SHARPNESS OF THE DIFFERENTIABILITY ALMOST EVERYWHERE AND CAPACITARY ESTIMATES FOR SOBOLEV MAPPINGS

. We give sharp conformal conditions for the diﬀerentiability in the Sobolev space W 1 ,n − 1 loc (Ω , R n ). Furthermore, we show that the space W 1 ,n − 1 loc (Ω , R n ) can be considered as the borderline space for some capacitary inequalities.


STANISLAV HENCL AND VILLE TENGVALL
Abstract. We give sharp conformal conditions for the differentiability in the Sobolev space W 1,n−1 loc (Ω, R n ). Furthermore, we show that the space W 1,n−1 loc (Ω, R n ) can be considered as the borderline space for some capacitary inequalities.

introduction
Let Ω ⊂ R n , n ≥ 2, be a domain and suppose that f ∈ W 1,n−1 loc (Ω, R n ) is a homeomorphism. If n = 2, then the theorem of Gehring and Lehto [2] implies that f is differentiable almost everywhere. Moreover, if we proceed to the case n ≥ 3 then the Sobolev embedding theorem on spheres implies that every homeomorphism in W 1,p loc (Ω, R n ), p > n − 1, is differentiable almost everywhere, see [13,Theorem 1.2]. On the other hand, Sobolev embedding theorem on spheres does not apply when f ∈ W 1,n−1 loc (Ω, R n ), n ≥ 3, which may cause the failure of the differentiability in a set of positive measure. Indeed, in [1, Example 5.2] Csörnyei, Hencl and Malý constructed a nowhere differentiable homeomorphism f ∈ W 1,n−1 loc ((−1, 1) n , R n ), n ≥ 3, of finite distortion with nowhere differentiable inverse.
Differentiability of mappings in the Sobolev space W 1,n−1 loc (Ω, R n ), n ≥ 3, can be recovered by requiring some integrability of the distortion functions (see Preliminaries for the definition of mappings of finite distortion). Indeed, it follows from [11,  (Ω, R n ) with locally integrable inner distortion function is differentiable almost everywhere. The topological assumptions above can be further relaxed by assuming f to be only continuous, discrete (the set f −1 (y) is a discrete set in Ω for every y ∈ R n ) and open (f (A) is an open set in R n for every open set A in Ω), see [17]. It was asked in [17] whether the local integrability assumption of the inner distortion function above is sharp. We will now give a positive answer to this question by a novel construction: Theorem 1.1. Let 0 < δ < 1 and n ≥ 3. Then there is a homeomorphism f ∈ W 1,n−1 ((−1, 1) n , R n ) with K I ∈ L δ ((−1, 1) n ) such that f is not classically differentiable on a set of positive measure.
When we are studying analytical properties, such as differentiability almost everywhere and continuity, of mappings of finite distortion we are commonly lead to study the geometry of these mappings via weighted capacitary inequalities. One of the most well-known weighted capacitary inequalities in the study of mappings of finite distortion is the K I -inequality, also known as the Poletsky-type capacity inequality: Definition 1.2. Let f ∈ W 1,1 loc (Ω, R n ) be a continuous and open mapping of finite distortion with locally integrable inner distortion function. We say that f satisfies the K I -inequality if for every condenser (G, E) in Ω we have cap(f (G), f (E)) ≤ C I (G) cap K I (G, E) , (1.1) where the constant C I ≥ 0 depends only on the dimension n and on the maximum multiplicity N (f, G) := sup y∈R n card f −1 (y) ∩ G. For the terminology used above we refer the reader to the Preliminaries.
We remark that inequality (1.1) is enough to guarantee differentiability almost everywhere for a continuous, discrete and open mapping, and especially for a homeomorphism, of finite distortion, see [15,16]. It is also important to notice that the best possible constant C I (G) plays an important role in several applications. Moreover, inequalities similar to (1.1) has been applied widely to study properties of quasiregular mappings, see e.g. [9,8,10].
In the case of quasiregular mappings K I -inequality with the constant C I (G) ≡ 1 was first proved by Martio, Rickman and Väisälä [10]. Later Martio improved this result by improving the constant C I (G), see [9]. In [7] Koskela and Onninen generalized this inequality for continuous, discrete and open mappings of finite distortion in W 1,n loc (Ω, R n ) with a locally integrable inner distortion function. It was further shown that the regularity assumption f ∈ W 1,n loc (Ω, R n ) can be slightly relaxed, say to |Df | n log −1 (e+|Df |) ∈ L 1 loc (Ω). These results by Koskela and Onninen were based on a duality argument, relying on integration by parts against the Jacobian determinant. This method does not work if we assume f ∈ W 1,p loc (Ω, R n ) for some p ∈ [1, n). In [17,Lemma 4.4] it was shown by applying Ziemer's duality equation [20,21] that for spherical condensers the K I -inequality is true even for continuous, open and discrete mappings of finite distortion in W 1,n−1 loc (Ω, R n ) with locally integrable inner distortion function. We will show that the space W 1,n−1 loc (Ω, R n ) can be considered as the borderline space for the K I -inequality: Theorem 1.3. Let n ≥ 3 and ε > 0. Then there is a homeomorphism f ∈ W 1,n−1−ε ((−1, 1) n , R n ) with K I ∈ L 1 ((−1, 1) n ) for which the K I -inequality (1.1) fails.
The main idea for the proof of Theorem 1.3 is to construct a homeomorphism in W 1,n−1−ε ((−1, 1) n , R n ) with the given conformality conditions which does not satisfy the Lusin's condition (N ) on almost every hyperplane with respect to (n − 1)dimensional Hausdorff measure. In our construction we will actually destroy the Lusin's condition on every hyperplane. The reason for this comes from the proof of [17,Lemma 4.4] where the Sobolev regularity of a mapping was only used to show that mapping satisfy the condition (N ) on almost every hyperplane.
In the proof of Theorem 1.3 we will apply probability-based techniques to calculate L p -norms of distortion functions and differential matrices. More precisely, we will apply the notion of expected value and the famous Khintchine inequality [6] to calculate integrals in Theorem 1.3. As far as we know this is the first time that Khintchine inequality is used to study mappings of finite distortion.

Preliminaries
2.1. Notation. We will denote by C := C(p 1 , . . . , p k ) a positive constant which depends only on the given parameters p 1 , . . . p k . The constant C might change from line to line. Furthermore, for given functions f and g we denote f g if there exists a positive constant C > 0 such that f (x) ≤ Cg(x) for all points x. If both conditions f g and g f are satisfied we denote f ∼ g.

Mappings of finite distortion.
Let Ω ⊂ R n , n ≥ 2 be a domain. We recall that a mapping f ∈ W 1,1 With such a mapping f we may associate the distortion function K I : Ω → [1, ∞] as follows: We call the function K I as the inner distortion functions of f . Above, D f stands for the adjugate matrix of the differential matrix Df and |A| stands for the operator norm of a matrix A. When K I ∈ L ∞ (Ω) we call mapping f quasiregular.

2.3.
Condensers and the capacity. Suppose that Ω ⊂ R n , n ≥ 2, is a domain. A pair (G, E) of sets is called as a condenser in Ω if (1) G ⊂⊂ Ω is a domain, and (2) E ⊂ G is a non-empty, compact subset of G. Condenser (G, E) is called spherical if both G and E are balls centered at the same point.
We define the ω-weighted capacity of a condenser (G, E) for a non-negative weight function ω ∈ L 1 (G) as In the case ω ≡ 1 we write cap(G, E) instead of cap 1 (G, E).
will serve as the set of indices for our construction. Next, suppose that {a k } ∞ k=0 is a decreasing sequence such that 1 = a 0 ≥ a 1 ≥ · · · > 0, and define Set z 0 = 0. Then it follows that Q(z 0 , r 0 ) = (−1, 1) m and further we proceed by induction.
Then for the measure of the kth frame Formally we should write w(v(k)) instead of w(k) but for the simplification of the notation we will avoid this. It is not difficult to find out that the resulting Cantor set is a product of m Cantor sets C a in R, and the number of the cubes {Q v(k) : v(k) ∈ V k } is 2 mk . Therefore, the measure of the Cantor set C A := C[{a k } k ] can be calculated as 2.5. Canonical parametrizations. To prove Theorem 1.3 we need to find a way to map Cantor set onto another Cantor set by a homeomorphism. We will do this by using so-called canonical parametrizations and canonical transformations which were first introduced by Hencl, Koskela and Malý in [5].
For a given z ∈ R m and r > 0 we denote We define an affine map ϕ Qr(z) : Mapping ϕ Qr(z) is called the canonical parametrization of the cube Q r (z). Furthermore, if A = Q r (z)\Q r (z) is a cubical annuli, we define Mapping ϕ A is called as the canonical parametrization of a cubical annuli A.
To calculate Dϕ A we may distinguish 2 m different cases depending on which side of the annuli we are. However, by symmetry it suffices to deal with the case (t, x) ∈ {(t, x) ∈ [0, 1] × ∂Q 1 (0) : x 1 = 1}. In this case, if we denote d := (1 − t)r + tr , then we may write

Canonical transformation. Suppose that
A := Q r (z)\ int(Q r (z)) and B := Qr (z)\ int(Qr (z)) are two cubical annuli. We define the canonical transformation of A onto B as Then As in the Section 2.5 when we are calculating the matrix Dϕ A,B (ϕ A (t, x)) we may distinguish 2 m different cases depending on which side of the cubical annuli A we are. However, because of the symmetry we will again write the matrix formally only in the first case of these 2 m different cases. Then we have

2.7.
Mapping a Cantor set onto another. Consider two Cantor sets given by the algorithm introduced in the Section 2.4. Then we may define two sequences of cubical annuli where the cubes above are referring to the corresponding ones in the constructions of the Cantor sets C A and C B . We define the homeomorphism Φ : (−1, 1) m → (−1, 1) m which takes C A onto C B as the pointwise limit of mappings where χ E denotes the characteristic function of a set E, andΦ N is a linear mapping which takes each cube Q v(N ) linearly onto corresponding cubeQ v(N ) . To see that Φ defines a homeomorphism we refer the reader to [4,Section 4.3].
Mapping the Cantor set C A onto C B by using the mapping Φ.
As in the Sections 2.5 and 2.6 we may again distinguish 2 m different case when we are calculating DΦ. Again we calculate the differential matrix only in the first of these 2 m cases. In this case we have 2.8. Khintchine-type inequality. The Rademacher's distribution is a discrete probability distribution such that Furthermore, for a given 0 ≤ 1 ≤ 1 we define Bernoulli q-distribution to be the discrete probability distribution define by Next, assume that X 1 and X 2 are two independent random variables, X 1 has from the Rademacher's distribution and X 2 has the Bernoulli q-distribution. If we define Y := X 1 X 2 , then the probability distribution of Y can be written as otherwise. (2.5) From now on we will call this distribution as RB(q)-distribution. Then we can give a generalized version of the famous Khintchine's inequality [6] in the RB(q)-setting.
In the case q = 1 this result will be the usual Khintchine's inequality -see e.g. [6].
is a sequence of independent and identically distributed random variables from the RB(q)distribution, and let {d k } ∞ k=1 be a sequence of real numbers. Then for a given p > 0 we can estimate the expected value aŝ where the constantsÂ p,q andB p,q are depending only on p and q.
Proof. For the convenience of the reader we include the proof of this estimate in the case p > 1 which follows directly the approach of [19,Proposition 4.5]. There will be three steps in the proof: (i) Let us first assume that p = 2. Recall, that we may write where X k,1 are random variables from the Rademacher's distribution, X k,2 are random variables from the Bernoulli q-distribution, and all the random variables X k,j are independent. By using this fact and independence we get (ii) For the upper bound we notice that for every t > 0 we may estimate and by applying the numerical inequality . By applying Chebychev's inequality to this estimate, we get By applying Cavalieri's principle, we get where the constant C > 0 is depending only on p. This gives us the upper bound. (iii) The lower bound follows from (i), (ii) and Hölder's inequality: 2.9. Oscillating snowflake mapping on a Cantor set. Let which will play a role of a sign function in our construction.
-+ + - Suppose that {d k } ∞ k=1 is a sequence of real numbers converging to zero. Then we define a sequence {S k } ∞ k=1 of mappings S k : wherex := (x 1 , . . . , x m ) and Then we may define mappings S N : and set S(x) := lim N →∞ S N (x). This limit exists for all x ∈ (−1, 1) m+1 because Furthermore, it is not hard to see that S : 2.10. Properties of DS. To calculate DS(x) we define sets and indicator functions otherwise, for every i = 1, . . . , m.
For a given point x = (x, x m+1 ) ∈ Q v(k) × (−1, 1) the differential matrix DS k (x) can be written as . We also notice that δ j Qv (x) and δ(v) are independent random variables, and thus it is easy to see that the probability mass function of δ j v can be defined as in (2.5) with q = 1 n−1 .

Proof of Theorem 1.1
In this section we will prove Theorem 1.1. To capture the essential part of the proof we will first give a detailed proof in the case n = 3 and then sketch the proof in the case n ≥ 4. Before going to proofs we will describe a basic building block of our construction.
Let γ > 0 be sufficiently small as described later. Let 0 < a < γ, 0 < η <  Then h is continuous, strictly decreasing function, piecewise C 1 with h(0) = π and h(a) = 0. We set g(t) = − a η π h(t) + a η . Now |g (t)| = a η π |h (t)|, g(0) = 0 and g(a) = a η . We define a linear function Then ρ(2a η ) = 2a η and ρ(a) = a η . Let us consider three anuloids T ⊂ S ⊂ R given by We will define a homeomorphism F : R → R such that F (p) = p for every p ∈ ∂R and F (∂T ) = ∂S is given by the natural stretching. We will use the following system of coordinates: r radius in the anuloid: r 2 = ( y 2 + z 2 − 1) 2 + x 2 , α angle in the anuloid: sin α = x r , β angle around the anuloid: sin β = y y 2 + z 2 , and we set Lemma 3.1. Let F : R → R be as described above in (3.1), and let 0 < δ < 1. Then Moreover, we have where the constant C > 0 depends only on n and δ.
(2) As ρ(a) = a η = g(a) and h(a) = 0 it is easy to check that the mapping is continuous at ∂T . By the first line in (3.1), and by applying the facts ρ(2a η ) = 2a η and ρ(a) = a η we easily see that F is a homeomorphism of R \ T onto R \ S. It is also a homeomorphism of T onto S as g is increasing with g(0) = 0 and g(a) = a η and in the β coordinate we have a simple rotation by the angle h(r). (3) The property F (∂T ) = ∂S easily follows from the arguments in (2). (4) We note that the middle circle ({y 2 + z 2 = 1 and x = 0}) of R is rotated by the angle of π = h(0) and hence we conclude that |F (p) − p| = 2 there.
We still have to verify (3.2). We express the derivative in the system of coordinates given by [r, α, β]. The radial direction (α and β fixed and r increases), angular direction inside (r and β fixed and α increases) and angular direction around (r and α fixed and β increases) are orthogonal and hence we can compute |DF | or J F with respect to these directions. The derivative in the corresponding system of coordinates is given by (see [ Let us note that the middle term comes from the fact that the circle of radius 2πr is mapped to the circle of radius 2πρ(r) (or 2πg(r) in the second case) and hence the derivative around this circle which must be the same as the derivative in the tangential direction to this circle equals 2πρ(r) 2πr . The last term with M comes from the fact that the length of any circle around our anuloid (in the β direction) has length in [2π(1 − 2a η ), 2π(1 + 2a η )] and it is mapped to the similar circle around -as we are squeezing in the r direction this is not necessarily the same circle. Note that as 2a η < 1 2 we have M ∼ C for some constant C > 0. Analogously we can check that the normalization of the other terms in the matrix is done correctly.
It is easy to check that on R \ T we have ρ (r) ∼ C and ρ(r) r ≥ C and hence |DF | ∼ ρ(r) r there. Using polar coordinates it follows that As ρ (r) ∼ C it is easy to see that |D # F | ∼ ρ(r) r and J F ∼ ρ(r) r on R\T , and hence Now we estimate the derivative on T . In the small part around r = 0 where h(r) = π − r we have |g (r)| = a η π |h (r)| = a η π and also g(r) r = a η π . It follows that on this set we have |DF | ≤ C and It follows that on this set we have    It follows that It is not difficult to check that |D # F | ∼ |h (r)| g(r) r and J F ∼ |g (r)| g(r) r ∼ a η |h (r)| g(r) r on this set and hence By applying algorithm in Section 2.4 we construct a Cantor set Then, by (2.2) we get

Suppose that
are the corresponding cubes in the construction of the Cantor set C B .
We will now define our homeomorphism f . For this, set Given δ < 1 we find η > 0 such that For every k ∈ N we consider homeomorphism as in Lemma 3.1 applied to a = a k scaled in each direction (both in domain and in target) by a factor of 1 4 2 −k . We put a translated copy of such a homeomorphism in every As 2a η k < b k−1 − b k there is enough room there (the volume of the anuloid is roughly 2 −k−2 × 2 −k−2 × (2 −k−2 2(a k ) η )). In this way we obtain 2 3k copies of this homeomorphism for each k ∈ N and the supports of these maps are clearly disjoint. These homeomorphisms equal to identity on the boundaries and hence we can extend them by identity everywhere else and we obtain a homeomorphism of (−1, 1) 3 onto (−1, 1) 3 .
We have 2 3k cubes in the construction and by scaling by 2 −k−2 in all three directions we get by Lemma 3.1 that It remains to show that f is not differentiable at points of C A = C a × C a × C a . For every x ∈ C A and every k we can find v ∈ V k such that x ∈ Q v . In Q v \ Q v we have a translated and scaled copy of a homeomorphism as in Lemma 3.1 and hence the distance of x and its support is less than 10 · 2 −k . For p on the boundary of the corresponding anuloid we have f (p) = p and as f (x) = x we can see that if there is a classical derivative Df (x) it must be equal to identity matrix Df (x) = I as this happens on all scales k. On the other hand in the central circle of the anuloid our function is rotated and |f (p) − p| = 1 2 2 −k there. This happens on all scales k and it follows that Df (x) cannot be I and hence f is not differentiable at x.

3.2.
Proof of the Theorem 1.1 when n ≥ 4. The construction in the case n ≥ 4 is very similar and therefore we only briefly outline this. We work with anuloids like } and again S has radius a η and T has radius a. We use spherical (n − 1)-dimensional coordinates inside the annulus (variables r, α 1 , . . . , α n−2 ) and angle β around the annulus as before.
We only sketch the estimates of the most important terms. In T we have and hence |D # f | ∼ |h (r)|( g(r) r ) n−2 and J f ∼ |g (r)|( g(r) r ) n−2 there. The important part of the derivative (where h(t) = log  Given δ < 1 we can choose η > 0 such that −ηδ + n − 1 − (n − 1)δ > 0. Other integrals can be estimated analogously as before.
Similarly we construct a Cantor type set of positive measure in R n and in each of the 2 kn sets Q v \ Q v , v ∈ V k , we put a translated and scaled (by factor 2 −k in n directions) copy of the homeomorphism as above. Analogously to the computation in (3.3) we obtain using −ηδ + n − 1 − (n − 1)δ > 0 and (3.4) our conclusion.

4.1.
Construction of the mapping. Let ε > 0 and suppose that n ≥ 2. We will define a homeomorphism f : (−1, 1) n → R n of finite distortion as follows: (1) Define the sequences {a k } ∞ k=0 and {b k } ∞ k=0 by setting a k = 1 (k + 1) α and b k = where α > 0. It is easy to see that for every k ≥ 1 we have .
(2) Denote by ] the Cantor sets in R n−1 given by the sequences {a k } ∞ k=0 and {b k } ∞ k=0 above. For each index v(k) ∈ V k , k ∈ N, we denote by the corresponding (n − 1)-dimensional cubical annuli in the construction of the Cantor sets C A and C B . Then for k ≥ 1 we have L n (A v(k) × (−1, 1)) = 2 −k(n−1)+1 (a n−1 k−1 − a n−1 k ) (4.2) and similarly . Then, up to a permutation of (n − 1) first coordinate axes, for every point for some t ∈ [0, 1] and z is the corresponding center (see Section 2.7). Especially , .
(4) Define a sequence {d k } ∞ k=1 of real numbers by and suppose that S : (−1, 1) n → R n is the oscillating snowflake mapping on the Cantor set C B generated by the sequence {d k } ∞ k=1 (see Section 2.9). Then for every point x ∈ v∈V k A v(k) we have H(x) ∈ v∈V k B v(k) and thus Section 2.9), and therefore Recall that the distribution of δ j Q (v 1 ,...,v l ) (x)δ(v l ) is given by the RB(q)-distribution (see end of Section 2.10). By applying Khintchine's inequality Lemma 2.1 for a given p > 0, we get (5) Define f : (−1, 1) n → R n as Then, for almost every point x ∈ v∈V k A v(k) we have using (4.3) that Especially 4.2. Properties of the mapping. Next we will show that f is a homeomorphism for which Lusin's condition (N ) fails on hyperplanes, f ∈ W 1,n−1−ε ((−1, 1) n , R n ), K I ∈ L 1 ((−1, 1) n ) and f maps hyperplanes, perpendicular to n-th coordinate axis, to sets of infinite (n − 1)-dimensional Hausdorff measure: (1) It is clear that f is a homeomorphism as a composed mapping of two homeomorphisms. Moreover, it is easy to see that f cannot satisfy Lusin's condition (N ) on any hyperplane H t = {x ∈ (−1, 1) n : x n = t}, t ∈ (−1, 1), with respect to (n − 1)-dimensional Hausdorff measure. We know that the H n−1 measure of the (n − 1)-dimensional sets C t B = {x ∈ (−1, 1) n : (x 1 , . . . , x n−1 ) ∈ C B and x n = t} is positive. Since S may only increase H n−1 , we have for every set C t A = {x ∈ (−1, 1) n : (x 1 , . . . , x n−1 ) ∈ C A and x n = t} that H n−1 (C t A ) = 0 and H n−1 (f (C t A )) = H n−1 (S(C t B )) > 0.

4.3.
Failure of the K I -inequality for f . In this section we will show that K Iinequality (1.1) fails for f which will end the proof of Theorem 1.3. The proof of failure of the K I -inequality is given in Proposition 4.3. However, before this we will give two technical lemmata which we then apply in the proof of Proposition 4.3.
x ∈ A} is the graph of u h on A, and let n be the unit normal vector of the surface G u 0 . Denote by Then for a given ε > 0 there exist h ε > 0 such that Lemma 4.1 follows easily from the fact S h S 0 , and we leave the proof for the reader. In the next lemma we show that if we have some linear function and a small slab between a small shift of this function, then it actually almost minimizes some capacity among all continuous deformations of this slab. For h > 0 and for a given continuous function g : A → R define a number Then there exists h 0 > 0 such that Thus, by assuming 0 < ε < 1/2 we get H n−1 (S h ) 2 n−1 l n−1 ≤ D×R |∇w| n , and if we take infimum over all admissible test functions w, we conclude that Proof. Fix M ≥ 1. For t ∈ R denoteÊ t = (−1 + 2δ, 1 − 2δ) n−1 × {t}, and whenever t = s define where the infimum is taken over all functions u ∈ C ∞ ((−1, 1) n−1 × R) such that u| f (Êt) ≡ 0 and u| f (Ês) ≡ 1. By δ = 1 10 we know that C A ⊂ (−1 + δ, 1 − δ) n−1 , where C A is the given Cantor set of zero measure in the construction of f . Then it is easy to see that it suffices to show that there is h M > 0 such that for all 0 < h < h M .
As the mapping in the construction of S k are piecewise affine, we may divide the setsÊ 0 andÊ h into finitely many pairwise disjoint, open, convex and maximal (n−1)dimensional sets {E j 0 } l j=1 and {E j h } l j=1 such that: where the closure is taken in (−1, 1) n−1 × R instead of R n . (ii) E j h = E j 0 + h for all j = 1, . . . , l. (iii) Sets E j 0 , j = 1, . . . , l, are not depending on h. (iv) The restriction of S N on each set E j 0 and E j h can be written as S N (x) = (x, L j (x)) for allx ∈ E j 0 and S N (x) = (x, L j (x) + h) for allx ∈ E j h , where L j : E j 0 → R is a linear affine function. Fix ε ∈ (0, 1/2). Denote by n j the unit normal vector of G Then, if u is an admissible test function for cap(f N (Ê 0 ), f N (Ê h )), we have I |∇u| ≥ 1 for all I ∈ ∪ l j=1 I j .
Especially, by integrating over each (n − 1)-dimensional set S j h , applying Hölder's inequality and using the fact that the length of each line segment I ∈ ∪ l j=1 I j is at most h, we get  The first n − 1 coordinate functions of f agree with first n − 1 coordinate functions of f N and the last coordinate mapping of f agrees with that coordinate of piecewise linear mapping f N close to ∂(−1+2δ, 1−2δ) n−1 ×(0, h 0 ). Hence we can apply Lemma 4.2 (for the last coordinate mapping) and (4.18) to derive Finally, by putting together estimates in (4.15) and (4.19) we get and the claim follows.