Differentiability in the Sobolev space W 1 , n − 1

Let (cid:2) ⊂ R n be a domain, n ≥ 2. We show that a continuous, open and discrete mapping f ∈ W 1 , n − 1 loc ((cid:2), R n ) with integrable inner distortion is differentiable almost everywhere on (cid:2) . As a corollary we get that the branch set of such a mapping has measure zero.


Introduction
Suppose that ⊂ R n , n ≥ 2, is a domain and f : → R n a continuous, discrete and open mapping in the Sobolev space W 1,n−1 loc ( , R n ). A theorem of Gehring and Lehto asserts that if n = 2, then f is differentiable at almost every point [7]. For planar homeomorphisms the result was established earlier by Menchoff [29]. These results are false in higher dimensions. Indeed, if n ≥ 3, one can construct a nowhere differentiable homeomorphism in W 1,n−1 ( , R n ), see [2,Example 5.2].
In this paper we study sufficient conformality conditions that guarantee differentiability almost everywhere for discrete and open mappings in W 1,n−1 loc ( , R n ). Such conformality conditions are usually given in terms of a distortion function. There are several distortion functions, each having considerable interest in geometric function theory, see [16, §6.4]. The principal feature of these distortions is that they provide some control on the lower order minors of the differential matrix in terms of the determinant.
We call K I (·, f ) the inner distortion of f and K O (·, f ) the outer distortion of f . Note that a mapping of finite inner distortion does not have to be a mapping of finite outer distortion. For instance, the mapping f : R n → R n , n ≥ 3, f (x 1 , . . . , x n ) = (x 1 , 0, . . . , 0) is a mapping of finite inner distortion, but not a mapping of finite outer distortion. However, we are able to show that a discrete and open mapping with K I (·, f ) ∈ L 1 loc ( ) has a finite outer distortion almost everywhere. Our main result reads as follows.

Theorem 1.1 Suppose that
⊂ R n , n ≥ 2, is a domain. Let f ∈ W 1,n−1 loc ( , R n ) be a continuous, discrete and open mapping of finite inner distortion with K I ( · , f ) ∈ L 1 loc ( ). Then f is differentiable almost everywhere and has finite outer distortion. Theorem 1.1 is new even when f is assumed to be a homeomorphism. Moreover, we are able to relax the integrability assumption on the inner distortion, see Theorem 5.1. The sharpness of this refinement is given in Example 5.5. We show as a corollary of Theorem 1.1 that the branch set of f , i.e. the set of points where f is not a local homeomorphism, has measure zero, see Corollary 5.4. This leads to a natural generalization of theorem by Hencl and Koskela [ Our results have a strong connection to the study of the regularity of the inverse of a Sobolev homeomorphism. Under the minimal conformality assumptions these questions can be traced back to the works of Iwaniec and Šverák [17], Koskela and Onninen [21] and Astala et al. [1]. The study of the regularity of the inverse mapping under the natural Sobolev setting W 1,n loc goes back to the pioneering work of Hencl and Koskela [10], and has recently been further developed by several authors, see [11,12,30,31]. Regularity questions in the Sobolev space W 1,n−1 loc ( , R n ) were first studied by Csörnyei et al. [2]. For our purposes, one of the important observations that Csörnyei et al. made on their paper was that a homeomorphism in the Sobolev space W 1,n−1 loc ( , R n ) satisfies Lusin's condition (N ) on almost every hyperplane. By modifying their proof we are able to show that this is true also for a discrete and open mapping in the Sobolev space W 1,n−1 loc ( , R n ). It is important to notice that it is possible to study the regularity properties of mappings without assuming injectivity. For instance, by applying modulus and capacity methods, see [32,35,40], it is possible to answer to many regularity questions in terms of inner distortion without using any invertibility of mapping.
We say that a mapping f ∈ W 1,n loc ( , R n ) is a mapping of bounded distortion, or a quasiregular mapping, if it satisfies distortion inequality with some global constant 1 ≤ K < ∞. There has been a lot of study on mappings of bounded distortion, see [34] and [35], and mappings of finite distortion are natural generalizations of these mappings. By the fundamental theorem of Reshetnyak [34], every mapping of bounded distortion is continuous, i.e. has a continuous representative, and is either constant or discrete and open. However, this is not always the case with mappings of finite distortion. Especially, a mapping f ∈ W 1,n−1 loc ( , R n ) of finite inner distortion with K I (·, f ) ∈ L 1 loc ( ) does not have to be continuous, discrete or open. Thus, it is justifiable to assume these properties from f in Theorem 1.1. For more details about the sharp analytic assumptions that guarantee continuity, discreteness and openness for mappings of finite distortion, see [9,13,15,[17][18][19]25,26].
In the theory of mappings of bounded distortion one of the powerful tools to deal with noninjective mappings is the Poletsky inequality, see [ [21] by showing that a mapping of finite distortion with K I (·, f ) ∈ L 1 loc ( ) and f ∈ W 1,n loc ( , R n ) enjoys a Poletsky-type inequality. It was further shown that the regularity assumption f ∈ W 1,n loc ( , R n ) can be slightly relaxed, say to |D f | n log −1 (e + |D f |) ∈ L 1 loc ( ). These results were based on a duality argument, relying on integration by parts against the Jacobian determinant. This method does not work when we only assume f ∈ W 1, p loc ( , R n ), for some p < n. By using the duality of p-capacity we are now able to establish Poletsky-type inequality, Lemma 4.4, for mappings in W 1,n−1 loc ( , R n ) with integrable inner distortion. We apply this result to prove Theorem 1.1.

Preliminaries
Let be a domain in R n . We say that f : → R n belongs to the Sobolev space W 1, p loc ( , R n ), 1 ≤ p < ∞, if the coordinate functions of f are locally p-integrable and have locally pintegrable distributional derivatives.
For x ∈ R n we denote by x i , i = 1, 2, . . . , n, its coordinates, i.e. x = (x 1 , x 2 , . . . , x n ). Fix y ∈ R. Then {y} × R n−1 is a copy of R n−1 and thus its Hausdorff measure coincides with (n − 1)-dimensional Lebesgue measure and we might sometimes write dz instead of dH n−1 (z).
We denote the Euclidean norm of x ∈ R n by |x|, the open ball centered at x ∈ R n with radius r > 0 by B(x, r ) = {y ∈ R n : |x − y| < r }, and its closure by B(x, r ). Similarly, we denote the corresponding sphere by We define an n-dimensional open cube centered at x ∈ R n with radius r > 0 by For the convenience of the reader we recall the boxing lemma, proved in [8].

Lemma 2.1 (Gustin's boxing lemma) Every compact set K ⊂ R n can be covered by balls
where the constant C(n) > 0 depends only on dimension n. Moreover, for every compact set K we have Here H n−1 ∞ stands for the Hausdorff (n − 1)-content.
If E ⊂ R n is a measurable set, then we denote by |E| its Lebesgue measure. We say that a mapping f : → R n satisfies Lusin's condition (N ) on E if | f (A)| = 0 for every A ⊂ E such that |A| = 0.
The notation E ⊂⊂ means that the closure of E is compact and E ⊂ . For a mapping f : → R n and a Borel set E ⊂⊂ , the multiplicity function N (y, f, E) of f is defined by where ⊂ R n and m ≥ n, we say that the area formula holds for Next we collect some topological facts about open and discrete mappings. For more details we refer to [35, I.4].

A mapping f : → R n is open if it maps open sets in to open sets in
of preimages does not accumulate in for any y ∈ R n . Next we assume that f : → R n is a discrete and open mapping. From now on we always assume that f is continuous.
Let f : → R n be a continuous mapping from a domain ⊂ R n . Then we can define the local degree deg(y, f, U ) ∈ Z for any subdomain U ⊂⊂ and every y ∈ R n \ f (∂U ), see [35]. If y / ∈ f (∂U ), we say that y is an ( f, U )-admissible point. If f and g are two homotopic mappings via homotopy h t , t ∈ [0, 1], and y is (h t , U )-admissible for all t ∈ [0, 1], then deg(y, f, U ) = deg(y, g, U ).
If points y and z belong to the same component of By a condenser in ⊂ R n we understand a pair (E, G) of sets with G ⊂⊂ open and E compact in R n and with E ⊂ G. The p-capacity of a condenser (E, G) is defined as where the infimum is taken over all smooth functions u ≥ 0 with support in G such that u ≥ 1 on E. We call such functions admissible for cap p (E, G). The n-capacity is also called conformal capacity. A condenser (E, G) is ringlike if E and the complement of G are connected.
We will say that a set σ separates a set A from a set B if σ is a compact set in R n and if there are disjoint open sets A and B such that R n \σ = A ∪ B, A ⊂ A and B ⊂ B. Let denote the class of all sets that separate A from B. For every σ ∈ we associate a complete measure μ as follows. For a H n−1 -measurable set D ⊂ R n , we define It follows from the properties of Hausdorff measure that the Borel sets of R n are μ-measurable. The modulus of is defined as follows.
If now (E, G) is a condenser in R n and is the class of all the sets separating E form R n \G then, by [42], we have the following duality where n = n n−1 . More generally, where p = p p−1 . One can find more information about p-capacity and the relations (8) and (9) from [35,42,43].

Lusin's condition (N) on hyperplanes
In this section we show that a discrete and open mapping f ∈ W 1,n−1 loc ( , R n ) satisfies Lusin's condition (N ) on almost every hyperplane. This fact has been proven for homeomorphisms by Csörnyei et al. in [2], and we use methods similar to theirs.
Let ϕ be the standard smooth mollification kernel on R n i.e.
where the constant C is chosen such that R n ϕ(x) dx = 1. Then for δ > 0 we define the family of mollifiers ϕ δ by the formula This means that ϕ δ is a smooth nonnegative function supported in B(0, δ) and R n ϕ δ (x) dx = 1. Moreover, ϕ δ > 0 on B(0, δ). We also introduce a "crude family" of mollification kernels Here Q is a cube defined in (2) and χ E denotes the characteristic function of E i.e.
where ν D is the outer normal of D.
Because f is a continuous, discrete and open mapping, it is either sense-preserving or sense-reversing, see [28, p. 151]. Without loss of generality we may assume that f is sensepreserving. Then m ≥ 1, and we have 1

Then by Gustin's boxing lemma we have
Now the integrand above is continuous in W 1,n−1 , and thus as k → ∞. Moreover, because f k → f uniformly in D, for any y ∈ f (D) we can choose k y such that y is (h k t , D)-admissible for all t ∈ [0, 1] and for every homotopy Thus, by [35,Proposition 4.4] deg(y, f k , D) = deg(y, f, D) for every k ≥ k y , and we have By using (12) The claim follows now from (11) when k → ∞.
Let U ⊂⊂ be a domain. Then for every t ∈ R we denote For δ > 0 small enough, we define is satisfied. Then Proof For simplicity, let us assume that t = 0 and Q = [−R, R] n−1 . We denote Consider ρ > 0 such that I 2ρn ⊂ U . Because f |∂ I r ∈ W 1,n−1 for almost every r ∈ (0, ρ), by Lemma 3.1 we have for almost every r ∈ (0, ρ). We integrate over the interval (0, ρ) with respect to r and use Fubini's theorem to obtain Hence, using Fubini's theorem and the fact that y ∈ Q(x, 2ρ) if and only if x ∈ Q(y, 2ρ) we obtain Sinceφ 2ρ ≤ C(n) ϕ 2ρn , we have  Proof We may assume that U is a normal domain of f . By using a translation, if necessary, we may assume that U t = ∅ if and only if t ∈ (−u, u). First we show that f satisfies condition (13) for almost every t ∈ (−u, u). For this purpose, we denote By L 1 -convergence of the mollifications, Fatou's lemma and Fubini's theorem, we have This means that lim inf δ (t) = 0 for almost every t ∈ (−u, u), and each such t satisfies (13).
Next we fix t ∈ (−u, u) in such a way that (13) is satisfied and U t |D f (x)| n−1 dH n−1 (x) < ∞. This is true for almost every t ∈ (−u, u). By Lemma 3.2 holds for every closed cube Q × {t} ⊂ U t . Let E ⊂ U t be a set of (n − 1)-measure zero. Given > 0, we find an open set G ⊂ U t such that E ⊂ G and Let {Q j } j be a sequence of non-overlapping cubes in U t such that G = j Q j . Then Letting → 0 the claim follows.
The following corollary follows easily from Proposition 3.3.  show that if f ∈ W 1,n−1 loc has integrable inner distortion, then it satisfies a Poletsky-type inequality, see Lemma 4.4. The importance of this result is based on the fact that it allows us to use modulus and capacity methods in the study of non-homeomorphic mappings. For more details, see [35].
To prove Theorem 1.1 it would be enough to use only the duality of conformal capacity, introduced in (8). However, to prove Theorem 5.1 in Chapter 5 we have to use a more general duality of p-capacity, introduced in (9). The duality of capacity is also a strong tool in the applications that use modulus and capacity methods. Gehring [6] showed that the conformal capacity is related to the extremal length of a family of surfaces that separate the boundary components of a ring. Also other authors have dealt with the extremal length of separating surfaces, see [5,6,14,36,42,43]. In this paper we follow closely the work of Ziemer, see [42,43].
We give next a change of variables -type formula for subsets of spheres. This result follows from [24, Theorem 9.2], see also [33]. a continuous mapping, and B(x, r ) ⊂⊂ . If f satisfies condition (N ) on almost every (n − 1)-dimensional sphere S(x, t), 0 < t < r, with respect to measure H n−1 , then for almost every t ∈ (0, r ) we have for all H n−1 -measurable subsets E ⊂ S(x, t) and for all H n−1 -measurable functions u : The next lemma is crucial in our proof of Poletsky type-inequality for rings. This lemma is a modified version of [33,Theorem 3.3].
Proof Let ρ be a test function as in the statement of the lemma. Define a Borel functioñ ρ : B(x 0 , r ) → [0, +∞] by setting By Corollary 3.4 the restriction of f to the sphere S t satisfies condition (N ) with respect to H n−1 for almost every t ∈ (0, r ). Lemma 4.1 shows that for almost every t ∈ I . On the other hand This completes the proof.
Next we will prove our duality result, Lemma 4.3. To formulate this result, we have to define some new concepts that are not in common use in literature. For this purpose, let U ⊂⊂ be a normal domain of f and B(x 0 , 2r ) ⊂⊂ U . Let M : → R be a weight function. Then we can define the capacity of the condenser (B(x 0 , r ), B(x 0 , 2r )) with respect to symmetric test functions and weight M as where with Ad(sym) we denote the set of admissible Borel functions for capacity cap p (B(x 0 , r ), B(x 0 , 2r )) with the additional assumption that for every s ∈ (0, 1) the set u −1 (s) is a sphere S(x 0 , t), for some r < t < 2r .
where A p is as in Lemma 4.2 and Proof Choose > 0 and let ρ be any Borel function in U with property σ ρ(x) dH n−1 (x) ≥ 1 for every σ ∈ .
Let u be any admissible function for C := cap sym p, B p (B(x 0 , r ), It is clear that u −1 (s) ∈ for every 0 < s < 1. Hence, by Hölder's inequality and coarea formula [41], we have ⎛ Since > 0 was arbitrary, which is also true when the infimum is taken on the left hand side over all admissible test functions ρ.
We could have formulated Lemma 4.3 also for more general weights. However, for our purpose it is enough to use the weight function B p defined above.
The following Poletsky-type inequality is crucial for our proof of Theorem 1.1. It would be interesting to know if it is possible to prove this result also for more general condensers.

where C(n, m) > 0 is a constant depending only on dimension n and on the degree m of the mapping f on normal domain U , and B p is defined as in Lemma 4.3.
Proof Now ( f (B(x 0 , r )), f (B(x 0 , 2r ))) is a condenser in U and we can define to be the family of sets separating R n \ f (B(x 0 , 2r )) from f (B(x 0 , r ))). Notice that because f is an open mapping, we have = ∅. Moreover, openness of f implies that f (B(x 0 , 2r )) is an open set and continuity of f implies that f (B(x 0 , r )) is a compact set. Thus  ( f (B(x 0 , r )), f (B(x 0 , 2r ))) is a condenser in f (U ). We set L := {S t : t ∈ (r, 2r )} and l := {S t : t ∈ (r, 2r )}, where S t := S n−1 (x 0 , t). Because every admissible test function for the modulus M S p ( f ( l )) is also admissible for the modulus M S p ( f ( L )), we have that M S p ( f ( L )) ≤ M S p ( f ( l )). Moreover, we also have ⊃ f ( l ). Thus we have by (8), using the monotonicity of modulus and by Lemmas 4.2 and 4.3 cap p ( f (B(x 0 , r )), f (B(x 0 , 2r ) where A p is as in Lemmas 4.2 and 4.3.

Differentiability in the Sobolev space W 1,n−1
In this section we prove our main result which says that a discrete and open mapping f ∈ W 1,n−1 loc ( , R n ) with integrable inner distortion is differentiable almost everywhere in . The integrability assumption of the distortion function can be relaxed if we assume higher integrability of the Jacobian determinant. Indeed, we will prove the following result. Theorem 1.1 follows from Theorem 5.1 when we choose p = n. In the proof of Theorem 5.1 we apply the Rademacher-Stepanov theorem [38], see also [23].
The following lemma can be originally found from [22,Proposition 6], see also [27,Lemma 5.9]. A big part of the geometric structure of the proof of Theorems 5.
We are now ready to prove our main result Theorem 5.1. We will follow the idea of the proof of Theorem 1 by Salimov and Sevost'yanov in [37], see also [28, §4.3].
Proof of Theorem 5.1 If n = 2, the result follows from the theorem of Gehring and Lehto [7], and from the fact that in the planar case every mapping of finite inner distortion is always a mapping of finite outer distortion.
It is also easy to see that J (x, f ) = 1 for every x ∈ (−1, 1) n and J (y, f −1 ) = 1 for every y ∈ f ((−1, 1) n ). Moreover, we notice that f and f −1 are nowhere differentiable. Next we calculate for any compact set F ⊂⊂ (−1, 1) n and the claim follows.
To construct our mapping in Example 5.5 we had to assume that n ≥ 3. Indeed, every continuous mapping f ∈ W 1,1 loc ( , R 2 ), ⊂ R 2 , has finite partial derivatives almost everywhere, because it is absolutely continuous on almost every line parallel to coordinate axes, see [35, I.1.2]. Therefore, as a consequence of theorem by Gehring and Lehto [7], we have that every planar homeomorphism f ∈ W 1,1 loc ( , R 2 ), ⊂ R 2 , is differentiable almost everywhere. We do not know what is the sharp integrability assumption for K I (·, f ) that guarantees differentiability at almost every point in Theorem 1.1. However, we believe that the correct assumption should be K I (·, f ) ∈ L 1 loc ( ).