Note on coarea formulae in the Heisenberg group

Publ. Mat. 48 (2004), 409–422 NOTE ON COAREA FORMULAE IN THE HEISENBERG GROUP Valentino Magnani Abstract We show a first nontrivial example of coarea formula for vectorvalued Lipschitz maps defined on the three dimensional Heisenberg group. In this coarea formula, integration on level sets is performed with respect to the 2-dimensional spherical Hausdorff measure, built by the Carnot-Carath´eodory distance. The standard jacobian is replaced by the so called “horizontal jacobian”, corresponding to the jacobian of the Pansu differential of the Lipschitz map. Joining previous results, we achieve all possible coarea formulae for Lipschitz maps defined on the Heisenberg group. 1. Introduction The study of sub-Riemannian Geometry is recently carried out in several areas of Mathematics, such as Differential Geometry, PDEs, Geometric Measure Theory, Sobolev spaces and Geometric Control Theory. An account on these developments can be found for instance in [1], [10], [11] and [17]. Aim of this note is to show the first nontrivial example of coarea formula for vector-valued maps, whose domain is a noncommutative stratified group endowed with its natural sub-Riemannian structure. Coarea formulae for real-valued maps on stratified groups and the more general Carnot-Carath´eodory spaces have been largely studied by several authors in different contexts, [8], [9], [15], [16], [18], [20], [21]. Most of these results hold for functions of bounded variation, where the notion of perimeter measure plays a central role. In our case this notion cannot be employed since level sets have codimension higher than one. Moreover, the choice of target may affect even the existence of nontrivial coarea formulae, [13]. As main result of this note we obtain the following 2000 Mathematics Subject Classification. 28A75 (22E25). Key words. Coarea formula, Heisenberg group. 410 V. Magnani coarea formula (1) ZA u(x)JHf(x)dx =ZR2 Zf−1(t)∩A u(y)dS2 H3(y)!dt, where u:A−→ [0,+∞] is a measurable function, Ais a measurable subset of the Heisenberg group H3, and f:A−→ R2is a Lipschitz function with respect to the Euclidean distance. Heisenberg group certainly is the simplest model of stratified group, [24]. The “sub-Riemannian” features of (1) are the horizontal jacobian JHfand the spherical Hausdorff measure S2 H3with respect to the Carnot-Carath´eodory distance. The horizontal jacobian corresponds to the jacobian of the matrix representing the Pansu differential (Definition 2.1) and the Carnot-Carath´eodory distance is the control distance associated to the horizontal distribution of H3(Section 2). These two objects are strictly related, as formulae (13) and (14) show. The measure S2 H3only detects the non-horizontal part of level sets and the choice of JHfsurprisingly fits this property. Lipschitz functions with respect to the Euclidean distance are also Lipschitz with respect to the Carnot-Carath´eodory distance, but the converse is not true. This naturally raises the question of extending (1) to Lipschitz maps with respect to the Carnot-Carath´eodory distance of H3. The difficulty of this problem clearly appears in examples of Lipschitz maps with respect to the Carnot-Carath´eodory distance which are nowhere differentiable on a set of full measure, [15]. Coarea formula (1) fits into the general coarea formula stated in [13], whose validity for arbitrary stratified groups is still an open problem. Nonetheless, formula (1) allows us to complete the picture of all possible coarea formulae for Lipschitz maps defined on H3, as we show in Theorem 5.2. In ending, although our proof of coarea formula suggests a clear pattern for its extension to higher dimensional Heisenberg groups, a number of new difficulties appears in this case, as we explain in Remark 4.4. In this perspective, the present note becomes the first step to understand more general coarea formulae in higher dimensional stratified groups, where the intriguing geometry of higher codimensional sets is a new terrain for further investigations. 2. A digest of basic notions We begin this section introducing the 3-dimensional Heisenberg group. This is a simply connected Lie group H3whose Lie algebra h3is endowed with a basis (X1, X2) satisfying the nontrivial bracket relations [X1, X2] = 2 X3. We will identify the Lie algebra h3with the Coarea Formulae in the Heisenberg Group 411 isomorphic Lie algebra of left invariant vector fields of H3. The exponential map exp: h3−→ H3is a diffeomorphism, then it is possible to introduce global coordinates on H3. We consider F:R3−→ H3defined by (2) F(x) = expx1X1+x2X2+x3X3. We will assume throughout that a system of coordinates defined by (2) is fixed. This allows us to identify H3with R3 . The vector fields (X1,X2,X3) with respect to our coordinates read as X1=∂x1−x2∂x3,X2=∂x2+ x1∂x3and X3=∂x3. The group operation is represented by the formula (3) x·y=x1+y1, x2+y2, x3+y3+x1y2−x2y1. A natural family of dilations which respects the group operation (3) can be defined by setting δr(x) = (rx1, rx2, r2x3) for every r > 0. In fact, the map δr:H3−→ H3defined above is a group homomorphism with respect to the operation (3). Our frame (X1, X2, X3) admits a dual basis (dx1, dx2, ϑ) of one-forms on H3, where the contact form ϑcan be explicitly written as (4) ϑ=dx3+x2dx1−x1dx2. The vector fields X1,X2span a smooth distribution of 2-dimensional planes, which define all horizontal directions of H3. A point γ(t) of a differentiable curve γ: [a, b]−→ H3is characteristic if γ0(t) is a horizontal direction and it is called transverse otherwise. Absolutely continuous curves which are a.e. characteristic are called horizontal curves, [1]. The sub-Riemannian metric structure of H3is obtained fixing a left invariant Riemannian metric on H3and defining the Carnot-Carath´eodory distance between two points as the infimum over Riemannian lengths of horizontal curves joining these points. Vector fields X1and X2satisfy the Lie bracket generating condition, therefore the Chow theorem implies that every couple of points is joined by at least one horizontal curve, see for instance [1, p. 15]. As a result, the Carnot-Carath´eodory distance is well defined. Through coordinates (2) we can introduce the one dimensional Hausdorff measure H1on H3with respect to the Euclidean distance in R3. This measure clearly depends on our coordinates, however our final results will be formulated in intrinsic terms. We will assume throughout that Lipschitz functions on subsets of H3are considered with respect to the Euclidean distance of H3. The symbol |·| will denote the Euclidean norm. By contrast with Analysis in Euclidean spaces, where the Euclidean distance is the most natural choice, in the Heisenberg group 412 V. Magnani several distances have been introduced for different purposes. However, all of these distances are “homogeneous”, namely, they are left invariant and satisfy the relation ρ(δry, δrz) = r ρ(y, z) for every y, z ∈H3 and r > 0. To simplify notations we write ρ(x, 0) = ρ(x), where 0 denotes either the origin of R3or the unit element of H3. The open ball of center xand radius r > 0 with respect to a homogeneous distance is denoted by Bx,r. The Carnot-Carath´eodory distance is an important example of homogeneous distance. However, all of our computations hold for a general homogeneous distance, therefore in the sequel ρwill denote a homogeneous distance, if not stated otherwise. Note that the Hausdorff dimension of H3with respect to any homogeneous distance is four. Before introducing the next definition we recall that any L:H3−→ Rk is a G-linear map if it is a group homomorphism satisfying the homogeneity property L(δrx) = rL(x) for every x∈H3and r > 0. Note that G-linear maps are also linear in the usual sense, as we identifiy H3 with R3. G-linear maps constitute the family of intrinsic differentials, as we clarify in the following definition. Definition 2.1 (P-differentiability).Let f: Ω −→ Rk, where Ω is an open subset of H3. We say that fis P-differentiable at x∈Ω if there exists a G-linear map L:H3−→ Rksuch that |f(x·h)−f(x)− L(h)|ρ(h)−1−→ 0 as ρ(h)→0. The G-linear map Lwith the previous property is uniquely defined and it is called the P-differential of fat x. We use the notation Df(x) to indicate the P-differential L. The notion of P-differentiability has been introduced by Pansu in the more general framework of stratified groups, [22]. One can check by direct computation that f: Ω −→ Rkis P-differentiable at x∈Ω if it is differentiable at xin the usual sense. Note that the converse is not true. The k×3 matrix representing Df(x) can be written as follows (5) Df(x) =       X1f1(x)X2f1(x) 0 X1f2(x)X2f2(x) 0 . . .. . .. . . X1fk(x)X2fk(x) 0       . We denote by ∇f(x) the k×3 matrix (fi xj)i=1,...,k j=1,2,3representing the standard differential df(x) of fat x. The horizontal jacobian JHf(x) of f at xis defined by taking the standard jacobian of the matrix (5). The standard jacobian of fat xis denoted by Jf(x). The Lebesuge measure Coarea Formulae in the Heisenberg Group 413 of a measurable subset Ain H3is denoted by |A|and the d-dimensional spherical Hausdorff measure Sdis always considered with respect to the fixed homogeneous distance ρ. Note that our definition of spherical Hausdorff measure differs from the standard one of [6], in that the volume of the d-dimensional ball ωdis replaced by one. The reason for this choice clearly appears in Corollary 3.2, where the “natural” dimensional constant 2/ρ(0,0,1)2in the definition of S2 H3replaces ω1= 2. 3. Intrinsic measure of transverse curves The present section is devoted to the blow-up of C1curves with respect to a homogeneous distance. As a consequence, we achieve formula (13), corresponding to the integral representation of the 2-dimensional spherical Hausdorff measure of a transverse curve. This formula has been first obtained by Pansu, [20]. To make this note more selfcontained, here we recall its proof. In the sequel, Ω will denote an open subset of H3and coordinates (2) will be understood, then we will identify Ω with an open subset of R3. Theorem 3.1. Let γ⊂H3be a one-dimensional immersed submanifold of class C1and let x∈γ. If γis transverse at x, then kϑ(x)k>0and the limit (6) lim r→0+ H1(γ∩Bx,r) r2=c kϑ(x)k holds, where c= 2/ρ(0,0,1)2. Proof: Let us denote by the same symbol γ:J−→H3a local parametrization of the immersed submanifold γnear the point x, such that γ(0) = x and Jis an open neighbourhood of zero. Defining the subset Ix,r = {t∈J|ρ(γ(t), x)< r}, we have H1(γ∩Bx,r) = ZIx,r |γ0(t)|dt, then the change of variable t=r2τyields (7) H1(γ∩Bx,r) r2=Zr−2Ix,r |γ0(r2τ)|dτ, where we have defined r−2Ix,r ={τ∈r−2J|ρ(γ(r2τ), x)< r}. The left invariance of ρand the homogeneity of dilations yield r−2Ix,r ={τ∈r−2J|ρδ1/r x−1γ(r2τ)<1}. 414 V. Magnani The group law (3) allows us to compute the components of δ1/rx−1γ(r2τ)  in R3, obtaining (8) [δ1/r x−1γ(r2τ)]j=γj(r2τ)−γj(0) r−→ 0 as r→0+, for every j= 1,2. Computing the third component we get [δ1/r x−1γ(r2τ)]3=γ3(r2τ)−γ3(0) −γ1(0)γ2(r2τ) + γ2(0)γ1(r2τ) r2 =γ3(r2τ)−γ3(0) −γ1(0)(γ2(r2τ)−γ2(0))+γ2(0)(γ1(r2τ)−γ1(0)) r2 and from the expression of the contact form (4) we conclude that (9) [δ1/r x−1γ(r2τ)]3−→ τ(γ0 3(0) −γ1(0)γ0 2(0) + γ2(0)γ0 1(0)) =τ ϑ(γ(0), γ0(0)). By definition of contact form a vector V∈TxH3is horizontal if and only if ϑ(x, V ) = 0, then ϑ(x, γ0(0)) 6= 0 and kϑ(x)k>0, because γis transverse at x. Limits (8) and (9) imply that for every t∈R\{τ0,−τ0} we have (10) 1r−2Ix,r (t)−→ 1I0 x,0(t) as r→0+, where τ0=|ϑ(x, γ0(0))|−1ρ(0,0,1)−2and (11) I0 x,0={τ∈R| |τϑ(x, γ0(0))|ρ(0,0,1)2<1}= (−τ0, τ0). Finally, formulae (7), (11) and limit (10) along with Lebesgue convergence theorem yield (12) H1(γ∩Bx,r) r2−→ 2τ0|γ0(0)|as r→0+. This completes the proof. Corollary 3.2 (Integral representation).Let γ⊂H3be a one-dimensional immersed submanifold of class C1which is S2-a.e. transverse. Then we have the formula (13) S2 H3(γ) = Zγ kϑ(x)kdH1(x), where c= 2/ρ(0,0,1)2and S2 H3=cS2. Coarea Formulae in the Heisenberg Group 415 Proof: It suffices to define the new measure µ=kϑ(x)kH1, then Theorem 3.1 along with standard differentiability theorems applied to µ, see for instance Theorem 2.10.17(2) and Theorem 2.10.18(1) of [6], lead us to our claim. Remark 3.3.Note that formula (13) can be also expressed with respect to any left invariant metric g, replacing the role of the Euclidean distance. In fact, we have the equalities Zγ kϑ(x)kgdH1 g=ZJ |ϑ(γ(t), γ0(t))| |γ0(t)|g |γ0(t)|gdt =ZJ |ϑ(γ(t), γ0(t))| |γ0(t)||γ0(t)|dt =Zγ kϑ(x)kdH1=S2 H3(γ), where H1 gis the one dimensional Hausdorff measure with respect to the Riemannian distance and | · |gdenotes the Riemannian norm. This remark emphasizes the auxiliary role of the Euclidean distance. 4. Coarea formula for vector valued maps The purpose of this section is to prove our main result stated in Theorem 4.3. To do this, the next theorem constitutes the key tool. Theorem 4.1. Let f: Ω −→ R2be a C1function, x∈Ωand assume that df(x): H3−→ R2is surjective. Then there exists a neighbourhood U of xsuch that for every ybelonging to the one-dimensional submanifold f−1(f(x)) ∩Uwe have (14) JHf(y) = kϑ(y)kJf(y). Proof: We denote by (∇f)i1i2the 2×2 submatrix of ∇fwith columns i1 and i2, and by Mi1i2(∇f) the minor det(∇f)i1i2. By hypothesis the matrix ∇f(x) has rank two, therefore we assume for instance that M13(∇f(x)) 6= 0. The implicit function theorem yields a C1immersion γ:J−→ H3such that γ(0) = xand f(γ(t)) = f(x) for every tbelonging to the open interval Jcontaining the origin. In addition, the curve γcan be represented as γ(t) = (γ1(t), t, γ3(t)), where γj:J−→ R is a C1function for j= 1,2. By a simple and elementary calculation, the differentation of equality f((γ1(t), t, γ2(t))) = f(x) leads us to the 416 V. Magnani formula (15) "γ0 1 γ0 3#=−1 M13 (∇f)"f2 x3−f1 x3 −f2 x1f1 x1#"f1 x2 f2 x2#, where we have explicitly written the inverse matrix ((∇f)13)−1. Expression (15) yields (16) γ0 1=−M23(∇f) M13(∇f)and γ0 3=−M12(∇f) M13(∇f). Using the definition of JHfand the explicit expressions of operators Xj one can achieve the following equality (17) JHf(x) = |M12(∇f(x)) + x1M13(∇f(x)) −x2M32(∇f(x))|. As a consequence of this formula, dividing both terms of the quotient JHf/Jf by |M13(∇f)|and using (16), we obtain (18) JHf(γ(t)) Jf (γ(t)) =|γ0 3(t)−γ1(t) + tγ0 1(t)| |γ0(t)|=|ϑ(γ(t), γ0(t))| |γ0(t)|=kϑ(γ(t)) k. Clearly, either possible cases M12(∇f(x)) 6= 0 or M23(∇f(x)) 6= 0 would lead us to the same formula, due to its intrinsic form. Remark 4.2.Note that in the statement of the next theorem the horizontal jacobian JHfis considered when fis defined on a measurale set instead of an open set. This refers to a slightly more general notion of P-differentiability, where interior points of the domain Aare replaced with density points. Even in this case the P-differential is uniquely defined, see Definition 7 and Proposition 2.2 of [12] for more details. Theorem 4.3 (Coarea formula).Let f:A−→ R2be a Lipschitz map, where A⊂H3is a measurable subset. Then for every measurable function u:A−→ [0,+∞]the formula (19) ZA u(x)JHf(x)dx =ZR2 Zf−1(t)∩A u(y)dS2 H3(y)!dt holds, where c= 2/ρ(0,0,1)2and S2 H3=cS2. Proof: We first prove (19) in the case fis defined on all of H3and is of class C1. Let Ω be an open subset of H3. In view of the Euclidean coarea formula we have (20) ZΩ u(x)Jf(x)dx =ZR2 Zf−1(t)∩Ω u(y)dH1(y)!dt, Coarea Formulae in the Heisenberg Group 417 where u: Ω −→ [0,+∞] is a measurable function, see for instance [6]. Now we define u(x) = JHf(x)1{Jf6=0}∩Ω(x)/Jf(x) and use (20), obtaining (21) ZΩ JHf(x)dx =ZR2 Zf−1(t)∩Ω JHf(x)1{Jf6=0}(x) Jf(x)dH1(y)!dt. The validity of (20) also implies that for a.e. t∈R2the set of points of f−1(t) where Jf vanishes is H1-negligible, then the previous formula becomes (22) ZΩ JHf(x)dx =ZR2 Zf−1(t)∩Ω JHf(x) Jf(x)dH1(y)!dt. By Theorem 2.7 of [13], for a.e. t∈R2we have that S2(Ct∩Ω) = 0, where we have defined Ct={y∈f−1(t)∩Ω|JHf(y) = 0}. As a result, from formulae (13) and (14) we have proved that for a.e. t∈R2the equalities Zf−1(t)∩Ω JHf(x) Jf(x)dH1(y) = S2 H3(f−1(t)∩Ω\Ct) = S2 H3(f−1(t)∩Ω) hold, therefore we have achieved (23) ZΩ JHf(x)dx =ZR2 S2 H3(f−1(t)∩Ω) dt. The arbitrary choice of Ω yields the validity of (23) also for arbitrary closed sets. Then, approximation of measurable sets by closed ones, Borel regularity of S2 H3and the coarea estimate 2.10.25 of [6] extend the validity of (23) to the following one (24) ZA JHf(x)dx =ZR2 S2 H3(f−1(t)∩A)dt, where Ais a measurable subset of H3. Now we consider the general case, where f:A−→ R2is a Lipschitz map defined on a measurable bounded subset Aof H3. Let f1:H3−→ R2be a Lipschitz extension of f, namely, f1|A=fholds. Due to the Whitney extension theorem (see for instance 3.1.15 of [6]) for every arbitrarily fixed ε > 0 there exists a C1function f2:H3−→ R2such that the open subset O= {z∈H3|f1(z)6=f2(z)}has Lebesgue measure less than or equal to ε. The map fis a.e. differentiable in the Euclidean sense, then it is also