© Hindawi Publishing Corp. HAUSDORFF MEASURE OF THE SINGULAR SET OF QUASIREGULAR MAPS ON CARNOT GROUPS

Recently, the theory of quasiregular mappings on Carnot groups has been developed intensively. Let ν stand for the homogeneous dimension of a Carnot group and let m be the index of the last vector space of the corresponding Lie algebra. We prove that the ( ν − m − 1 ) -dimensional Hausdorff measure of the image of the branch set of a quasiregular mapping on the Carnot group is positive. Some estimates of the local index of quasiregular mappings are also obtained.

1. Introduction and statement of main results.Quasiregular mappings, or, in another terminology, mappings with bounded distortion, were firstly introduced and investigated by Reshetnyak in a series of papers that began to appear in 1966 [29,31].His work was handed and furthered by several analysts, among them Martio, Rickman, Väisälä [22,23,32], and others [1,28,33] later.The analytic definition of a quasiregular mapping is similar to the analytic definition of a quasiconformal one, with the exception of the requirement of homeomorphism.
Recently, the analysis on homogeneous groups (the Carnot groups) has been developed intensively.The fundamental role of such groups in analysis was pointed out by Stein [34] in his address to the International Congress of Mathematicians in 1970 (see also his monograph [35]).Briefly, a homogeneous group is a simply connected nilpotent Lie group, whose Lie algebra admits a grading.There is a natural family of dilations on the group under which the metric behaves like the Euclidean metric under the Euclidean dilation [7,10].The analysis on the homogeneous group is also a test ground for the study of general subelliptic problems arising from vector fields X 1 ,...,X k satisfying the Hörmander hypoellipticity condition [17].
Quasiconformal mappings on a homogeneous group of a special type were initially considered by Mostow [24] in 1971 in connection with rigidity theorems for the rank one symmetric space.Various definitions of quasiconformal mappings can be found in [12,15,18,37,38,39].In the works [13,16,19,20,37,39,40,42], analytical foundations of the theory of quasiregular mappings on Carnot groups were developed.We should note that in [13,16] quasiregular mappings were studied with an additional smoothness condition.An alternative approach to this theory on the Heisenberg group was given in the works [8,9].
The next analytic definition of a quasiregular mapping on a Carnot group G is the basic in this paper.Definition 1.1.Let Ω ⊂ G be a domain.A mapping f : Ω → G is said to be a quasiregular mapping (mapping with bounded distortion) if (1) f is continuous open and discrete, (2) f belongs to HW 1 ν,loc (Ω), (3) the formal horizontal differential D H f satisfies the condition for almost all x ∈ Ω.
The smallest constant K is called the outer coefficient of quasiregularity and is denoted by K O (f ).If f : Ω → G is a quasiregular mapping, then there is a constant K such that for almost all x ∈ Ω.The smallest constant K is called the inner coefficient of quasiregularity and is denoted by K I (f ).The relationships K O (f ) ≤ CK ν−1 I (f ) and K I (f ) ≤ CK ν−1 O (f ) are realized, where C is some constant.In the case G = R n , the constant C is equal to 1. Also we use the notation 1 is compatible with the definition of the quasiregular mapping in the Euclidean space.The definition of D H f and more details can be found in Section 2.
We denote by B f the branch set of f , that is, Homotopy considerations (see [23]) imply that the (n − 2)-dimensional Hausdorff measure of f (B f ) is strictly positive.By definition, a quasiregular mapping on the Carnot group is a discrete open map, that yields immediately a similar inequality for the set f (B f ).In the present paper, we give a more precise estimate for the Hausdorff measure of f (B f ), namely, we prove the next theorem.
Theorem 1.2.Let Ω ⊂ G be an open connected set, and let f : Ω → G be a nonconstant quasiregular mapping such that the set of branch points B f is not empty.Then Here m is an index of the last vector space in the graduated Lie algebra Ᏻ of G.
Some estimates for the local index of quasiregular mappings on Carnot groups are also obtained.

Notes and preliminary results.
Let Ᏻ be a Lie algebra and let G be a corresponding simply connected Lie group.If U and V are some sets from Ᏻ, then we denote by [U , V ] the subspace of the algebra Ᏻ generated by the elements By induction, we define the following series: (2.1) We call a Lie algebra to be graduated if it splits into the direct sum of vector spaces and only finitely many elements do not vanish.A Lie algebra Ᏻ is called stratified if Ᏻ is graduated and the subspace V 1 ⊂ Ᏻ generates Ᏻ as an algebra.For the step m nilpotent Lie algebra Ᏻ, we have A Lie group is stratified and nilpotent if the corresponding Lie algebra is so.
A Carnot group G is a stratified simply connected nilpotent Lie group with the Lie algebra Ᏻ.Let X 11 ,...,X 1n 1 be a basis of the vector space From now on, we call V 1 the horizontal space.Since the vector fields X 11 ,...,X 1n 1 generate the Lie algebra Ᏻ, we can choose a basis We see that the collection X 11 ,X 12 ,...,X 1n 1 satisfies the Hörmander hypoellipticity condition [17].
It is known [10] that if G is a simply connected nilpotent Lie group with the Lie algebra Ᏻ, then the exponential map exp : Ᏻ → G is a diffeomorphism from the Lie algebra Ᏻ to the Lie group G. Thus, dx • exp −1 is a bi-invariant Haare measure on G, where dx is the Lebesgue measure on Ᏻ.We can identify the elements x ∈ G of the group with the elements x ∈ Ᏻ of the algebra, and thus, with are called the coordinates of the point x.There is a natural group of dilations, which is defined by the rule We use the Carnot-Carathéodory metric based on the length of horizontal curves.A piecewise curve γ : [0,b] → G is said to be horizontal if its tangent vector γ(s) belongs to the space V 1 , that is, there exist functions a j (s), such that γ(s) = n 1 j=1 a j (s)X 1j γ(s) . (2. 3) The result of [6] implies that one can connect two arbitrary points x, y ∈ G by a horizontal curve.We fix on V 1 a nondegenerate quadratic form •, • so that the vector fields X 11 (x),...,X 1n 1 (x) are orthonormal with respect to this form at every x ∈ G.The length l(γ) of a curve γ is defined by the formula The Carnot-Carathéodory distance d c (x, y) is the infimum of the length over all horizontal curves connecting x and y ∈ G. Since the quadratic form is left invariant, the Carnot-Carathéodory metric is left invariant as well.The group G is connected, therefore the metric d c (x, y) is finite (see [36]).We also use a metric, which is generated by a homogeneous norm.The homogeneous norm on G is, by definition, a continuous nonnegative function ρ(•) on G such that ρ(x) = ρ(x −1 ), ρ(δ r (x)) = r ρ(x), and ρ(x) = 0 if and only if x = 0, where 0 denotes the identity element of G.The homogeneous norm is not uniquely defined, however, any two homogeneous norms are equivalent.We fix the homogeneous norm ρ(x) of the element x = (x 1 ,...,x m ), x i ∈ V i , which is prescribed by the formula The homogeneous norm defines the homogeneous metric by the rule ρ( the open ball with the center at x and of radius r > 0 in the metric ρ.Note that B(x, r ) is the left translation by x of the ball B(0,r ), which is the image under δ r of the "unit ball" B(0, 1).The Hausdorff dimension of the metric space (G,d c ) coincides with its homogeneous dimension ν.By |E|, we denote the measure of the set E: |E| = E dx.Our normalizing condition is such that the balls of radius one are of measure one; |B(0, 1)| = B(0,1) dx = 1.Since the Jacobian determinant of the dilation δ r is r ν , we have that The Euclidean space R n with the standard structure is an example of the Abelian group; the exponential map is the identity and the vector fields X i = ∂/∂x i , i = 1,...,n, have only trivial commutative relations and form the basis of the corresponding Lie algebra.
The simplest example of a non-Abelian group is the Heisenberg group H n .The underlying space is R 2n+1 with the group law of multiplication x n+j x j − x j x n+j . (2.6) Here, x, x ∈ R 2n , t, t ∈ R. The Lie algebra Ᏻ of the Heisenberg group H n is generated by the left-invariant vector fields the tangent space at each point of the group.The vector fields X j , j = 1,...,2n, form a basis of the horizontal vector space V 1 and span{T } = V 2 .Thus, the Hiesenberg group is a two-step Carnot group.An example of the homogeneous norm is ρ(x, t) = (|x| 4 + t 2 ) 1/4 .The homogeneous dimension ν is equal to 2n + 2. Now we define an absolutely continuous function on curves of the horizontal fibration that is formed by the horizontal vector fields X 1j , j = 1,...,n 1 (see terminology, for instance, in [18,37,43]).Definition 2.1.A function u : Ω → R, Ω ⊂ G, is said to be absolutely continuous on lines (u ∈ ACL(Ω)) if for any domain U,U ⊂ Ω, and any fibration ᐄ defined by the left-invariant vector fields X 1j , j = 1,...,n 1 , the function u is absolutely continuous on γ ∩ U with respect to the Ᏼ 1 -Hausdorff measure for dγ-almost all curves γ ∈ ᐄ.
The Sobolev space for any test function ϕ ∈ C ∞ 0 and of finite norm (2.9) The quantity ∇ ᏸ u = (X 11 u,...,X 1n 1 u) is called the subgradient of u.For a smooth function, the subgradient is equal to the projection of the usual Riemannian gradient on V 1 .We will say that u belongs to (2) the coordinate functions f ij belong to ACL(Ω) for all i and j, (3) belongs to the subspace V 1 for almost all x ∈ Ω and k = 1,...,n 1 .
In [37,40], the reader can find various definitions of the Sobolev space and their relationships.
The matrix X 1k f = (X 1k f ij ) defines an operator D H f which is called the formal horizontal differential.It maps the horizontal space into the horizontal space [27].A mapping f ∈ HW 1  p,loc (Ω) which possesses this property is called the (weakly) contact mapping.The norm of the operator D H f is defined by It has been proved in [43] (see also [26,40]) that the formal horizontal differential D H f generates a homomorphism Df : Ᏻ → Ᏻ of the algebra Ᏻ, which is called a formal differential.The norm of the operator Df is defined in a usual way as  [30,31] and G = H n [8,39], inequality (1.1) implies the openness and discreteness of quasiregular mappings.The question about topological properties of quasiregular mappings on an arbitrary Carnot group satisfying only inequality (1.1) is open (see discussion in [9,13,39]).
The definition and main results on differentiability on Carnot groups are due to Pansu [26,27].
Theorem 2.4.A quasiregular mapping f : Ω → G, Ω ⊂ G, possesses the following properties: (1) ᏼ-differentiable almost everywhere in Ω; (2) Lusin's ᏺ-property: We use the notation U(x,s) for the x-component of the set Making use of the method of the paper [22], we can show that if C ⊂ Ω is a compact, then the function (x, s) l (x, s) is continuous and (x, s) L (x, s) is lower semicontinuous on the set C × (0,t).
For the mapping f : Ω → G, we denote by the linear distortion and the inverse linear distortion of f at x ∈ Ω, respectively.A subdomain D ⊂ Ω is called a normal domain under a mapping f : The next topological facts can be proved for a rather broad class of metric spaces.For the Euclidean space R n , the proof can be found in [14,22,32].By the symbol CD we will denote the complement to the domain D. Lemma 2.5.Let f : Ω → G be a continuous open discrete and sense preserving mapping.Then for a point x ∈ Ω, there exists σ x > 0 such that for any 0 < s ≤ σ x , the following conditions hold: (1) U(x,s) is the normal neighborhood of x; (2) (4) the sets CU(x,s) and CU(x,s) are connected; (5) U(x,s)\ U(x,t) is a ring-type domain for 0 < t < s ≤ σ x ; (6) l (x, L(x, r )) = L (x, l(x, r )) = r for r ∈ (0,l (x, σ x )).
If U = U(x,s) is a normal neighborhood of x, then the degree µ(y, f , U) of continuous open discrete mapping f with respect to y = f (x) is constant for any point z ∈ f (U) and is denoted by µ(f , U).The definition of the mapping degree can be found in [30].In the same work, it was proved that for a compact subdomain G ⊂ U , the degree µ(y, f , G), y = f (x), x ∈ U , has the same value as for U , which is called the local index i(x, f ) of the point x ∈ U.The index i(x, f ) is equal to 1 if and only if x is not a branch point.
We define the quantity and call it a multiplicity function.If a set A is such that A is compact and A ⊂ Ω, then the discreteness implies that N(f , A) is finite.Some relations between the multiplicity function, the mapping degree, and the local index are expressed in the following lemma.Its proof for the Carnot group can be obtained as in [

and only if U is the normal neighborhood of a point
x ∈ U.

Estimates of the measure of the image of the branch set. The value
is called the Hausdorff α-measure.We denote by diam ρ (B i ) the diameter of the set B i with respect to the homogeneous metric, and by diam • (B i ) the diameter under the Euclidean metric.The Hausdorff α-measure with respect to these different metrics we note by Ᏼ α ρ and Ᏼ α • , respectively.For the beginning, we give a simple proposition on the positiveness of the Hausdorff (N −2)-measure of the image of the branch set of quasiregular mappings on Carnot groups.

Lemma 3.1. Let f : Ω → G be a quasiregular mapping on a Carnot group with
Proof.From [23], it follows that for an open discrete map in the Euclidean space R N , the following estimate Ᏼ N−2 • (f (B f )) > 0 holds.We cover f (B f ) by open balls {B i }, i = 1, 2,....We have the inequality • ≤ ρ by the definition of the homogeneous metric ρ.Then Now, we give more precise estimates for the Hausdorff measure of the image of the branch set under quasiregular mappings on Carnot groups.First of all, we describe a set P z (E).We need this construction in the following lemma.Let z, ω be disjoint points on the group G.We denote by σ z ω (t) = zδ t (ω) the oneparametric curve which starts from the point z and passes through the point ω.We let E be a set on G and let z ∉ E. The collection of the one-parametric curves σ z ω (t), ω ∈ E, we denote by P z (E).
Here m is the index of the last vector space in the graduated Lie algebra Ᏻ of G.
Proof.Since the Haar measure is invariant with respect to the left translation, we can assume that z = 0, and from now on we omit the letter z.Moreover, up to the end of the proof, the notations B ρ (x, r ), S ρ (x, r ) mean the ball and the sphere, respectively, under the homogeneous metric ρ, and B • (x, r ), S • (x, r ) mean the ball and the sphere with respect to the Euclidean metric.
We will use the notations the lemma will be proved if we establish the equality Ᏼ α+m ρ (P k ) = 0 for any k ∈ N. Since Ᏼ α ρ (E k ) = 0 for any k, we cover E k by a system of balls B ρ (x i ,r i ), such that (1) i r α i < ε, for any ε ∈ (0, 1).We can assume that the intersection E k ∩ B ρ (x i ,r i ) is not empty for any index i, otherwise we omit such balls.Let H i denote the set P (B ρ (x i ,r i )) ∩ R k .Then P k ⊂ i H i .We fix i ∈ N. Since the homogeneous norm of the point x i satisfies the equalities then a point x ∈ σ x i (t) is located between the points x i ∈ S ρ (0, 1/k) and x i ∈ S ρ (0,k).We want to estimate the Euclidean length of the part of the curve σ x i (t) which lies in R k .The parametric equation of σ x i (t) in the Euclidean coordinates of R N is of the form σ (1)  x i (t) where ) are the coordinates of the point x i in the vector space V l .The Euclidean length l(σ k and does not depend on i ∈ N.Here we have used the estimates We divide the arc σ k x i into parts of equal Euclidean length which does not exceed C • (k 2 r i ) m < 1 by points x i = s i0 ,s i1 ,...,s iN i = x i .Here C is a constant from [25, Proposition 1.1], which depends on k.Let N i be the biggest positive integer such that N i ≤ L/(C • (k 2 r i ) m ) + 1.To this partition of the curve corresponds the following partition of the interval J i : The collection of the balls The first assertion follows (see [25,Proposition 1.1]) from the inclusions for some t ∈ [T j−1 ,T j ].Furthermore, the first statement implies that the point s belongs to B ρ (s ij ,k 2 r i ).From this and from the triangle inequality, we have ρ(s ij ,y) ≤ ρ(s ij , s) + ρ(s, y) ≤ 2k 2 r i .Thus y ∈ B ρ (s ij , 2k 2 r i ) and the second assertion is proved.From the statement above, we obtain that the countable collection of the balls with respect to the homogeneous metric {B ρ (s ij , 2k 2 r i )}, j = 1,...,N i , i ∈ N, covers the set P k .Finally, we have The proof of the next theorem is based on the idea of [11, Theorem 1] and [30,Theorem 10.3].

Theorem 3.3.
Let Ω ⊂ G be an open connected set, and let f : Ω → G be a nonconstant quasiregular mapping such that the set of branch points B f is not empty.Then Here m is the index of the last vector space in the graduated Lie algebra Ᏻ of G.
Proof.We suppose, contrary to our claim, that Ᏼ ν−(m+1) ρ (f (B f )) = 0. We choose x 0 ∈ B f and we put y 0 = f (x 0 ).Let σ x 0 > 0 be the number from Lemma 2.5 such that B(x 0 ,σ x 0 ) ⊂ Ω, and let l 0 = (1/2)l(x 0 ,σ x 0 ).Then U = U(x 0 ,l 0 ) is the normal neighborhood of x 0 , and the inclusions B(y We may assume that {t k } converges to t 0 ∈ U. Since B f is closed in Ω and U ⊂ Ω, then t 0 ∈ B f and we have f (t 0 ) = z by the continuity of f .The ball B(y 0 ,l 0 ) does not contain points of the boundary ∂U, We set E = f (B f ∩ U).By the assumption Ᏼ ν−(m+1) ρ (E) = 0, there is a point z ∈ B(y 0 ,l 0 ) such that z ∉ E. We construct the set P z (E).Lemma 3.2 implies that Ᏼ ν−1 ρ (P z (E)) = 0.The set E is closed in B(y 0 ,l 0 ), and the intersection B(y 0 ,l 0 ) ∩ P z (E) is also closed.Thus H = B(y 0 ,l 0 ) \ P z (E) is open.We show that the set H is simple connected.Really, since z ∉ E, any closed curve γ can be shrunk into the point z along the one-parametric curves σ z ω (t), which are contained in H and connect z with the points ω ∈ γ.
Let W ⊂ U be a connected component of f −1 (H), which is contained in U. Then W ⊂ B(x 0 ,σ x 0 ), f (W ) = H, f (∂W ) ⊂ ∂H, and the restriction f | W is a quasiconformal mapping [42,44].
If W 1 ,W 2 ,...,W k are connected components of f −1 (H) ∩ U and f −1 k is the inverse mapping to the restriction f | W k , then f −1 k is a quasiconformal mapping on H = B(y 0 ,l 0 ) \ P z (E).Since we have Ᏼ ν−1 ρ (P z (E)) = 0, [37,Theorem 8] is a homeomorphism.This contradicts the fact that the set U contains branch points of f .4. Bounds for the local index.We mention some auxiliary definitions and results.Let U be an open set on the Carnot group G and let V be a compact in U .The ordered pair (U , V ) is called a condenser.
is called the p-capacity of the condenser (U , V ).The infimum is taken over all nonnegative functions v ∈ C ∞ 0 (U ) such that v| V ≥ 1.
The properties of the p-capacity in the spaces with the geometry of vector fields satisfying the Hörmander hypoellipticity condition can be found in [4,5].If U is an open ball and V is a closed ball with the same center and of smaller radius, then such a condenser (U , V ) is called the ring condenser.Lemma 4.2.Let Ω be a domain in G. Then for the condenser (B(x, R), B(x, r )), r < R, there are constants θ 1 , θ 2 , independent of r and R, such that The proof of Lemma 4.2 follows from [5, Theorems 6.6 and 6.9].
In the next lemma, some properties of the ν-capacity of some condensers are given.They have been established in [42] for more general mappings with bounded s-distortion.We formulate them for the case s = ν.
We show that the inverse linear distortion H (x, f ) of the quasiregular mapping f is bounded above, and the estimate does not depend on x ∈ Ω. Lemma 4.4.Let f : Ω → G be a quasiregular mapping and K(f ) be a coefficient of quasiregularity.Then where the constant C depends on the homogeneous dimension ν and on K(f ).
From the equality i(x k ,f ) = i(x 0 ,f ) and Lemma 2.6, we deduce that U(x k , L k ), L k = L(x k ,r k ), is the normal neighborhood of x k .Our supposition and Lemma 4. The images of B(x k ,r k ) under the mapping f cover the set f (F i,j ∩ U(x 0 ,s)).From (4.18) and property (P2), we conclude that Thus Ᏼ α ρ (f (F i,j ∩ U(x 0 , s))) = 0, which contradicts (1.3).
Some similar estimates of the index of a point were obtained in [21] for the case G = R n .Corollary 4.6.Let f : Ω → G be a nonconstant quasiregular mapping, and let F ⊂ B f be a continuum.Then and set E = {x ∈ B f : i(x, f ) ≥ ν ν−1 } is totally disconnected.
Proof.If F is a continuum, then Ᏼ 1 ρ (F ) > 0 and the first assertion follows from Theorem 4.5 with α = 1.
If we assume that E contains a continuum, then we get the contradiction with the first statement.Here m is the index of the last vector space of the graduated Lie algebra Ᏻ of G.

Corollary 4 . 7 .
Let Ω ⊂ G be an open connected set, and let f : Ω → G be a nonconstant quasiregular mapping such that B f is not empty.Theninf x∈F i(x, f ) < K I (f map is called open if the image of an open set is an open set, and discrete if the preimage of any point consists of isolated points.It is known that for [40,43]is equal to |D H f | for almost all x ∈ Ω[40,43].The determinant of the matrix Df (x) is denoted by J(x, f ) and is called the Jacobian of f .A