A C m Lusin Approximation Theorem for Horizontal Curves in the Heisenberg Group
aa r X i v : . [ m a t h . M G ] A ug A C m LUSIN APPROXIMATION THEOREM FOR HORIZONTALCURVES IN THE HEISENBERG GROUP
MARCO CAPOLLI, ANDREA PINAMONTI, AND GARETH SPEIGHT
Abstract.
We prove a C m Lusin approximation theorem for horizontal curvesin the Heisenberg group. This states that every absolutely continuous hori-zontal curve whose horizontal velocity is m − L differentiable almosteverywhere coincides with a C m horizontal curve except on a set of small mea-sure. Conversely, we show that the result no longer holds if L differentiabilityis replaced by approximate differentiability. This shows our result is optimaland highlights differences between the Heisenberg and Euclidean settings. Introduction
In mathematical analysis it is often useful to understand when a rough map canbe approximated by a smoother one. For instance, Lusin’s theorem asserts that ev-ery measurable function on R n is continuous after removing a set of small measurefrom the domain. Another useful result states that every absolutely continuouscurve in R n has the 1-Lusin property, which means that it coincides with a C curve except for a set of small measure (Theorem 2.4). The position and velocityof absolutely continuous curves are related according to the Fundamental Theoremof Calculus, so these curves are important in analysis and geometry. Concerninghigher regularity, if a curve in R n is approximately differentiable of order m almosteverywhere (Definition 2.8), then it has the m -Lusin property allowing approxima-tion by C m curves (Theorem 2.5) [20]. In the present article we study the m -Lusinproperty for horizontal curves in the Heisenberg group, a non-Euclidean space withmuch geometric structure. The key difference between the Heisenberg group andEuclidean space is that in the Heisenberg group both the initial and the approxi-mating curve must be horizontal, which means they are constrained to move in asmaller but still rich, family of directions.In recent years, it has become clear that a large part of geometric analysis,geometric measure theory and real analysis in Euclidean spaces may be generalizedto more general settings, see for example [4, 6, 7, 13, 14, 16, 18, 22, 21, 23, 25,26, 27]. Carnot groups are Lie groups whose Lie algebra admits a stratification.This stratification gives rise to dilations and implies that points can be connectedby absolutely continuous curves with tangents in a distinguished subbundle of thetangent bundle. These are the so called horizontal curves. Considering lengthsof horizontal curves gives rise to the Carnot-Carath´eodory distance and endowsevery Carnot group with a metric space structure. Moreover, every Carnot grouphas a natural Haar measure which respects the group translations and dilations.This plethora of structure makes the study of analysis and geometry in Carnotgroups highly interesting [4, 7, 22]. However, results in the Carnot setting can bevery different to Euclidean ones since all such results must respect the horizontal structure of the Carnot group. The Heisenberg group is the simplest non-EuclideanCarnot group and admits an explicit representation in R n +1 (Definition 2.1) with2 n horizontal directions and one vertical direction.In the Heisenberg group, the 1-Lusin property is known to be true for all abso-lutely continuous horizontal curves with the requirement that the approximatingcurve can be chosen both C and horizontal. More precisely, every absolutely con-tinuous horizontal curve can be approximated by a C horizontal curve (Theorem2.7) [28]. A similar result holds in step two Carnot groups [19] and more generalpliable Carnot groups [17, 29]. However the natural analogue is not true in theEngel group, a Carnot group of step three [28]. This highlights that the approxi-mation depends on the space considered and Euclidean results do not always extendto the Carnot group setting. Proving that smooth approximations exist is closelyconnected to validity of a Whitney extension theorem. In Euclidean spaces, theWhitney extension theorem (Theorem 2.14) [3, 31] characterizes when a collectionof continuous functions defined on a compact set can be extended to a C m func-tion on some larger set. To prove a Lusin approximation result from a Whitneyextension theorem, one typically restricts to a large compact set where the originalmapping satisfies the hypotheses of the Whitney extension theorem and then ob-tains a C m mapping which agrees with the starting map on a large set. To applythis idea in the Heisenberg group it is important to have an analogue of the Whitneyextension theorem for curves in the Heisenberg group. Such a theorem is indeedknown for C horizontal curves in the Heisenberg group [32] and in more generalspaces [17, 29]. Very recently it was also understood for C m horizontal curves inthe Heisenberg group [24].In the present paper we focus on the m -Lusin property in the Heisenberg group(Definition 2.6), investigating which horizontal curves can be approximated by C m horizontal curves. We now describe our main results.Our first main result is Theorem 4.1. This asserts that if Γ = ( f, g, h ) is anabsolutely continuous horizontal curve in H with f ′ , g ′ almost everywhere m − L differentiable (Definition 2.10), then Γ has the m -Lusin property. Thisshould be compared with the previously known analogue in Euclidean space (The-orem 2.5), which has the weaker hypothesis that f, g, h are m times approximatelydifferentiable almost everywhere (Definition 2.8). Our arguments adapt the proofof Theorem 2.5 from [20] to the Heisenberg group using our stronger hypothesis torestrict to a compact set on which we can apply the C m Whitney extension theoremfor horizontal curves in the Heisenberg group (Theorem 2.15) recently proved [24].Our second main result is Theorem 5.1, which illustrates the difference betweenthe Heisenberg setting and the Euclidean setting. It also justifies the hypothesesof Theorem 4.1. In Theorem 5.1 we construct an absolutely continuous horizontalcurve Γ in H such that f, g, h are almost everywhere twice L p differentiable forall p ≥ f ′ , g ′ , h ′ are almost everywhere once approximately differentiable, yet Γdoes not have the 2-Lusin approximation property. This shows that the Euclideanhypothesis of twice approximate differentiability is not sufficient in H and that onereally needs to assume differentiability properties of the derivatives f ′ , g ′ , h ′ ratherthan only on f, g, h . Our argument is an explicit construction of a horizontal curve.In the main results of this paper we restrict our attention to the first Heisenberggroup H = H . We expect the natural analogue of Theorem 4.1 is also true in USIN APPROXIMATION IN THE HEISENBERG GROUP 3 higher dimensional Heisenberg groups H n with similar proofs but more cumbersomenotation.We now describe the structure of the paper. In Section 2 we recall key definitionsinvolving the Heisenberg group, approximate derivatives, L p derivatives, m -Lusinproperty and Whitney extension theorems. In Section 3 we prove preliminaryresults describing how L differentiability behaves under integration or lifting to ahorizontal curve and how approximate differentiability almost everywhere can beused to obtain a Whitney field. In Section 4 we prove our first main result (Theorem4.1). Finally in Section 5 we prove our second main result (Theorem 5.1). Acknowledgements:
Part of this work was done while A. Pinamonti and M.Capolli were visiting the University of Cincinnati (supported by funding from theTaft Research Center and the University of Trento) and while G. Speight was visit-ing the University of Trento (supported by funding from the University of Trento).A. Pinamonti and M. Capolli are members
Gruppo Nazionale per l’Analisi Mate-matica, la Probabilit`a e le loro Applicazioni (GNAMPA) of the
Istituto Nazionaledi Alta Matematica (INdAM). This work was also supported by a grant from theSimons Foundation (
Preliminaries
Heisenberg Group and Horizontal Curves.Definition 2.1.
The
Heisenberg group H n is the Lie group represented in coordi-nates by R n +1 , whose points we denote by ( x, y, t ) with x, y ∈ R n and t ∈ R . Thegroup law is given by:( x, y, t )( x ′ , y ′ , t ′ ) = x + x ′ , y + y ′ , t + t ′ + 2 n X i =1 ( y i x ′ i − x i y ′ i ) ! . We equip H n with left invariant vector fields X i = ∂ x i + 2 y i ∂ t , Y i = ∂ y i − x i ∂ t , ≤ i ≤ n, T = ∂ t . Here ∂ x i , ∂ y i and ∂ t denote the coordinate vectors in R n +1 , which may be inter-preted as operators on differentiable functions. If [ · , · ] denotes the Lie bracket ofvector fields, then [ X i , Y i ] = − T . Thus H n is a Carnot group with horizontal layerSpan { X i , Y i : 1 ≤ i ≤ n } and second layer Span { T } . In this paper we will mostlyrestrict ourselves to the first Heisenberg group H which we also denote by H . Definition 2.2.
A vector in R n +1 is horizontal at p ∈ R n +1 if it is a linearcombination of the vectors X i ( p ) , Y i ( p ) , ≤ i ≤ n .An absolutely continuous curve γ in the Heisenberg group is horizontal if, atalmost every point t , the derivative γ ′ ( t ) is horizontal at γ ( t ). Lemma 2.3.
An absolutely continuous curve γ : [ a, b ] → R n +1 is a horizontalcurve in the Heisenberg group if and only if, for t ∈ [ a, b ] : γ n +1 ( t ) = γ n +1 ( a ) + 2 n X i =1 Z ta ( γ ′ i γ n + i − γ ′ n + i γ i ) . We will use Lemma 2.3 repeatedly throughout the paper. In the first Heisenberggroup H = H , the relevant equations for an absolutely continuous curve to be MARCO CAPOLLI, ANDREA PINAMONTI, AND GARETH SPEIGHT horizontal simplify to(2.1) γ ( t ) = γ ( a ) + 2 Z ta ( γ ′ γ − γ ′ γ ) . Clearly Lemma 2.3 implies that for any horizontal curve γ we have γ ′ n +1 ( t ) = 2 n X i =1 ( γ ′ i ( t ) γ n + i ( t ) − γ ′ n + i ( t ) γ i ( t )) for a.e. t ∈ [ a, b ] . If we assume that γ is C , this equality holds for every t ∈ [ a, b ]. If we furtherassume that γ is C m for some m >
1, then, for 1 ≤ k ≤ m , we may write(2.2) D k γ n +1 ( t ) = n X i =1 P k (cid:0) γ i ( t ) , γ n + i ( t ) , γ ′ i ( t ) , γ ′ n + i ( t ) , . . . , D k γ i ( t ) , D k γ n + i ( t ) (cid:1) for all t ∈ [ a, b ] where P k is a polynomial determined by the Leibniz rule. For a C m horizontal curve γ in the first Heisenberg group H = H , the equations simplify to(2.3) γ k = 2 k − X i =0 (cid:18) k − i (cid:19) ( γ k − i γ i − γ k − i γ i ) for 1 ≤ k ≤ m. Before defining the m -Lusin approximation property in H n we recall two resultswhich are known in Euclidean spaces. In each case we recall only what is useful forour purposes and not the most general result. The first is classical [2, 8]. Theorem 2.4.
Let γ : [ a, b ] → R n be absolutely continuous. Then for every ε > there exists a C map Γ : [ a, b ] → R n such that m ( { x ∈ [ a, b ] : Γ( t ) = γ ( t ) } ) < ε. The second is more recent and a special case of results from [20].
Theorem 2.5.
Suppose γ : [ a, b ] → R n is measurable and approximately differen-tiable of order m almost everywhere. Then for every ε > there exists a C m map Γ : [ a, b ] → R n such that m ( { x ∈ [ a, b ] : Γ( t ) = γ ( t ) } ) < ε. Definition 2.6.
An absolutely continuous horizontal curve Γ : [ a, b ] → H n is saidto have the Lusin property of order m if for every ǫ > C m horizontalcurve e Γ : [ a, b ] → H n such that L ( { x ∈ [ a, b ] : e Γ( x ) = Γ( x ) } ) < ε. We will also refer to the Lusin property of order m as the m -Lusin property. Werecall the following result for the 1-Lusin property in Heisenberg groups [28]. Theorem 2.7.
Absolutely continuous horizontal curves in H n have the -Lusinproperty. It is also known that absolutely continuous horizontal curves have the 1-Lusinproperty in step two Carnot groups [19], in pliable Carnot groups [17] and in suitablesub-Riemannian manifolds [29].
USIN APPROXIMATION IN THE HEISENBERG GROUP 5
Approximate Differentiability and Integral Differentiability.
Recallthat if f : R d → R , x ∈ R d and l ∈ R , then aplim y → x f ( x ) = l means that for every ε > { y ∈ R d : | f ( y ) − l | ≤ ǫ } has density one at x , i.e.lim R → L d ( B ( x, R ) ∩ { y ∈ R d : | f ( y ) − l | ≤ ǫ } ) L d ( B ( x, R )) = 1 . Definition 2.8.
Given x ∈ R d and k ∈ N , we say that a function u : R d → R is m times approximately differentiable at x if there exists a polynomial P mu,x of degreeat most m such that(2.4) aplim y → x | u ( y ) − P mu,x ( y ) || y − x | m = 0 . Remark . The polynomial P mu,x in Definition 2.8 is uniquely determined and canbe expressed in the form(2.5) P mu,x ( y ) = X | α |≤ m u α ( x ) | α | ! ( y − x ) α for some u α ( x ) ∈ R [20].Throughout this paper we use the usual notation for integral averages − Z A f = 1 L d ( A ) Z A f for any A ⊂ R d and f : A → R for which the expression is well defined. Definition 2.10.
Let u : R d → R , x ∈ R d , p ∈ [1 , ∞ ), and m ∈ N .We say that u is m times L p differentiable at x if there exists a polynomial P mu,x on R d of degree at most m such that(2.6) " − Z B ( x,ρ ) | u ( y ) − P mu,x ( y ) | p d y /p = o ( ρ m ) . Remark . As noted for instance in [1], if u is m times L p differentiable at x then u is also m times approximately differentiable at x with the same derivativepolynomial.2.3. Jets and Whitney Extension.Definition 2.12. A jet of order m ∈ N on a set K ⊂ R consists of a collection of( m + 1) − continuous functions F = ( F k ) mk =0 on K .Given such a jet F and a ∈ K , the Taylor polynomial of order m of F at a is T ma F ( x ) = m X k =0 F k ( a ) k ! ( x − a ) k for all x ∈ R . If m or a are clear from the context, we may write T F for the Taylor polynomial.We will also use the notation F ( x ) for F ( x ).Given a jet F of order m on K ⊂ R , for a ∈ K and 0 ≤ k ≤ m we define( R ma F ) k ( x ) = F k ( x ) − m − k X ℓ =0 F k + ℓ ( a ) ℓ ! ( x − a ) ℓ for all x ∈ R . MARCO CAPOLLI, ANDREA PINAMONTI, AND GARETH SPEIGHT
Definition 2.13.
A jet F of order m on K is a Whitney field of class C m on K if, for every 0 ≤ k ≤ m , we have( R ma F ) k ( b ) = o ( | a − b | m − k )as | a − b | → a, b ∈ K .We now recall the classical Whitney extension theorem in the special case thatthe domain is a subset of R [31]. Theorem 2.14 (Classical Whitney extension theorem) . Let K be a closed subsetof an open set U ⊂ R . Then there is a continuous linear mapping W from the spaceof Whitney fields of class C m on K to C m ( U ) such that D k ( W F )( x ) = F k ( x ) for ≤ k ≤ m and x ∈ K, and W F is C ∞ on U \ K . We now recall the Whitney extension theorem for C m horizontal curves in H from [24]. Suppose F, G, H are jets of order m on K ⊂ R . For a, b ∈ K , we definethe area discrepancy A ( a, b ) := H ( b ) − H ( a ) − Z ba (( T ma F ) ′ ( T ma G ) − ( T ma G ) ′ ( T ma F ))(2.7) + 2 F ( a )( G ( b ) − T ma G ( b )) − G ( a )( F ( b ) − T ma F ( b ))and the velocity (2.8) V ( a, b ) := ( b − a ) m + ( b − a ) m Z ba ( | ( T ma F ) ′ | + | ( T ma G ) ′ | ) . Theorem 2.15.
Let K ⊂ R be compact and F, G, H be jets of order m on K . Then ( F, G, H ) extends to a C m horizontal curve ( f, g, h ) : R → H if and only if(1) F, G, H are Whitney fields of class C m on K ,(2) For ≤ k ≤ m the following equation holds at all points of K (2.9) H k = 2 k − X i =0 (cid:18) k − i (cid:19) ( F k − i G i − G k − i F i ) , (3) A ( a, b ) /V ( a, b ) → uniformly as ( b − a ) → with a, b ∈ K . Finally we state for future use the following fact about polynomials from [24].
Lemma 2.16.
Let P be a polynomial of degree n , a < b , and k P k ∞ := max [ a,b ] | P | .Then n k P k ∞ ≤ − Z ba | P | ≤ k P k ∞ . Facts about Approximate Derivatives and L Derivatives
In this section we prove several lemmas which will be useful later in the paper.
Lemma 3.1.
Let f : [ a, b ] → R be absolutely continuous and m ≥ .Suppose f ′ is m − times L differentiable at a point x ∈ ( a, b ) with L derivativegiven by the polynomial P m − f,x of degree at most m − . Then f is m times L differentiable at x with L derivative Q mf,x of degree at most m defined by Q mf,x ( y ) := f ( x ) + R yx P m − f,x ( t ) d t . USIN APPROXIMATION IN THE HEISENBERG GROUP 7
Proof.
Denote P = P m − f,x and define Q = Q mf,x by Q mf,x ( y ) := f ( x ) + R yx P ( t ) d t .Let ε >
0. From the definition of L differentiability we have for all sufficientlysmall ρ > − Z B ( x,ρ ) | f ′ ( t ) − P ( t ) | d t ≤ ερ m − / . Absolute continuity gives for all y ∈ B ( x, ρ ), | f ( y ) − Q ( y ) | = (cid:12)(cid:12)(cid:12)(cid:12) f ( x ) + Z yx f ′ ( t ) d t − (cid:18) f ( x ) + Z yx P ( t ) d t (cid:19)(cid:12)(cid:12)(cid:12)(cid:12) = (cid:12)(cid:12)(cid:12)(cid:12)Z yx ( f ′ ( t ) − P ( t )) d t (cid:12)(cid:12)(cid:12)(cid:12) ≤ Z B ( x,ρ ) | f ′ ( t ) − P ( t ) | d t ≤ ερ m . Hence given ε >
0, we have for all sufficiently small ρ > − Z B ( x,ρ ) | f ( y ) − Q ( y ) | d y ≤ − Z B ( x,ρ ) ερ m = ερ m . This proves the lemma. (cid:3)
Lemma 3.2.
Suppose ( f, g, h ) : [ a, b ] → H is an absolutely continuous horizontalcurve in H and f ′ , g ′ are m − times L differentiable at a point x ∈ [ a, b ] for some m ≥ . Then h is m times L differentiable at x . More precisely, denote R ( y ) := h ( x ) + 2 Z yx ( P ′ Q − Q ′ P ) , where P, Q are the L derivatives of order m of f, g respectively which exist byLemma 3.1. Let e R be the polynomial of degree at most m such that R ( y ) − e R ( y ) isdivisible by ( y − x ) m +1 . Then e R is the L derivative of h of order m at x .Proof. Let R be defined as in the statement of the lemma. Fix 0 < ε <
1. Thenthere exists δ > < ρ < δ we have − Z B ( x,ρ ) | f − P | ≤ ερ m , − Z B ( x,ρ ) | f ′ − P ′ | ≤ ερ m − , and − Z B ( x,ρ ) | g − Q | ≤ ερ m , − Z B ( x,ρ ) | g ′ − Q ′ | ≤ ερ m − . Let 0 < ρ < δ and y ∈ B ( x, ρ ). We estimate as follows, using the fact ( f, g, h ) is ahorizontal curve and (2.1), h ( y ) − R ( y ) = h ( x ) + 2 Z yx h ′ − h ( x ) − Z yx ( P ′ Q − Q ′ P )= 2 Z yx (( f ′ g − P ′ Q ) + ( Q ′ P − g ′ f )) . MARCO CAPOLLI, ANDREA PINAMONTI, AND GARETH SPEIGHT
We estimate the first term as follows (cid:12)(cid:12)(cid:12)(cid:12) Z yx ( f ′ g − P ′ Q ) (cid:12)(cid:12)(cid:12)(cid:12) ≤ Z B ( x,ρ ) | f ′ g − P ′ Q | = 4 ρ − Z B ( x,ρ ) | f ′ g − P ′ Q | . Since f ′ g − P ′ Q = ( f ′ − P ′ ) g + P ′ ( g − Q ) and g, P ′ are continuous hence boundedon [ a, b ], we can continue our estimate as follows4 ρ − Z B ( x,ρ ) | f ′ g − P ′ Q | ≤ ρ k g k ∞ − Z B ( x,ρ ) | f ′ − P ′ | + k P ′ k ∞ − Z B ( x,ρ ) | g − Q | ! ≤ ρ (cid:0) k g k ∞ ερ m − + k P ′ k ∞ ερ m (cid:1) ≤ Cερ m for a constant C independent of y and ρ . The estimate of 2 R yx ( Q ′ P − g ′ f ) is similar.Hence we obtain | h ( y ) − R ( y ) | ≤ Cερ m for all 0 < ρ < δ . Consequently − Z B ( x,ρ ) | h − R | ≤ Cερ m . To conclude we notice that if e R is the polynomial of degree at most m defined inthe statement of the lemma then for some constant C independent of ρ < − Z B ( x,ρ ) | h − e R | ≤ − Z B ( x,ρ ) | h − R | + − Z B ( x,ρ ) | e R − R |≤ − Z B ( x,ρ ) | h − R | + Cρ m +1 . Hence − Z B ( x,ρ ) | h − e R | = o ( ρ m )so h is m times L differentiable at x with derivative e R . (cid:3) We next prove Proposition 3.4 which shows that approximate differentiabilityalmost everywhere leads to Whitney fields on large compact sets. Our argument isadapted from [20] where a similar result is proved under slightly different assump-tions. As in [20] we need the following lemma by De Giorgi [5].
Lemma 3.3 (De Giorgi) . Let E be a measurable subset of the ball B ( x, r ) in R n such that L n ( E ) ≥ Ar n for some constant A > . Then for each m ∈ N there is apositive constant C , depending only on n,m and A , such that for each polynomial p of degree at most m and for every multi-index α | D α p ( x ) | ≤ Cr n + | α | Z E | p ( y ) | dy. Proposition 3.4.
Let u : [ a, b ] → R be measurable and m times approximatelydifferentiable almost everywhere. Let the approximate derivative at almost everypoint x be denoted by P mu,x ( y ) = m X i =0 u i ( x ) i ! ( y − x ) i . USIN APPROXIMATION IN THE HEISENBERG GROUP 9
Then for every ε > there exists a compact set K ⊂ [ a, b ] with L ([ a, b ] \ K ) ≤ ε such that Γ = ( u i ) mi =0 is a C m Whitney field on K .Proof. It is proven in [20] that all the functions u i are measurable under the givenhypotheses. Let 0 < δ < < ε < x ∈ [ a, b ] where u is approximately differentiable and r >
0, define W ( x, r ) := { y ∈ [ a, b ] ∩ [ x − r, x + r ] : | u ( y ) − P mu,x ( y ) | > δ | x − y | m } . Each set W ( x, r ) is measurable because all the u i are measurable. We can write W ( x, r ) = { y ∈ [ a, b ] : ( x, y ) ∈ T ( r ) } , where T ( r ) := { ( x, y ) ∈ [ a, b ] × [ a, b ] : | x − y | < r, | u ( y ) − P mu,x ( y ) | > δ | x − y | m } . Since T is measurable, it follows x
7→ L ( W ( x, r )) is a measurable function of x .For n ∈ N define the sets(3.1) B n := { x ∈ [ a, b ] : L ( W ( x, r )) ≤ r/ r ≤ /n } . Since x
7→ L ( W ( x, r )) is a measurable function of x and L ( W ( x, r )) is monotonicin r for each fixed x , it is easy to show that the sets B n are measurable. Clearly B n ⊂ B n +1 for every n . Since u is m times approximately differentiable almosteverywhere, it follows L ([ a, b ] \ S ∞ n =1 B n ) = 0. Consider two points x, y ∈ B n with x ≤ y and | x − y | ≤ /n . Let r = | y − x | and define the measurable sets S ( x, y ) := [ x, y ] \ ( W ( x, r ) ∪ W ( y, r )) . Then L ( S ( x, y )) ≥ | y − x | − L ( W ( x, r )) − L ( W ( y, r )) ≥ r/ . Define the polynomial q := P mu,y − P mu,x . For z ∈ S ( x, y ) we estimate | q ( z ) | as follows | q ( z ) | ≤ | P mu,y ( z ) − u ( z ) | + | u ( z ) − P mu,x ( z ) | ≤ δ ( | z − y | m + | x − z | m ) ≤ δr m . We apply De Giorgi’s Lemma to the polynomial q with E = S ( x, y ) and A = 1 / k | D k q ( y ) | = | u k ( y ) − D k P mu,x ( y ) | ≤ Cr k Z S ( x,y ) | q ( z ) | d z ≤ Cδr m − k . Recall ε > L ([ a, b ] \ S ∞ n =1 B n ) = 0 and the sets B n areincreasing, we may choose N ∈ N such that L ([ a, b ] \ B N ) ≤ ε/
2. We then choose K a compact subset of B N with L ([ a, b ] \ K ) ≤ ε . Now we recall the dependenceof K on ε, δ and denote K = K ( ε, δ ) and N = N ( ε, δ ). The set K ( ε, δ ) has thefollowing two properties for a constant C depending only on m :(1) L ([ a, b ] \ K ( ε, δ )) ≤ ε ,(2) For every 0 ≤ k ≤ m and x, y ∈ K ( ε, δ ) with | x − y | ≤ /N ( ε, δ ) we have | u k ( y ) − D k P mu,x ( y ) | ≤ Cδ | x − y | m − k . We now put our compact sets together. Fix ε > K = ∞ \ n =1 K ( ε/ n , /n ) . Using (1) for the sets K ( ε/ n , /n ), we estimate the measure of K as follows L ([ a, b ] \ K ) ≤ ∞ X n =1 L ([ a, b ] \ K ( ε/ n , /n )) ≤ ∞ X n =1 ε/ n = ε. Using (2) for the sets K ( ε/ n , /n ), we see that K has the following property.Whenever 0 ≤ k ≤ m and x, y ∈ K satisfy | x − y | ≤ N ( ε/ n , /n ) for some n ∈ N , | u k ( y ) − D k P mu,x ( y ) | ≤ C | x − y | m − k /n. In other words, for every 0 ≤ k ≤ m we have | u k ( y ) − D k P mu,x ( y ) | = o ( | x − y | m − k )as | x − y | → x, y ∈ K . Hence Γ = ( u i ) mi =0 is a C m Whitney field on K . (cid:3) C m Horizontal Lusin Approximation for Horizontal Curves with L Differentiable Velocity
In this section we prove our first main theorem. Before giving the statement wefirst recall that if f : [ a, b ] → R is m times L differentiable at a point x ∈ [ a, b ],then we denote the L derivative at x by P mf,x ( y ) = m X i =0 f i ( x ) i ! ( y − x ) i , where f i ( x ) ∈ R for 0 ≤ i ≤ m . Also, if a function f : [ a, b ] → R is absolutelycontinuous and f ′ is m − L differentiable at a point x ∈ [ a, b ], then f is m times L differentiable at x with derivative given by Lemma 3.1. Theorem 4.1.
Let
Γ = ( f, g, h ) : [ a, b ] → H be an absolutely continuous horizontalcurve such that f ′ and g ′ are m − times L differentiable at almost every pointof [ a, b ] . Then Γ has the m -Lusin property. Further, for every η > there is a C m horizontal curve e Γ = ( e f , e g, e h ) : [ a, b ] → H such that L m [ k =0 { x ∈ [ a, b ] : e f k ( x ) = f k ( x ) or e g k ( x ) = g k ( x ) or e h k ( x ) = h k ( x ) } ! < η. Proof.
Using Lemma 3.1 it follows that f and g are m times L differentiable almosteverywhere. By Lemma 3.2 we also know that h is m times L differentiable almosteverywhere. At almost every x ∈ [ a, b ] denote the L derivative of f by P mf,x ( y ) = m X k =0 f k ( x ) k ! ( y − x ) k where the f k are measurable functions by [20]. Similarly for the L derivatives p mg,x and p mh,x with coefficients g k ( x ) and h k ( x ) at almost every point x which aremeasurable functions of x .Fix η >
0. Choose a compact set K ⊂ [ a, b ] satisfying L ([ a, b ] \ K ) < η withthe following properties: USIN APPROXIMATION IN THE HEISENBERG GROUP 11 (1) the jets
F, G, H defined on K by F k = f k | K , G k = g k | K and H k = h k | K for 0 ≤ k ≤ m are Whitney fields of class C m on K .(2) For every ε > δ > a, b ∈ K with | b − a | < δ then(4.1) − Z ba | f ′ − ( T ma F ) ′ | ≤ ǫ ( b − a ) m − and − Z ba | g ′ − ( T ma G ) ′ | ≤ ǫ ( b − a ) m − . The first property above is possible using from Proposition 3.4. To obtain thesecond property we use the almost everywhere ( m −
1) times L differentiability of f ′ and g ′ , Lemma 3.1, elementary measure theory, and the fact that P mf,a = T ma F and P mg,a = T ma G . We now show that the hypotheses of Theorem 2.15 hold for thejets F, G, H on the compact set K . Verification of Theorem 2.15(1).
This follows directly from the definition of K . Verification of Theorem 2.15(2).
We need to check (2.9), which we recall states H k = 2 k − X i =0 (cid:18) k − i (cid:19) ( F k − i G i − G k − i F i ) on K for 1 ≤ k ≤ m. Fix a ∈ K and let T F = T ma F , T G = T ma G , T H = T ma H for simplicity. UsingLemma 3.2, we know( P mh,a ) ′ = 2(( P mf,a ) ′ ( P mg,a ) − ( P mg,a ) ′ ( P mf,a )) + S ′ a ( y ) , where S a ( y ) is a polynomial divisible by ( y − a ) m +1 . Hence( T H ) ′ = 2(( T F ) ′ ( T G ) − ( T G ) ′ ( T F )) + S ′ a ( y ) . Differentiating the Taylor polynomials as was done to derive (2.9) yields(
T H ) k = 2 k − X i =0 (cid:18) k − i (cid:19) (( T F ) k − i ( T G ) i − ( T G ) k − i ( T F ) i )+ S ka on K for 1 ≤ k ≤ m, where the polynomial S ka ( y ) is divisible by ( y − a ) m +1 − k . In particular, S ka ( a ) = 0for 1 ≤ k ≤ m . Since the Taylor polynomials are based at a , ( T F ) i ( a ) = F i ( a ) for0 ≤ i ≤ m and similarly for G and H . Hence substituting in a we obtain H k ( a ) = 2 k − X i =0 (cid:18) k − i (cid:19) ( F k − i ( a ) G i ( a ) − G k − i ( a ) F i ( a ))for all a ∈ K and 1 ≤ k ≤ m as required. Verification of Theorem 2.15(3).
Given 0 < ε < δ > a, b ∈ K with 0 < b − a < δ . For ease of notation we write T F = T ma F and T G = T ma G . For simplicity we will consider only the case F ( a ) = G ( a ) = H ( a ) = 0 . Otherwise one can use the translation invariance of A ( a, b ) and V ( a, b ) as in [24].Since F ( a ) = G ( a ) = H ( a ) = 0, A ( a, b ) is of the form A ( a, b ) = H ( b ) − H ( a ) − Z ba (( T F ) ′ T G − T F ( T G ) ′ ) . Since ( f, g, h ) is a horizontal curve, we have H ( b ) − H ( a ) = h ( b ) − h ( a ) = 2 Z ba ( f ′ g − f g ′ ) . We estimate | A ( a, b ) | as follows (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) H ( b ) − H ( a ) − Z ba (( T F ) ′ T G − T F ( T G ) ′ ) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ≤ Z ba | f ′ g − ( T F ) ′ T G | + Z ba | f g ′ − T F ( T G ) ′ | ! . We will show how to estimate the first term after the inequality, the second onewill follow by changing the roles of f and g . First we pass to the average Z ba | f ′ g − ( T F ) ′ T G | = ( b − a ) − Z ba | f ′ g − ( T F ) ′ T G | and then we decompose the argument as f ′ g − ( T F ) ′ T G = ( f ′ − ( T F ) ′ )( g − T G ) + ( f ′ − ( T F ) ′ ) T G + ( g − T G )( T F ) ′ . We then obtain( b − a ) − Z ba | f ′ g − ( T F ) ′ T G | ≤ ( b − a ) " − Z ba | f ′ − ( T F ) ′ | ! || g − T G || ∞ + − Z ba | f ′ − ( T F ) ′ | ! || T G || ∞ + − Z ba | g − T G | ! || ( T F ) ′ || ∞ . From (4.1) we obtain − Z ba | f ′ − ( T F ) ′ | ! ≤ ǫ ( b − a ) m − . Absolute continuity of g , the Fundamental Theorem of Calculus, and (4.1) gives || g − T G || ∞ ≤ ( b − a ) − Z ba | g ′ − ( T G ) ′ | ! ≤ ǫ ( b − a ) m . Using Lemma 2.16 we have || ( T F ) ′ || ∞ ≤ C − Z ba | ( T F ) ′ | for some constant C ≥ m . Using T G ( a ) = G ( a ) = g ( a ) = 0and again the Fundamental Theorem of Calculus, we have || T G || ∞ ≤ Z ba | ( T G ) ′ | . USIN APPROXIMATION IN THE HEISENBERG GROUP 13
Combining all together we get Z ba | f ′ g − ( T F ) ′ T G | ≤ ǫ ( b − a ) m + ǫ ( b − a ) m Z ba | ( T G ) ′ | + Cǫ ( b − a ) m Z ba | ( T F ) ′ | . ≤ CεV ( a, b )By doing the same computation with f and g switched we obtain | A ( a, b ) | ≤ CεV ( a, b )whenever a, b ∈ K with 0 < b − a < δ . This yields Theorem 2.15(3). Conclusion.
We have shown that the jets
F, G, H satisfy the hypotheses ofTheorem 2.15 on the compact set K . Hence Γ = ( F, G, H ) extends to a C m horizontal curve e Γ = ( e f , e g, e h ) : [ a, b ] → H satisfying e f k | K = F k , e g k | K = G k , e h k | K = H k for 0 ≤ k ≤ m. From the definition of the compact set K and the jets F, G, H we have L m [ k =0 { x ∈ [ a, b ] : e f k ( x ) = f k ( x ) or e g k ( x ) = g k ( x ) or e h k ( x ) = h k ( x ) } ! ≤ L ([ a, b ] \ K ) < η. This completes the proof of the theorem. (cid:3) A Horizontal Curve with no Lusin Approximation
In this section we prove our second main theorem, which justifies the hypothesesof Theorem 5.1 and highlights the difference between the settings of Euclidean spaceand the Heisenberg group.
Theorem 5.1.
There exists
Γ = ( f, g, h ) : [0 , → H which is absolutely continuousand horizontal with the following properties:(1) Almost everywhere the maps f, g, h are twice L p differentiable for all p ≥ ,(2) Almost everywhere the maps f ′ , g ′ , h ′ are once approximately differentiable,(3) Γ does not admit a C horizontal Lusin approximation. We use the remainder of this section to prove Theorem 5.1.5.1.
Construction of the Horizontal Curve.
Parameters for the Construction.
Fix decreasing sequences h n , λ n > ∞ X n =1 n λ n < ∞ , h n /λ n → , n h n → ∞ , λ n +1 ∞ X k = n +1 k − n h k → . One possible choice is h n = 1 / n and λ n = (2 / n . A consequence of (5.1) is(5.2) ∞ X n =1 n h n < ∞ . We next fix a decreasing sequence w n > w n ≤ / n , λ n +1 ∞ X k = n +1 k − n w k → , and(5.4) 1 λ p +1 n +1 ∞ X k = n +1 k − n w k h pk → p ≥ . This is possible since w n can be chosen very small compared to h n and λ n .5.1.2. The Sets I n and I . For each n ≥ I n ⊂ [0 , I is the openinterval with center 1 / w . Once I , I , . . . , I n are defined, we define I n +1 as the union of those open intervals J with the following properties: • J has center k/ n +1 for some integer k with 0 < k < n +1 , • J has radius w n +1 , • J does not intersect I ∪ I ∪ · · · ∪ I n .Define I = ∪ ∞ n =1 I n . The set I n is a disjoint union of at most 2 n − intervals oflength 2 w n . Hence, since w n ≤ / n ,(5.5) m ( I ) ≤ ∞ X n =1 m ( I n ) ≤ ∞ X n =1 n w n ≤ / . Definition of the Horizontal Components.
We now define f, g : [0 , → R which will be the first two components of the curve. In [0 , \ I we set f and g tobe identically 0. Otherwise we proceed as follows. Suppose J is one of the finitelymany disjoint open intervals chosen in the definition of I n for some n ≥
1. Divide J into 4 adjacent disjoint equally sized intervals labelled from left to right J = ( p , p ) , J = [ p , p ] , J = [ p , p ] , J = ( p , p ) . The maps f, g are piecewise linear functions in J defined as follows:(1) In J , f is identically 0 and g is linear with g ( p ) = 0, g ( p ) = h n .(2) In J , f is linear with f ( p ) = 0, f ( p ) = h n and g is identically h n .(3) In J , f is identically h n and g is linear with g ( p ) = h n , g ( p ) = 0.(4) In J , f is linear with f ( p ) = h n , f ( p ) = 0 and g is identically 0.5.1.4. Absolute Continuity of the Horizontal Components.
Clearly f and g are dif-ferentiable at all but finitely many points of I n for each n , hence at all but countablymany points of I . Our first task is to prove that f and g are differentiable at almostevery point of [0 , \ I . Before doing so we prove a lemma which roughly statesthat at almost every point of [0 , \ I the maps f and g do not see ‘big jumps’unexpectedly close to x .For x ∈ R and S ⊂ R we denote d ( x, S ) := inf {| x − y | : y ∈ S } . For n ≥
1, define A n = { x ∈ [0 , \ I : d ( x, I ∪ · · · ∪ I n ) < λ n } and let A := lim sup A n ⊂ [0 , \ I. By definition of the limit superior, for any x ∈ [0 , \ ( I ∪ A ), there exists N ( x ) > n > N ( x ) implies d ( x, I ∪ · · · ∪ I n ) ≥ λ n . Roughly speaking, this states that if x ∈ [0 , \ ( I ∪ A ) then on small scales nearto x one sees only relatively small intervals. We will use this fact repeatedly later. Lemma 5.2.
The set A has Lebesgue measure zero. USIN APPROXIMATION IN THE HEISENBERG GROUP 15
Proof.
The set I i consists of 2 i − intervals and is contained in I . Hence m ( { x / ∈ I : d ( x, I i ) < λ n } ) ≤ i − λ n = 2 i λ n . Hence m ( A n ) = (2 + 2 + · · · + 2 n ) λ n = 2 λ n (2 n − . Since P ∞ n =1 n λ n < ∞ it follows P ∞ n =1 m ( A n ) < ∞ . The Borel Cantelli lemmagives the conclusion. (cid:3) Lemma 5.3.
For every x ∈ (0 , \ ( I ∪ A ) , f and g are differentiable at x with f ′ ( x ) = g ′ ( x ) = 0 .Proof. Fix a point x as in the statement of the lemma and corresponding N ( x ) > n > N ( x ) implies d ( x, I ∪ · · · ∪ I n ) ≥ λ n . For all t sufficiently small there is n > N ( x ) such that λ n +1 ≤ | t | < λ n . Then d ( x, I ∪ · · · ∪ I n ) ≥ λ n > | t | . This implies x + t / ∈ I ∪ · · · ∪ I n . By definition of f we see 0 ≤ f ( x + t ) ≤ h n +1 .Since x / ∈ I we have f ( x ) = 0 and so (cid:12)(cid:12)(cid:12)(cid:12) f ( x + t ) − f ( x ) t (cid:12)(cid:12)(cid:12)(cid:12) ≤ h n +1 λ n +1 . Since h n /λ n →
0, it follows that f is differentiable at x with f ′ ( x ) = 0. Theargument is the same for g . (cid:3) We have now shown that f and g are differentiable almost everywhere on [0 , Proposition 5.4.
The maps f, g : [0 , → R are absolutely continuous.Proof. Suppose J is one of the intervals chosen in the construction of I n for some n ≥
1. Then for any x ∈ J we have | f ′ ( x ) | ≤ h n / ( w n /
4) = 4 h n /w n . Since m ( J ) ≤ w n it follows that R J | f ′ | ≤ h n . Since there are at most 2 n − disjoint intervals in the construction of I n , we have R I n | f ′ | = 2 n +2 h n for every n ≥
1. Since f ′ = 0 almost everywhere outside I , we deduce, Z | f ′ | ≤ ∞ X n =1 n +2 h n < ∞ . Hence f ′ is integrable on [0 , f ( b ) − f ( a ) = Z ba f ′ whenever a < b. Clearly (5.6) is satisfied if a and b belong to a common chosen interval J fromthe definition of I . Indeed, f is piecewise linear and hence absolutely continuousinside any such interval. Suppose this is not the case. By splitting the integral ifnecessary, to prove (5.6) we may assume a, b / ∈ I . If J = [ c, d ] is any interval chosenin the construction of I which is contained in ( a, b ), then Z J f ′ = f ( d ) − f ( c ) = 0 . There are countably many such intervals and f ′ = 0 at almost every point outside I . Hence R ba f ′ = 0. Since a, b / ∈ I we have f ( b ) = f ( a ) = 0. Hence (5.6) holds.This proves that f is absolutely continuous. The argument for g is the same. (cid:3) Vertical Component of the Curve.
Since f, g are bounded and f ′ , g ′ are inte-grable, the products f ′ g and g ′ f are integrable. We define h : [0 , → R by h ( x ) := 2 Z x ( f ′ g − g ′ f ) for x ∈ [0 , . Clearly h is absolutely continuous on [0 , f, g, h ) is anabsolutely continuous horizontal curve. It is easy to check that h is piecewise linearsince each interval chosen in the construction of I . We also record the followingfact for later. Lemma 5.5.
Suppose J = [ a, b ] is one of the connected components of I n . Then h ( b ) − h ( a ) = 4 h n . Proof.
Since f ( a ) = f ( b ) = 0 we know h ( b ) − h ( a ) = 2 Z ba ( f ′ g − g ′ f ) = 4 Z ba f ′ g. From the construction of f and g and the fact ( b − a ) / w n / h ( b ) − h ( a ) = 4( w n / h n / ( w n / h n = 4 h n . (cid:3) Differentiability of the Horizontal Curve.Proposition 5.6.
At almost every point x ∈ [0 , , the maps f, g, h : [0 , → R aretwice L p differentiable at x for all p ≥ . For every point x ∈ (0 , \ ( I ∪ A ) , thesecond order L p derivatives of f, g, h at x are identically f ( x ) = 0 , g ( x ) = 0 , and h ( x ) (possibly non-zero) respectively.Proof. Recall that f, g, h are piecewise linear inside each of the countably manyintervals whose disjoint union is I . Hence f, g, h are twice L p differentiable for all p ≥ I . Suppose x / ∈ ( I ∪ A ). To show f istwice L p differentiable at x we will show that for every p ≥ t → t p +1 Z [ x − t,x + t ] | f ( y ) | p d y = 0 . Using the definition of A , we may choose N ( x ) > n > N ( x ) implies d ( x, I ∪ · · · ∪ I n ) ≥ λ n . Recall that P ∞ n =1 n λ n < ∞ which implies λ n ≤ / n for all sufficiently large n .Given any t > n > N ( x ) with λ n +1 ≤ t < λ n ≤ / n . This implies [ x − t, x + t ] ∩ ( I ∪ · · · ∪ I n ) = ∅ . The interval [ x − t, x + t ] has length at most 2 λ n ≤ / n − . Since the intervalsin I k have centers separated by at least distance 1 / k , it follows that [ x − t, x + t ] USIN APPROXIMATION IN THE HEISENBERG GROUP 17 can intersect at most 2 k − n +1 intervals from I k for k > n . Recall that x / ∈ I gives f ( x ) = 0, | f ( y ) | ≤ h k for y in an interval from I k , and that t > λ n +1 . We have1 t p +1 Z [ x − t,x + t ] | f ( y ) | p d y = 1 t p +1 Z [ x − t,x + t ] ∩ I | f ( y ) | p d y ≤ t p +1 ∞ X k = n +1 k − n +1 w k h pk ≤ λ p +1 n +1 ∞ X k = n +1 k − n +1 w k h pk . The previous line converges to 0 as n → ∞ for every p ≥ w k , h k , λ k . The argument for g is exactly the same. Finally to show h istwice L p differentiable at x we will show that for every p ≥ t → t p +1 Z [ x − t,x + t ] | h ( y ) − h ( x ) | p d y = 0 . Recall that [ x − t, x + t ] can intersect at most 2 k − n +1 intervals from I k for k > n .Hence, using Lemma 5.5, for any y ∈ [ x − t, x + t ] we have | h ( y ) − h ( x ) | ≤ ∞ X k = n +1 k − n +3 h k . Hence 1 t p +1 Z [ x − t,x + t ] | h ( y ) − h ( x ) | p d y ≤ t p ∞ X k = n +1 k − n +3 h k ! p ≤ λ n +1 ∞ X k = n +1 k − n h k ! p . We conclude by noticing the last line converges to 0 as n → ∞ for every p ≥ (cid:3) Proposition 5.7.
The maps f ′ , g ′ , h ′ are once approximately differentiable almosteverywhere. In particular, f ′ and g ′ have approximate derivative at every pointof (0 , \ ( I ∪ A ) .Proof. Approximate differentiability of f ′ , g ′ , h ′ at all but countably many pointsof I is clear since f, g, h are piecewise linear inside each interval chosen during theconstruction of I . Recall that f ′ ( x ) = g ′ ( x ) = 0 for every point x ∈ (0 , \ ( I ∪ A ).Fix such an x . Choose corresponding N ( x ) > n > N ( x ) implies d ( x, I ∪ · · · ∪ I n ) ≥ λ n . As in the proof of Proposition 5.6, given any t > n > N ( x ) such that λ n +1 ≤ t < λ n ≤ / n , which implies [ x − t, x + t ] ∩ ( I ∪ · · · ∪ I n ) = ∅ . Again it follows that [ x − t, x + t ] can intersect at most 2 k − n +1 intervals from I k for k > n . Recalling that f ′ ( x ) = 0 at every point of (0 , \ ( I ∪ A ), we have m { y ∈ [ x − t, x + t ] : f ′ ( y ) > } t ≤ m ([ x − t, x + t ] ∩ I )2 t ≤ λ n +1 ∞ X k = n +1 k − n +2 w k . Since the previous line converges to 0 as n → ∞ , it follows f ′ is approximatelydifferentiable at x with approximate derivative 0. The argument for g is the same.For h we recall that h ′ = 2( f ′ g − g ′ f ) almost everywhere. Combining this with thefact f ′ ( x ) = g ′ ( x ) = 0 for every point x ∈ (0 , \ ( I ∪ A ) gives h ′ ( x ) = 0 for almostevery x ∈ (0 , \ ( I ∪ A ). For such x the same argument as above applies, givingthe desired conclusion. (cid:3) No C Horizontal Lusin Approximation.Proposition 5.8.
The curve Γ does not have the C horizontal Lusin approxima-tion property. We will prove Proposition 5.8 by contradiction. Suppose Γ does have the C horizontal Lusin approximation property. Fix θ > / /
31 and a C horizontalcurve e Γ = (
F, G, H ) : [0 , → H such that the set E := { t ∈ [0 ,
1] : e Γ( t ) = Γ( t ) } satisfies m ( E ) > θ . Since m ( I ) < /
31 by (5.5), we have m ( E \ I ) > / Lemma 5.9.
Suppose x ∈ E \ I is a Lebesgue density point of E \ I . Then F ( x ) = F ′ ( x ) = F ′′ ( x ) = 0 and G ( x ) = G ′ ( x ) = G ′′ ( x ) = 0 . Proof.
Let x be as in the statement of the lemma. Then F ( x ) = f ( x ) because x ∈ E and f ( x ) = 0 because x / ∈ I ; hence F ( x ) = 0. Since x is a Lebesgue densitypoint of E \ I there exist x n ∈ E \ I with x n → x . By the same argument as beforewe have F ( x n ) = 0 for every n . Hence x n → x and F ( x n ) = 0 = F ( x ) for every n .Since F is C this implies F ′ ( x ) = F ′′ ( x ) = 0. The argument for G is the same. (cid:3) Since e Γ is C , F ′′ and G ′′ are uniformly continuous on [0 , δ > | F ′′ ( x ) − F ′′ ( y ) | < | G ′′ ( x ) − G ′′ ( y ) | < | x − y | < δ. Lemma 5.10.
Suppose a, b ∈ E \ I are Lebesgue density points of E \ I and | b − a | < δ . Then | H ( b ) − H ( a ) | ≤ | b − a | . Proof.
Since F ( a ) = F ( b ) = 0 by Lemma 5.9 and e Γ is horizontal, we have H ( b ) − H ( a ) = 2 Z ba ( F ′ G − G ′ F ) = 4 Z ba F ′ G. We have F ( a ) = F ′ ( a ) = F ′′ ( a ) = 0 by Lemma 5.9 and | F ′′ ( t ) − F ′′ ( a ) | < t ∈ [ a, b ] by (5.7). Hence | F ′ ( t ) | ≤ b − a and | F ( t ) | ≤ ( b − a ) for t ∈ [ a, b ]. Thesame estimates hold for G . This gives | H ( b ) − H ( a ) | ≤ b − a )( b − a )( b − a ) = 4( b − a ) . (cid:3) USIN APPROXIMATION IN THE HEISENBERG GROUP 19
Lemma 5.11.
For all sufficiently large n ∈ N , there exists a pair x, y ∈ (0 , withthe following properties: • x, y ∈ E \ I and are Lebesgue density points of E \ I , • | x − y | ≤ / n , • x, y are on opposite sides of an interval chosen in the construction of I n +1 .Proof. We argue by contradiction. Assume there exist arbitrarily large n ∈ N forwhich there is no pair x and y with the desired properties. Fix such an n . Weconsider intervals of the form [ L/ n , ( L + 1) / n ] for different integers 0 ≤ L < n .Suppose the interval [ L/ n , ( L + 1) / n ] has midpoint (2 L + 1) / n +1 which is thecenter of an interval J chosen in the construction of I n +1 . The interval J separates[ L/ n , ( L + 1) / n ] \ J into two subintervals J and J each of measure greater than(1 / / n ). Since there is no pair x, y ∈ (0 ,
1) with the properties in the statementof the lemma, in particular there is no such pair in the interval [ L/ n , ( L + 1) / n ].Hence either J or J does not contain any points of E \ I which are Lebesguedensity points of E \ I . Hence we have m (( E \ I ) ∩ [ L/ n , ( L + 1) / n ]) ≤ (2 / m ([ L/ n , ( L + 1) / n ]) . We now estimate the total measure of those intervals [ L/ n , ( L + 1) / n ] whosemidpoint (2 L + 1) / n +1 is not chosen in the construction of I n +1 . Fix such aninterval [ L/ n , ( L + 1) / n ]. Then B ((2 L + 1) / n +1 , w n +1 ) ∩ ( I ∪ · · · ∪ I n ) = ∅ . Different intervals of the form B ( k/ n +1 , w n +1 ) are separated by a distance1 / n +1 − w n +1 ≥ / n +1 − / n ≥ / n +2 . If an interval of length T intersects K intervals of the form B ( k/ n +1 , w n +1 ) thenwe must have T ≥ K/ n +2 , so K ≤ n +2 T . The set I i is a union of 2 i − intervalsof length 2 w i . Hence the number of intervals B ( k/ n +1 , w n +1 ) which intersect I ∪ · · · ∪ I n can be estimated by n X i =1 i − n +2 w i = 2 n +2 n X i =1 i w i . Hence the total measure of all those intervals [ L/ n , ( L + 1) / n ] whose midpoint isnot chosen in the construction of I n +1 can be estimated by(1 / n )2 n +2 n X i =1 i w i = 4 n X i =1 i w i ≤ ∞ X i =1 i / i = 4 / . Let I G be the collection of intervals [ L/ n , ( L + 1) / n ] whose midpoint is thecenter of an interval chosen in the construction of I n +1 . Let I B be those intervals [ L/ n , ( L + 1) / n ] whose midpoint is not chosen. We estimate as follows m ( E \ I ) = X L ∈ G m (( E \ I ) ∩ [ L/ n , ( L + 1) / n ])+ X L ∈ B m (( E \ I ) ∩ [ L/ n , ( L + 1) / n ]) ≤ X L ∈ G (2 / m ([ L/ n , ( L + 1) / n ]) + X L ∈ B m ([ L/ n , ( L + 1) / n ]) ≤ / / ≤ / . Since m ( E \ I ) > / (cid:3) We now derive a contradiction which proves Proposition 5.8. Recall δ > n h n → ∞ . Using Lemma 5.11, we may fix n with 1 / n < δ and 4 n h n +1 ≥ x, y ∈ (0 ,
1) with x < y such that • x, y ∈ E \ I and are Lebesgue density points of E \ I , • | x − y | ≤ / n , • x, y are on opposite sides of an interval chosen in the construction of I n +1 .Since | x − y | ≤ / n < δ and x, y ∈ E \ I are Lebesgue density points of E \ I , wehave by Lemma 5.10(5.8) | H ( y ) − H ( x ) | ≤ | y − x | ≤ / n . Since x < y are on opposite sides of an interval chosen in the construction of I n +1 ,we have by Lemma 5.5(5.9) h ( y ) − h ( x ) ≥ h n +1 . Since x, y ∈ E we have H ( y ) − H ( x ) = h ( y ) − h ( x ). Combining this with (5.8) and(5.9) gives h n +1 ≤ / n or equivalently 4 n h n +1 ≤
1. This contradicts the choiceof n with 4 n h n +1 ≥
2, proving Proposition 5.8 and hence proving Theorem 5.1.
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The Whitney Extension Theorem for C Horizontal Curvesin the Heisenberg Group , Journal of Geometric Analysis 28(1) (2018), 61–83. (Marco Capolli)
Department of Mathematics, University of Trento, Via Sommarive14, 38123 Povo (Trento), Italy
E-mail address , Marco Capolli: [email protected] (Andrea Pinamonti)
Department of Mathematics, University of Trento, Via Sommarive14, 38123 Povo (Trento), Italy
E-mail address , Andrea Pinamonti:
[email protected] (Gareth Speight)
Department of Mathematical Sciences, University of Cincinnati, 2815Commons Way, Cincinnati, OH 45221, United States
E-mail address , Gareth Speight:, Gareth Speight: