A C m Whitney Extension Theorem for Horizontal Curves in the - PowerPoint PPT Presentation

A C m Whitney Extension Theorem for Horizontal Curves in the Heisenberg Group Gareth Speight University of Cincinnati AMS Spring Central and Western Sectional Meeting 2019 Topics at the Interface of Analysis and Geometry C m Whitney Extension in H n Gareth Speight (Cincinnati) University of Hawaii, 2019 1 / 14

Jets If K ⊂ R n is compact, when can a function K → R be extended to a C m function R n → R with prescribed derivatives? C m Whitney Extension in H n Gareth Speight (Cincinnati) University of Hawaii, 2019 1 / 14

Jets If K ⊂ R n is compact, when can a function K → R be extended to a C m function R n → R with prescribed derivatives? Definition A jet of order m on K is a collection F = ( F k ) | k |≤ m of continuous functions on K . Here k = ( k 1 , . . . , k n ) ∈ N n is a multi index and | k | = k 1 + . . . + k n . C m Whitney Extension in H n Gareth Speight (Cincinnati) University of Hawaii, 2019 1 / 14

Jets If K ⊂ R n is compact, when can a function K → R be extended to a C m function R n → R with prescribed derivatives? Definition A jet of order m on K is a collection F = ( F k ) | k |≤ m of continuous functions on K . Here k = ( k 1 , . . . , k n ) ∈ N n is a multi index and | k | = k 1 + . . . + k n . For each C m function f : R n → R there is an associated jet on K : � ∂ | k | f � � f �→ . � ∂ x k � K | k |≤ m C m Whitney Extension in H n Gareth Speight (Cincinnati) University of Hawaii, 2019 1 / 14

Whitney Jets Given a jet ( F k ) | k |≤ m , we denote for | k | ≤ m : F k + l ( x ) ( R m x F ) k ( y ) = F k ( y ) − � ( y − x ) l . l ! | l |≤ m −| k | Here ( y − x ) l = ( y 1 − x 1 ) l 1 · · · ( y n − x n ) l n . C m Whitney Extension in H n Gareth Speight (Cincinnati) University of Hawaii, 2019 2 / 14

Whitney Jets Given a jet ( F k ) | k |≤ m , we denote for | k | ≤ m : F k + l ( x ) ( R m x F ) k ( y ) = F k ( y ) − � ( y − x ) l . l ! | l |≤ m −| k | Here ( y − x ) l = ( y 1 − x 1 ) l 1 · · · ( y n − x n ) l n . Definition ( F k ) | k |≤ m is a Whitney field of class C m on K if for all | k | ≤ m : ( R m x F ) k ( y ) | x − y | m −| k | → 0 uniformly as | x − y | → 0 with x , y ∈ K . C m Whitney Extension in H n Gareth Speight (Cincinnati) University of Hawaii, 2019 2 / 14

Whitney Jets Given a jet ( F k ) | k |≤ m , we denote for | k | ≤ m : F k + l ( x ) ( R m x F ) k ( y ) = F k ( y ) − � ( y − x ) l . l ! | l |≤ m −| k | Here ( y − x ) l = ( y 1 − x 1 ) l 1 · · · ( y n − x n ) l n . Definition ( F k ) | k |≤ m is a Whitney field of class C m on K if for all | k | ≤ m : ( R m x F ) k ( y ) | x − y | m −| k | → 0 uniformly as | x − y | → 0 with x , y ∈ K . The jet associated to a C m function is always a Whitney field of class C m on any compact set K . C m Whitney Extension in H n Gareth Speight (Cincinnati) University of Hawaii, 2019 2 / 14

Whitney Extension Theorem Theorem (Whitney) Let ( F k ) | k |≤ m be a Whitney field of class C m on K. Then there exists a C m map f : R n → R such that ∂ | k | f � K = F k for | k | ≤ m . � ∂ x k � C m Whitney Extension in H n Gareth Speight (Cincinnati) University of Hawaii, 2019 3 / 14

Whitney Extension Theorem Theorem (Whitney) Let ( F k ) | k |≤ m be a Whitney field of class C m on K. Then there exists a C m map f : R n → R such that ∂ | k | f � K = F k for | k | ≤ m . � ∂ x k � Corollary (Lusin Approximation of Curves) Suppose γ : [ a , b ] → R n is absolutely continuous and ε > 0 . Then there exists a C 1 curve Γ: [ a , b ] → R n such that L 1 { t ∈ [ a , b ]: Γ( t ) � = γ ( t ) or Γ ′ ( t ) � = γ ′ ( t ) } < ε. C m Whitney Extension in H n Gareth Speight (Cincinnati) University of Hawaii, 2019 3 / 14

Heisenberg Group Definition The first Heisenberg group H 1 is R 3 equipped with group law: ( x , y , t )( x ′ , y ′ , t ′ ) = ( x + x ′ , y + y ′ , t + t ′ − 2( xy ′ − yx ′ )) . C m Whitney Extension in H n Gareth Speight (Cincinnati) University of Hawaii, 2019 4 / 14

Heisenberg Group Definition The first Heisenberg group H 1 is R 3 equipped with group law: ( x , y , t )( x ′ , y ′ , t ′ ) = ( x + x ′ , y + y ′ , t + t ′ − 2( xy ′ − yx ′ )) . Left invariant horizontal vector fields on H 1 are defined by: X ( x , y , t ) = ∂ x + 2 y ∂ t , Y ( x , y , t ) = ∂ y − 2 x ∂ t . C m Whitney Extension in H n Gareth Speight (Cincinnati) University of Hawaii, 2019 4 / 14

Heisenberg Group Definition The first Heisenberg group H 1 is R 3 equipped with group law: ( x , y , t )( x ′ , y ′ , t ′ ) = ( x + x ′ , y + y ′ , t + t ′ − 2( xy ′ − yx ′ )) . Left invariant horizontal vector fields on H 1 are defined by: X ( x , y , t ) = ∂ x + 2 y ∂ t , Y ( x , y , t ) = ∂ y − 2 x ∂ t . The Haar measure on H 1 is L 3 : L 3 ( gA ) = L 3 ( A ). C m Whitney Extension in H n Gareth Speight (Cincinnati) University of Hawaii, 2019 4 / 14

Heisenberg Group Definition The first Heisenberg group H 1 is R 3 equipped with group law: ( x , y , t )( x ′ , y ′ , t ′ ) = ( x + x ′ , y + y ′ , t + t ′ − 2( xy ′ − yx ′ )) . Left invariant horizontal vector fields on H 1 are defined by: X ( x , y , t ) = ∂ x + 2 y ∂ t , Y ( x , y , t ) = ∂ y − 2 x ∂ t . The Haar measure on H 1 is L 3 : L 3 ( gA ) = L 3 ( A ). Dilations are defined by δ r ( x , y , t ) = ( rx , ry , r 2 t ). They satisfy δ r ( ab ) = δ r ( a ) δ r ( b ) and L 3 ( δ r ( A )) = r 4 L 3 ( A ) . C m Whitney Extension in H n Gareth Speight (Cincinnati) University of Hawaii, 2019 4 / 14

Horizontal Curves Definition An absolutely continuous curve γ : [ a , b ] → H 1 is horizontal if there exists h : [ a , b ] → R 2 such that for almost every t : γ ′ ( t ) = h 1 ( t ) X ( γ ( t )) + h 2 ( t ) Y ( γ ( t )) . C m Whitney Extension in H n Gareth Speight (Cincinnati) University of Hawaii, 2019 5 / 14

Horizontal Curves Definition An absolutely continuous curve γ : [ a , b ] → H 1 is horizontal if there exists h : [ a , b ] → R 2 such that for almost every t : γ ′ ( t ) = h 1 ( t ) X ( γ ( t )) + h 2 ( t ) Y ( γ ( t )) . The horizontal length of such a curve is defined by: � b L ( γ ) = | h | . a Any two points can be connected by a horizontal curve! C m Whitney Extension in H n Gareth Speight (Cincinnati) University of Hawaii, 2019 5 / 14

Horizontal Lift Lemma (Horizontal Lift) An absolutely continuous curve γ : [ a , b ] → H 1 is horizontal if and only if � t ( γ ′ 1 γ 2 − γ ′ γ 3 ( t ) = γ 3 ( a ) + 2 2 γ 1 ) a for every t ∈ [ a , b ] . C m Whitney Extension in H n Gareth Speight (Cincinnati) University of Hawaii, 2019 6 / 14

Horizontal Lift Lemma (Horizontal Lift) An absolutely continuous curve γ : [ a , b ] → H 1 is horizontal if and only if � t ( γ ′ 1 γ 2 − γ ′ γ 3 ( t ) = γ 3 ( a ) + 2 2 γ 1 ) a for every t ∈ [ a , b ] . Lemma (Height-Area Interpretation) Suppose σ : [ a , b ] → R 2 is a smooth curve with σ ( a ) = 0 . Let A σ denote the signed area of the region enclosed by σ and the straight line [0 , σ ( b )] . Then � b A σ = 1 ( σ 1 σ ′ 2 − σ 2 σ ′ 1 ) . 2 a C m Whitney Extension in H n Gareth Speight (Cincinnati) University of Hawaii, 2019 6 / 14

Horizontal Curves t y x C m Whitney Extension in H n Gareth Speight (Cincinnati) University of Hawaii, 2019 7 / 14

Whitney Extension for C 1 Horizontal Curves in H 1 Whitney extension for C m maps from K ⊂ H 1 or G to R are understood. C m Whitney Extension in H n Gareth Speight (Cincinnati) University of Hawaii, 2019 8 / 14

Whitney Extension for C 1 Horizontal Curves in H 1 Whitney extension for C m maps from K ⊂ H 1 or G to R are understood. Maps with target H 1 or G are harder to understand. C m Whitney Extension in H n Gareth Speight (Cincinnati) University of Hawaii, 2019 8 / 14

Whitney Extension for C 1 Horizontal Curves in H 1 Whitney extension for C m maps from K ⊂ H 1 or G to R are understood. Maps with target H 1 or G are harder to understand. Theorem (Zimmerman) Suppose ( f , f ′ ) , ( g , g ′ ) , ( h , h ′ ) are Whitney fields of class C 1 on K. Then there exists a C 1 horizontal curve Γ: R → H 1 such that Γ | K = ( f , g , h ) and Γ ′ | K = ( f ′ , g ′ , h ′ ) if and only if C m Whitney Extension in H n Gareth Speight (Cincinnati) University of Hawaii, 2019 8 / 14

Whitney Extension for C 1 Horizontal Curves in H 1 Whitney extension for C m maps from K ⊂ H 1 or G to R are understood. Maps with target H 1 or G are harder to understand. Theorem (Zimmerman) Suppose ( f , f ′ ) , ( g , g ′ ) , ( h , h ′ ) are Whitney fields of class C 1 on K. Then there exists a C 1 horizontal curve Γ: R → H 1 such that Γ | K = ( f , g , h ) and Γ ′ | K = ( f ′ , g ′ , h ′ ) if and only if h ′ ( s ) = 2( f ′ ( s ) g ( s ) − g ′ ( s ) f ( s )) for all s ∈ K C m Whitney Extension in H n Gareth Speight (Cincinnati) University of Hawaii, 2019 8 / 14