. Heisenberg Group Example The Heisenberg group H n (a step 2 Carnot group) is R 2 n +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 H n are given by: X i ( x , y , t ) = ∂ x i + 2 y i ∂ t , Y i ( x , y , t ) = ∂ y i − 2 x i ∂ t , 1 ≤ i ≤ n . Dilations are given by δ r ( x , y , t ) = ( rx , ry , r 2 t ). Haar measure on H n is L 2 n +1 . A. Pinamonti and G. Speight UDS in Carnot Groups Warwick GMT 2017 13 / 35

  8. Horizontal Curves in the Heisenberg Group t y x A. Pinamonti and G. Speight UDS in Carnot Groups Warwick GMT 2017 14 / 35

  9. Pansu Differentiability Definition A map L : G 1 → G 2 is group linear if for every x , y ∈ G 1 and r > 0: L ( xy ) = L ( x ) L ( y ) and L ( δ r ( x )) = δ r ( L ( x )) . A. Pinamonti and G. Speight UDS in Carnot Groups Warwick GMT 2017 15 / 35

  10. Pansu Differentiability Definition A map L : G 1 → G 2 is group linear if for every x , y ∈ G 1 and r > 0: L ( xy ) = L ( x ) L ( y ) and L ( δ r ( x )) = δ r ( L ( x )) . A map f : G 1 → G 2 is differentiable at x ∈ G 1 if there exists a group linear map L : G 1 → G 2 such that d ( f ( x ) − 1 f ( y ) , L ( x − 1 y )) lim = 0 . d ( x , y ) y → x A. Pinamonti and G. Speight UDS in Carnot Groups Warwick GMT 2017 15 / 35

  11. Pansu Differentiability Definition A map L : G 1 → G 2 is group linear if for every x , y ∈ G 1 and r > 0: L ( xy ) = L ( x ) L ( y ) and L ( δ r ( x )) = δ r ( L ( x )) . A map f : G 1 → G 2 is differentiable at x ∈ G 1 if there exists a group linear map L : G 1 → G 2 such that d ( f ( x ) − 1 f ( y ) , L ( x − 1 y )) lim = 0 . d ( x , y ) y → x 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

  12. Directional Derivatives Definition Suppose f : G → R is Lipschitz, x ∈ G and E ∈ V 1 . We define the directional derivative Ef ( x ) by: f ( x exp tE ) − f ( x ) Ef ( x ) = lim if the limit exists . t t → 0 A. Pinamonti and G. Speight UDS in Carnot Groups Warwick GMT 2017 16 / 35

  13. Directional Derivatives Definition Suppose f : G → R is Lipschitz, x ∈ G and E ∈ V 1 . We define the directional derivative Ef ( x ) by: f ( x exp tE ) − f ( x ) Ef ( x ) = lim if the limit exists . t t → 0 If f is differentiable at x , the derivative is df ( x )( v ) = ( p ( v ) , ∇ H f ( x )). Here p is the horizontal projection and ∇ H f ( x ) = ( X 1 f ( x ) , . . . , X m f ( x )). A. Pinamonti and G. Speight UDS in Carnot Groups Warwick GMT 2017 16 / 35

  14. Directional Derivatives Definition Suppose f : G → R is Lipschitz, x ∈ G and E ∈ V 1 . We define the directional derivative Ef ( x ) by: f ( x exp tE ) − f ( x ) Ef ( x ) = lim if the limit exists . t t → 0 If f is differentiable at x , the derivative is df ( x )( v ) = ( p ( v ) , ∇ H f ( x )). Here p is the horizontal projection and ∇ H f ( x ) = ( X 1 f ( x ) , . . . , X m f ( 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

  15. Directional Derivatives Definition Suppose f : G → R is Lipschitz, x ∈ G and E ∈ V 1 . We define the directional derivative Ef ( x ) by: f ( x exp tE ) − f ( x ) Ef ( x ) = lim if the limit exists . t t → 0 If f is differentiable at x , the derivative is df ( x )( v ) = ( p ( v ) , ∇ H f ( x )). Here p is the horizontal projection and ∇ H f ( x ) = ( X 1 f ( x ) , . . . , X m f ( 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

  16. Maximal Directional Derivatives in Carnot Groups Fix a Carnot group G with CC distance from a basis X 1 , . . . , X r of V 1 . A. Pinamonti and G. Speight UDS in Carnot Groups Warwick GMT 2017 17 / 35

  17. Maximal Directional Derivatives in Carnot Groups Fix a Carnot group G with CC distance from a basis X 1 , . . . , X r of V 1 . Let ω be an inner product norm on V 1 making X 1 , . . . , X r orthonormal. A. Pinamonti and G. Speight UDS in Carnot Groups Warwick GMT 2017 17 / 35

  18. Maximal Directional Derivatives in Carnot Groups Fix a Carnot group G with CC distance from a basis X 1 , . . . , X r of V 1 . Let ω be an inner product norm on V 1 making X 1 , . . . , X r orthonormal. Proposition Let f : G → R be a Lipschitz map. Then: Lip ( f ) = sup { Ef ( x ): x ∈ G , E ∈ V 1 , ω ( E ) = 1 , Ef ( x ) exists } . A. Pinamonti and G. Speight UDS in Carnot Groups Warwick GMT 2017 17 / 35

  19. Maximal Directional Derivatives in Carnot Groups Fix a Carnot group G with CC distance from a basis X 1 , . . . , X r of V 1 . Let ω be an inner product norm on V 1 making X 1 , . . . , X r orthonormal. Proposition Let f : G → R be a Lipschitz map. Then: Lip ( f ) = sup { Ef ( x ): x ∈ G , E ∈ V 1 , ω ( E ) = 1 , Ef ( x ) exists } . Definition Let x ∈ G and E ∈ V 1 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

  20. 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 ∈ V 1 \ { 0 } . A. Pinamonti and G. Speight UDS in Carnot Groups Warwick GMT 2017 18 / 35

  21. 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 ∈ V 1 \ { 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

  22. 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 ∈ V 1 \ { 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 ∈ V 1 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

  23. 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 ∈ V 1 \ { 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 ∈ V 1 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

  24. 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

  25. 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 ∈ V 1 \ { 0 } . A. Pinamonti and G. Speight UDS in Carnot Groups Warwick GMT 2017 19 / 35

  26. 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 ∈ V 1 \ { 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

  27. 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 ∈ V 1 \ { 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

  28. 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 ∈ V 1 \ { 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

  29. Free Carnot Groups of Step 2 Given integer r ≥ 2, let n = r + r ( r − 1) / 2. Denote coordinates in R n by x i , 1 ≤ i ≤ r , and x ij , 1 ≤ j < i ≤ r . A. Pinamonti and G. Speight UDS in Carnot Groups Warwick GMT 2017 20 / 35

  30. Free Carnot Groups of Step 2 Given integer r ≥ 2, let n = r + r ( r − 1) / 2. Denote coordinates in R n by x i , 1 ≤ i ≤ r , and x ij , 1 ≤ j < i ≤ r . Definition The free Carnot group of step 2 and r generators G r is R n with product: ( x · y ) ij = x ij + y ij + 1 ( x · y ) i = x i + y i , 2( x i y j − y i x j ) . A. Pinamonti and G. Speight UDS in Carnot Groups Warwick GMT 2017 20 / 35

  31. Free Carnot Groups of Step 2 Given integer r ≥ 2, let n = r + r ( r − 1) / 2. Denote coordinates in R n by x i , 1 ≤ i ≤ r , and x ij , 1 ≤ j < i ≤ r . Definition The free Carnot group of step 2 and r generators G r is R n with product: ( x · y ) ij = x ij + y ij + 1 ( x · y ) i = x i + y i , 2( x i y j − y i x j ) . We have V 1 = Span { X i : 1 ≤ i ≤ r } and V 2 = Span { X ij : 1 ≤ j < i ≤ r } , where x j x j � � X i := ∂ i + 2 ∂ ji − 2 ∂ ij X ij := ∂ ij . j > i j < i A. Pinamonti and G. Speight UDS in Carnot Groups Warwick GMT 2017 20 / 35

  32. Free Carnot Groups of Step 2 Given integer r ≥ 2, let n = r + r ( r − 1) / 2. Denote coordinates in R n by x i , 1 ≤ i ≤ r , and x ij , 1 ≤ j < i ≤ r . Definition The free Carnot group of step 2 and r generators G r is R n with product: ( x · y ) ij = x ij + y ij + 1 ( x · y ) i = x i + y i , 2( x i y j − y i x j ) . We have V 1 = Span { X i : 1 ≤ i ≤ r } and V 2 = Span { X ij : 1 ≤ j < i ≤ r } , where x j x j � � X i := ∂ i + 2 ∂ ji − 2 ∂ ij X ij := ∂ ij . j > i j < i Note [ X i , X j ] = X ij and G 2 = H 1 . A. Pinamonti and G. Speight UDS in Carnot Groups Warwick GMT 2017 20 / 35

  33. Free Carnot Groups of Step 2 Given integer r ≥ 2, let n = r + r ( r − 1) / 2. Denote coordinates in R n by x i , 1 ≤ i ≤ r , and x ij , 1 ≤ j < i ≤ r . Definition The free Carnot group of step 2 and r generators G r is R n with product: ( x · y ) ij = x ij + y ij + 1 ( x · y ) i = x i + y i , 2( x i y j − y i x j ) . We have V 1 = Span { X i : 1 ≤ i ≤ r } and V 2 = Span { X ij : 1 ≤ j < i ≤ r } , where x j x j � � X i := ∂ i + 2 ∂ ji − 2 ∂ ij X ij := ∂ ij . j > i j < i Note [ X i , X j ] = X ij and G 2 = H 1 . Horizontal curves in G r are lifts of curves in R r , 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

  34. Abstract View of Free Groups Definition Let r ≥ 2 and s ≥ 1 be integers. We say that F r , s is the free-nilpotent Lie algebra with r generators x 1 , . . . , x r of step s if: A. Pinamonti and G. Speight UDS in Carnot Groups Warwick GMT 2017 21 / 35

  35. Abstract View of Free Groups Definition Let r ≥ 2 and s ≥ 1 be integers. We say that F r , s is the free-nilpotent Lie algebra with r generators x 1 , . . . , x r of step s if: 1 F r , s is a Lie algebra generated by elements x 1 , . . . , x r , A. Pinamonti and G. Speight UDS in Carnot Groups Warwick GMT 2017 21 / 35

  36. Abstract View of Free Groups Definition Let r ≥ 2 and s ≥ 1 be integers. We say that F r , s is the free-nilpotent Lie algebra with r generators x 1 , . . . , x r of step s if: 1 F r , s is a Lie algebra generated by elements x 1 , . . . , x r , 2 F r , 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

  37. Abstract View of Free Groups Definition Let r ≥ 2 and s ≥ 1 be integers. We say that F r , s is the free-nilpotent Lie algebra with r generators x 1 , . . . , x r of step s if: 1 F r , s is a Lie algebra generated by elements x 1 , . . . , x r , 2 F r , 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: { x 1 , . . . , x r } → g , there is a unique homomorphism of Lie algebras ˜ Φ: F r , s → g that extends Φ. A. Pinamonti and G. Speight UDS in Carnot Groups Warwick GMT 2017 21 / 35

  38. Abstract View of Free Groups Definition Let r ≥ 2 and s ≥ 1 be integers. We say that F r , s is the free-nilpotent Lie algebra with r generators x 1 , . . . , x r of step s if: 1 F r , s is a Lie algebra generated by elements x 1 , . . . , x r , 2 F r , 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: { x 1 , . . . , x r } → g , there is a unique homomorphism of Lie algebras ˜ Φ: F r , 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 F r , s for some r ≥ 2 and s ≥ 1. A. Pinamonti and G. Speight UDS in Carnot Groups Warwick GMT 2017 21 / 35

  39. Rotations in Free Groups Lemma Suppose y ∈ G r with L = | p ( y ) | � = 0 . Then there exists a group isometric isomorphism F : G r → G r such that F 1 ( y ) = L and F i ( y ) = 0 for i > 1 . A. Pinamonti and G. Speight UDS in Carnot Groups Warwick GMT 2017 22 / 35

  40. Rotations in Free Groups Lemma Suppose y ∈ G r with L = | p ( y ) | � = 0 . Then there exists a group isometric isomorphism F : G r → G r such that F 1 ( y ) = L and F i ( y ) = 0 for i > 1 . Such a map can be chosen of the form F ( x , y ) = ( A ( x ) , B ( y )) , where A : R r → R r is a linear isometry and B is linear. A. Pinamonti and G. Speight UDS in Carnot Groups Warwick GMT 2017 22 / 35

  41. Rotations in Free Groups Lemma Suppose y ∈ G r with L = | p ( y ) | � = 0 . Then there exists a group isometric isomorphism F : G r → G r such that F 1 ( y ) = L and F i ( y ) = 0 for i > 1 . Such a map can be chosen of the form F ( x , y ) = ( A ( x ) , B ( y )) , where A : R r → R r 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 ∈ G r with y 1 > 0 and y i = 0 for i > 1. A. Pinamonti and G. Speight UDS in Carnot Groups Warwick GMT 2017 22 / 35

  42. Useful Horizontal Curve in G r Lemma (Le Donne, Pinamonti, Speight) Fix y ∈ G r with y 1 > 0 and y i = 0 for i > 1 . A. Pinamonti and G. Speight UDS in Carnot Groups Warwick GMT 2017 23 / 35

  43. Useful Horizontal Curve in G r Lemma (Le Donne, Pinamonti, Speight) Fix y ∈ G r with y 1 > 0 and y i = 0 for i > 1 . Let A = max 2 ≤ i ≤ r | y i 1 | and B = max i > j > 1 | y ij | . A. Pinamonti and G. Speight UDS in Carnot Groups Warwick GMT 2017 23 / 35

  44. Useful Horizontal Curve in G r Lemma (Le Donne, Pinamonti, Speight) Fix y ∈ G r with y 1 > 0 and y i = 0 for i > 1 . Let A = max 2 ≤ i ≤ r | y i 1 | and B = max i > j > 1 | y ij | . Then there exists a Lipschitz horizontal curve γ : [0 , 1] → G r which is a concatenation of horizontal lines such that: A. Pinamonti and G. Speight UDS in Carnot Groups Warwick GMT 2017 23 / 35

  45. Useful Horizontal Curve in G r Lemma (Le Donne, Pinamonti, Speight) Fix y ∈ G r with y 1 > 0 and y i = 0 for i > 1 . Let A = max 2 ≤ i ≤ r | y i 1 | and B = max i > j > 1 | y ij | . Then there exists a Lipschitz horizontal curve γ : [0 , 1] → G r which is a concatenation of horizontal lines such that: γ (0) = 0 and γ (1) = y, 1 A. Pinamonti and G. Speight UDS in Carnot Groups Warwick GMT 2017 23 / 35

  46. Useful Horizontal Curve in G r Lemma (Le Donne, Pinamonti, Speight) Fix y ∈ G r with y 1 > 0 and y i = 0 for i > 1 . Let A = max 2 ≤ i ≤ r | y i 1 | and B = max i > j > 1 | y ij | . Then there exists a Lipschitz horizontal curve γ : [0 , 1] → G r which is a concatenation of horizontal lines such that: γ (0) = 0 and γ (1) = y, 1 The Lipschitz constant of γ satisfies 2 � 1 / 2 �� � 1 / 2 � 1 + CA 2 � 1 + CB Lip ( γ ) ≤ y 1 max , , y 4 y 2 1 1 A. Pinamonti and G. Speight UDS in Carnot Groups Warwick GMT 2017 23 / 35

  47. Useful Horizontal Curve in G r Lemma (Le Donne, Pinamonti, Speight) Fix y ∈ G r with y 1 > 0 and y i = 0 for i > 1 . Let A = max 2 ≤ i ≤ r | y i 1 | and B = max i > j > 1 | y ij | . Then there exists a Lipschitz horizontal curve γ : [0 , 1] → G r which is a concatenation of horizontal lines such that: γ (0) = 0 and γ (1) = y, 1 The Lipschitz constant of γ satisfies 2 � 1 / 2 �� � 1 / 2 � 1 + CA 2 � 1 + CB Lip ( γ ) ≤ y 1 max , , y 4 y 2 1 1 γ ′ ( t ) exists for all t ∈ [0 , 1] except finitely many points and satisfies 3 � A √ � | ( p ◦ γ ) ′ ( t ) − p ( y ) | ≤ C max , B . y 1 A. Pinamonti and G. Speight UDS in Carnot Groups Warwick GMT 2017 23 / 35

  48. Useful Horizontal Curve in G r γ (0) = 0 y γ (1) = y = ( y 1 , 0 , . . . , 0 , y 21 , y 31 , . . . ) ( y 1 , 0 , . . . , 0) 0 X 1 In each subinterval we fix some vertical coordinate ij and leave the rest unchanged A. Pinamonti and G. Speight UDS in Carnot Groups Warwick GMT 2017 24 / 35

  49. The Engel Group Definition The Engel group E (a Carnot group of step 3) is R 4 with group law x 1 + y 1 , x 2 + y 2 , x 3 + y 3 − x 1 y 2 , x 4 + y 4 − x 1 y 3 + 1 � � 2 x 2 x · y = 1 y 2 . A. Pinamonti and G. Speight UDS in Carnot Groups Warwick GMT 2017 25 / 35

  50. The Engel Group Definition The Engel group E (a Carnot group of step 3) is R 4 with group law x 1 + y 1 , x 2 + y 2 , x 3 + y 3 − x 1 y 2 , x 4 + y 4 − x 1 y 3 + 1 � � 2 x 2 x · y = 1 y 2 . We have V 1 = Span { X 1 , X 2 } , V 2 = Span { X 3 } and V 3 = Span { X 4 } , where X 2 = ∂ 2 − x 1 ∂ 3 + x 2 1 X 1 = ∂ 1 , 2 ∂ 4 , X 3 = ∂ 3 − x 1 ∂ 4 , X 4 = ∂ 4 . A. Pinamonti and G. Speight UDS in Carnot Groups Warwick GMT 2017 25 / 35

  51. The Engel Group Definition The Engel group E (a Carnot group of step 3) is R 4 with group law x 1 + y 1 , x 2 + y 2 , x 3 + y 3 − x 1 y 2 , x 4 + y 4 − x 1 y 3 + 1 � � 2 x 2 x · y = 1 y 2 . We have V 1 = Span { X 1 , X 2 } , V 2 = Span { X 3 } and V 3 = Span { X 4 } , where X 2 = ∂ 2 − x 1 ∂ 3 + x 2 1 X 1 = ∂ 1 , 2 ∂ 4 , X 3 = ∂ 3 − x 1 ∂ 4 , X 4 = ∂ 4 . Non-trivial bracket relations are X 3 = [ X 2 , X 1 ] and X 4 = [ X 3 , X 1 ]. A. Pinamonti and G. Speight UDS in Carnot Groups Warwick GMT 2017 25 / 35

  52. The Engel Group Definition The Engel group E (a Carnot group of step 3) is R 4 with group law x 1 + y 1 , x 2 + y 2 , x 3 + y 3 − x 1 y 2 , x 4 + y 4 − x 1 y 3 + 1 � � 2 x 2 x · y = 1 y 2 . We have V 1 = Span { X 1 , X 2 } , V 2 = Span { X 3 } and V 3 = Span { X 4 } , where X 2 = ∂ 2 − x 1 ∂ 3 + x 2 1 X 1 = ∂ 1 , 2 ∂ 4 , X 3 = ∂ 3 − x 1 ∂ 4 , X 4 = ∂ 4 . Non-trivial bracket relations are X 3 = [ X 2 , X 1 ] and X 4 = [ X 3 , X 1 ]. 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

  53. 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

  54. 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( X 2 ) , but X 2 d (¯ p ) = 1 . A. Pinamonti and G. Speight UDS in Carnot Groups Warwick GMT 2017 26 / 35

  55. 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( X 2 ) , but X 2 d (¯ p ) = 1 . Proof. Suppose d has differential L at ¯ p . Then L ( h ) = � X 1 d (¯ p ) , X 2 d (¯ p ) � · � h 1 , h 2 � , so L ((0 , 0 , 0 , ε )) = 0 for ε ∈ R . Hence for ε ∈ ( − 1 , 0): p ) − L ((0 , 0 , 0 , ε )) = d ((0 , 1 , 0 , ε )) − 1 ≥ C | ε | 1 / 3 d (¯ p · (0 , 0 , 0 , ε )) − d (¯ ≥ Cd ((0 , 0 , 0 , ε )) , which contradicts differentiability at ¯ p . A. Pinamonti and G. Speight UDS in Carnot Groups Warwick GMT 2017 26 / 35

  56. 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

  57. 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

  58. 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 G r of step 2 and r generators. Fix a G δ 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

  59. 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 G r of step 2 and r generators. Fix a G δ 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 ∈ V 1 with ω ( E ) = 1. A. Pinamonti and G. Speight UDS in Carnot Groups Warwick GMT 2017 27 / 35

  60. 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 G r of step 2 and r generators. Fix a G δ 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 ∈ V 1 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

  61. 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 G r of step 2 and r generators. Fix a G δ 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 ∈ V 1 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

  62. 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

  63. Horizontal Curves for Almost Maximality Lemma (Le Donne, Pinamonti, Speight) Given η > 0 , the following holds for sufficiently small ∆ . Suppose: x , u ∈ G r with d ( u ) ≤ 1 , A. Pinamonti and G. Speight UDS in Carnot Groups Warwick GMT 2017 28 / 35

  64. Horizontal Curves for Almost Maximality Lemma (Le Donne, Pinamonti, Speight) Given η > 0 , the following holds for sufficiently small ∆ . Suppose: x , u ∈ G r with d ( u ) ≤ 1 , E ∈ V 1 with ω ( E ) = 1 , A. Pinamonti and G. Speight UDS in Carnot Groups Warwick GMT 2017 28 / 35

  65. Horizontal Curves for Almost Maximality Lemma (Le Donne, Pinamonti, Speight) Given η > 0 , the following holds for sufficiently small ∆ . Suppose: x , u ∈ G r with d ( u ) ≤ 1 , E ∈ V 1 with ω ( E ) = 1 , 0 < r < ∆ and s := r / ∆ . A. Pinamonti and G. Speight UDS in Carnot Groups Warwick GMT 2017 28 / 35

  66. Horizontal Curves for Almost Maximality Lemma (Le Donne, Pinamonti, Speight) Given η > 0 , the following holds for sufficiently small ∆ . Suppose: x , u ∈ G r with d ( u ) ≤ 1 , E ∈ V 1 with ω ( E ) = 1 , 0 < r < ∆ and s := r / ∆ . Then there is a Lipschitz horizontal curve g : R → G r , which is a concatenation of horizontal lines, such that: A. Pinamonti and G. Speight UDS in Carnot Groups Warwick GMT 2017 28 / 35

  67. Horizontal Curves for Almost Maximality Lemma (Le Donne, Pinamonti, Speight) Given η > 0 , the following holds for sufficiently small ∆ . Suppose: x , u ∈ G r with d ( u ) ≤ 1 , E ∈ V 1 with ω ( E ) = 1 , 0 < r < ∆ and s := r / ∆ . Then there is a Lipschitz horizontal curve g : R → G r , 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

  68. Horizontal Curves for Almost Maximality Lemma (Le Donne, Pinamonti, Speight) Given η > 0 , the following holds for sufficiently small ∆ . Suppose: x , u ∈ G r with d ( u ) ≤ 1 , E ∈ V 1 with ω ( E ) = 1 , 0 < r < ∆ and s := r / ∆ . Then there is a Lipschitz horizontal curve g : R → G r , which is a concatenation of horizontal lines, such that: 1 g ( t ) = x + tE ( x ) for | t | ≥ s, 2 g ( ζ ) = x δ r ( u ) , where ζ := � δ r ( u ) , E (0) � , A. Pinamonti and G. Speight UDS in Carnot Groups Warwick GMT 2017 28 / 35

  69. Horizontal Curves for Almost Maximality Lemma (Le Donne, Pinamonti, Speight) Given η > 0 , the following holds for sufficiently small ∆ . Suppose: x , u ∈ G r with d ( u ) ≤ 1 , E ∈ V 1 with ω ( E ) = 1 , 0 < r < ∆ and s := r / ∆ . Then there is a Lipschitz horizontal curve g : R → G r , which is a concatenation of horizontal lines, such that: 1 g ( t ) = x + tE ( x ) for | t | ≥ s, 2 g ( ζ ) = x δ r ( u ) , where ζ := � δ r ( u ) , E (0) � , 3 Lip ( g ) ≤ 1 + η ∆ , A. Pinamonti and G. Speight UDS in Carnot Groups Warwick GMT 2017 28 / 35

  70. Horizontal Curves for Almost Maximality Lemma (Le Donne, Pinamonti, Speight) Given η > 0 , the following holds for sufficiently small ∆ . Suppose: x , u ∈ G r with d ( u ) ≤ 1 , E ∈ V 1 with ω ( E ) = 1 , 0 < r < ∆ and s := r / ∆ . Then there is a Lipschitz horizontal curve g : R → G r , which is a concatenation of horizontal lines, such that: 1 g ( t ) = x + tE ( x ) for | t | ≥ s, 2 g ( ζ ) = x δ r ( u ) , where ζ := � δ r ( 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

  71. Horizontal Curves for Almost Maximality g ( ζ ) = xδ r ( u ), where ζ := � δ r ( u ) , E (0) � xδ r ( u ) E x x − sE ( x ) x + sE ( x ) Lip( g ) ≤ 1 + η ∆ | ( p ◦ g ) ′ ( t ) − p ( E ) | ≤ C ∆ A. Pinamonti and G. Speight UDS in Carnot Groups Warwick GMT 2017 29 / 35

  72. Almost Maximality implies Differentiability Let D f := { ( x , E ) ∈ N × V 1 : ω ( E ) = 1 , Ef ( x ) exists } . A. Pinamonti and G. Speight UDS in Carnot Groups Warwick GMT 2017 30 / 35

  73. Almost Maximality implies Differentiability Let D f := { ( x , E ) ∈ N × V 1 : ω ( E ) = 1 , Ef ( x ) exists } . Theorem (Le Donne, Pinamonti, Speight) Let f : G r → R be Lipschitz and ( x 0 , E 0 ) ∈ D f . A. Pinamonti and G. Speight UDS in Carnot Groups Warwick GMT 2017 30 / 35

  74. Almost Maximality implies Differentiability Let D f := { ( x , E ) ∈ N × V 1 : ω ( E ) = 1 , Ef ( x ) exists } . Theorem (Le Donne, Pinamonti, Speight) Let f : G r → R be Lipschitz and ( x 0 , E 0 ) ∈ D f . Let M denote the set of pairs ( x , E ) ∈ D f such that Ef ( x ) ≥ E 0 f ( x 0 ) and | ( f ( x + tE 0 ( x )) − f ( x )) − ( f ( x 0 + tE 0 ( x 0 )) − f ( x 0 )) | 1 ≤ 6 | t | (( Ef ( x ) − E 0 f ( x 0 )) Lip ( f )) 4 for every t ∈ ( − 1 , 1) . A. Pinamonti and G. Speight UDS in Carnot Groups Warwick GMT 2017 30 / 35

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