The Trace Theorem, the Luzin N- and Morse–Sard Properties for the Sharp Case of Sobolev–Lorentz Mappings

We prove Luzin N- and Morse–Sard properties for mappings \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$v:{\mathbb {R}}^n \rightarrow {\mathbb {R}}^d$$\end{document}v:Rn→Rd of the Sobolev–Lorentz class \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mathrm {W}^{k}_{p,1}$$\end{document}Wp,1k, \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$p=\frac{n}{k}$$\end{document}p=nk (this is the sharp case that guaranties the continuity of mappings). Our main tool is a new trace theorem for Riesz potentials of Lorentz functions for the limiting case \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$q=p$$\end{document}q=p. Using these results, we find also some very natural approximation and differentiability properties for functions in \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mathrm {W}^{k}_{p,1}$$\end{document}Wp,1k with exceptional set of small Hausdorff content.


Introduction
In this paper, we continue the study of the Luzin N -and Morse-Sard properties for the Sobolev mappings under minimal integrability assumptions initiated in our previous M. V. Korobkov was partially supported by the Russian Foundations for Basic Research and (Grants No. 14-01-00768 and 15-01-08275) and by the Dynasty Foundation.  [11,12,24], see also [22]. Of course, it is in this context very natural to restrict attention to continuous mappings, and so require from the considered function spaces that the inclusion v ∈ W k p (R n , R d ) should guarantee at least the continuity of v. For values k ∈ {1, . . . , n − 1} it is well known that v ∈ W k p (R n , R d ) is continuous for p > n k and could be discontinuous for p ≤ n k . So the borderline case is p = p • = n k . It is well known (see for instance [22]) that v ∈ W k p • (R n , R d ) is continuous if the derivatives of k-th order belong to the Lorentz space L p • ,1 , we will denote the space of such mappings by W k p • ,1 (R n , R d ). We refer to Sect. 2 for relevant definitions and notation.
Here H q ∞ (E) is as usual the q-dimensional Hausdorff content: Note that the case k = 1 was considered in the paper [22], and the case k > 1, q > p • in [24], so we omit them and consider here only the remaining limiting case q = p • , k > 1.
To study this limiting case, we need a new version of the Sobolev Embedding Theorem that gives inclusions in Lebesgue spaces with respect to suitably general positive measures. For β ∈ (0, n) denote by M β the space of all nonnegative Borel measures μ on R n such that where the supremum is taken over all n-dimensional cubic intervals I ⊂ R n and (I ) denotes sidelength. Recall the following classical theorem proved by Adams [2] (see also [5] and [28,Sect. 4.1] ).
Theorem A Let μ be a positive Borel measure on R n and α > 0, 1 < p < q < ∞, αp < n. Then for any f ∈ L p (R n ) the estimate holds with β = (n − αp) q p , where C depends on n, p, q, α only. Here |y − x| n−α dy is the Riesz potential of order α. The above estimate (1.2) fails for the limiting case q = p. Namely, there exist functions f ∈ L p (R n ) such that I α f (x) = +∞ on some set of positive (n − αp)-Hausdorff measure 1 , see for instance [23] and also for further background and history on the question [4]. Nevertheless, we prove the following result for this limiting case q = p: Theorem 1. 2 Let μ be a positive Borel measure on R n and α > 0, 1 < p < ∞, αp < n. Then for any f ∈ L p,1 (R n ) the estimate holds with β = n − αp, where C depends on n, p, α only.
In view of the definition of the Lorentz spaces, it is sufficient to prove the above assertion for the simplest case when f coincides with the indicator function of some compact set: Theorem 1.3 Let μ be a positive Borel measure on R n and α > 0, 1 < p < ∞, αp < n. Then for any compact set E ⊂ R n the estimate where 1 E is the indicator function of the set E and C depends on n, p, α only.
Note that our proof of the trace theorem is self-contained, is independent of the previous proofs of these type of results, and uses only very natural and elementary arguments.
From the definition of the space W k p • ,1 (R n , R d ) of Sobolev-Lorentz mappings and the classical estimate |∇v| ≤ C|I k−1 ∇ k v|, Theorem 1.2 implies Theorem 1.4 Let μ be a positive Borel measure on R n , k ∈ {1, . . . , n}. Then for any holds, where C depends on n, k only.
From these results, we deduce also some new differentiability and approximation properties of Sobolev-Lorentz mappings v ∈ W k p • ,1 (R n ). Namely, for m ≤ n the m-order derivatives ∇ m v are well-defined H m p • -almost everywhere, a function v is m-times differentiable (in the classical Fréchet-Peano sense) H m p • -almost everywhere, and, finally, it coincides with C m -smooth function on R n \ U , where the open exceptional set U has small H m p • ∞ -Hausdorff content (see Theorems 3.9, 3.11-3.12 ). Note that for mappings of the classical Sobolev space W k p • (R n ), the corresponding exceptional set U has small Bessel capacity B k−m, p (U ) < ε, and, respectively, the gradients ∇ m v are well defined in R n except for some exceptional set of zero Bessel capacity B k−m, p (see, e.g., [9] and Chap. 3 in [36]).
In the last Subsect. 3.5, we discuss Morse-Sard-type theorems for Sobolev-Lorentz mappings. Namely, for an open set ⊂ R n and a mapping v ∈ W k p • ,1,loc ( , R d ) denote Z v,m = {x ∈ : v is differentiable at x and rank ∇v(x) < m} (recall, that by previous results v is differentiable H p • a.e.). We state: is an open subset of R n , and v ∈ The theorem was proved for C k -smooth functions by Morse [30] in 1939 for the case k = n, m = d = q • = 1, and subsequently by Sard [32] in 1942 for k = n − m + 1, m = d = q • . For arbitrary natural values k, n, m, and C k -smooth functions, the result was proved almost simultaneously by Dubovitskiȋ [15] in 1967 and Federer [18,Theorem 3.4.3] in 1969 2 .
The Morse-Sard Theorem for Sobolev spaces W k p (R n , R m ) with p > n (i.e., when W k p (R n ) → C k−1 (R n ) ) was obtained in [13] (see also [20] for a simple proof), and for Lipschitz and Hölder continuous mappings C k,λ -see, e.g., in [7,8], respectively. More facts about history of this issue could be found in our papers [11,12,24], where the above Theorem 1.5 was proved in the Sobolev context W k p • (R n ) for k, m ∈ {2, . . . , n}. For k = 1 (i.e., q • = n) it is folklore (see, e.g., [33]), so in the present paper, we need to only consider the case m = 1, q • = p • = n k . Let us remark, in conclusion, that an interesting phenomenon occurs for functions of the Sobolev-Lorentz space W k p • ,1 (R n , R d ). On the one hand, the order of integrability is very sharp-the minimal order, that guaranties a priori only continuity of mappings. On the other hand, these mappings a posteriori have many additional analytical regularity properties: the Luzin N -property, differentiability and approximation properties, and the Morse-Sard property (see above).
For instance, if k = n − m + 1, then almost all level sets of mappings v ∈ W k p • ,1 (R n , R m ) are C 1 -smooth manifolds [24]. (The result should be contrasted with the fact that mappings of class W k p • ,1 (R n , R m ) are continuous only and need not to be C 1 -smooth in general. ) This property recently found some applications in mathematical fluid mechanics (see [25]).

Preliminaries
By an n-dimensional cubic interval, we mean a closed cube in R n with sides parallel to the coordinate axes. If Q is an n-dimensional cubic interval, then we write (Q) for its sidelength.
For a subset S of R n , we write L n (S) for its outer Lebesgue measure. The mdimensional Hausdorff measure is denoted by H m and the m-dimensional Hausdorff content by H m ∞ . Recall that for any subset S of R n we have by definition where for each 0 < α ≤ ∞, It is well known that H n (S) = H n ∞ (S) ∼ L n (S) for sets S ⊂ R n . To simplify the notation, we write is as usual defined as consisting of those R d -valued functions f ∈ L p (R n ) whose distributional partial derivatives of orders l ≤ k belong to L p (R n ) (for detailed definitions and differentiability properties of such functions see, e.g., [14,16,28,36]). Denote by ∇ l f the vector-valued function consisting of all l-th order partial derivatives of f arranged in some fixed order. However, for the case of first-order derivatives l = 1, we shall often think of ∇ f (x) as the Jacobi matrix of f at x, thus the d × n matrix whose r -th row is the vector of partial derivatives of the r -th coordinate function.
We use the norm and unless otherwise specified all norms on the spaces R s (s ∈ N) will be the usual euclidean norms. Working with locally integrable functions, we always assume that the precise representatives are chosen. If w ∈ L 1,loc ( ), then the precise representative w * is defined for all x ∈ by the limit exists and is finite,   0 otherwise, where the dashed integral as usual denotes the integral mean, and B(x, r ) = {y : |y − x| < r } is the open ball of radius r centered at x. Henceforth, we omit special notation for the precise representative writing simply w * = w. We will say that x is an L p -Lebesgue point of w (and simply a Lebesgue point If k < n, then it is well known that functions from Sobolev spaces W k p (R n ) are continuous for p > n k and could be discontinuous for p ≤ p • = n k (see, e.g., [28,36]). The is a refinement of the corresponding Sobolev space that for our purposes turns out to be convenient. Among other things functions that are locally in W k p • ,1 on R n are in particular continuous. Given a measurable function f : Since | f | and f * are equimeasurable, we have for every 1 ≤ p < ∞, is finite. We refer the reader to [27,35,36] for information about Lorentz spaces. However, let us remark that in view of the definition of · L p,q and the equimeasurability of f and f * we have f L p = f L p, p so that in particular L p, p (R n ) = L p (R n ). Further, for a fixed exponent p and q 1 < q 2 , we have the estimate f L p,q 2 ≤ f L p,q 1 , and, consequently, the embedding L p, Finally, we recall that · L p,q is a norm on L p,q (R n ) for all q ∈ [1, p] and a quasi-norm in the remaining cases q ∈ ( p, ∞] (see [27,Proposition 3.3]).
Here we shall mainly be concerned with the Lorentz space L p,1 , and in this case, one may rewrite the norm as (see for instance [27,Proposition 3.6]) We record the following subadditivity property of the Lorentz norm for later use.
Lemma 2.1 (see, e.g., [27,31]) Suppose that 1 ≤ p < ∞ and E = j∈N E j , where E j are measurable and mutually disjoint subsets of R n . Then for all f ∈ L p,1 we have where 1 E denotes the indicator function of the set E.
of degree at most m by the following rule: The following well-known bound will be used on several occasions.
Then v is a continuous mapping and for any n-dimensional cubic interval Q ⊂ R n the estimate holds. Taking into account the identity ∇ k u ≡ ∇ k v on Q and (2.6), we obtain the required estimate (2.5).
Then v is a continuous mapping and for any n-dimensional cubic interval Q ⊂ R n the estimate The above results can easily be adapted to give that v ∈ C 0 (R n ), the space of continuous functions on R n that vanish at infinity (see for instance [27,Theorem 5.5]).
Analogously, from previous estimates one could deduce Then for all m ∈ {1, . . . , k} and for any n-dimensional cubic interval Q ⊂ R n the estimate holds.
Here, we list some standard facts about the Lorentz spaces.
Theorem 2.5 (Boundedness of the maximal operator, see [27] Here is the usual Hardy-Littlewood maximal function of f . Corollary 2.6 (Regularization in Lorentz spaces [27]) Here and henceforth C ∞ 0 (R n ) means the space of C ∞ smooth and compactly supported functions on R n .
We need also the following classical fact (cf. with [10] ).
Lemma 2.10 (see Lemma 2 in [14]) Let u ∈ W m 1 (R n ), m ≤ n. Then for any ndimensional cubic interval Q ⊂ R n , x ∈ Q, and for any j = 0, 1, . . . , m − 1 the estimate holds, where the constant C depends on n, m only.

The Trace Theorem
Theorem 1.3 plays the key role among other results. Its proof splits into a number of lemmas. Fix parameters m > 0, 1 < p < ∞, 0 < αp < n, and a positive Borel measure μ on R n satisfying μ (B(x, r )) ≤ r n−αp (3.1) for every ball B(x, r ) ⊂ R n . Fix also a compact set E ⊂ R n . Denote by I E the corresponding Riesz potential I α (1 E ). It is very easy to check by standard calculation that where the constant C 0 depends on n, α only. Denote also t m = 2 m (here m ∈ Z), In this section, we will write f g, if f ≤ Cg, where C depends on n, α, p only (really, most of the corresponding constants below up to Lemma 3.6 depends on n, α only). For any z ∈ E \ B we have |z − y| ≥ r = |x − y|, thus, |x − z| ≤ |x − y| + |z − y| ≤ 2|z − y|, consequently, 2r ), by elementary estimates we have Denote F m = {x ∈ R n : I E (x) ∈ [t m , 2t m ]}, μ m = μ(F m ), μ m (·) = μ F m . By construction, (3.6) So our main purpose below is to estimate t m μ m . Of course, t m μ m ≤ R n I E (x) dμ m (x). By Fubini's Theorem, we have holds, where m 0 is a constant from Lemma 3.1.
, and the last inequality implies in conjunction with Fubini's Theorem (3.7).

Lemma 3.4 There exists a constant m
Proof Let m 1 ∈ N, its exact value will be specified below. We have |E ∩ B(x, ρ)| ≤ ω n ρ n , where ω n is a volume of a unit ball in R n . Thus So the target estimate (3.9) follows from (3.8) provided that 1 α ω n 2 −m 1 is sufficiently small. Proof Let i ≥ m − m 1 , ρ α ≤ t i+1 , (3.11) and x ∈ F m = supp μ m . Then by definitions

Lemma 3.5 There exists a constant i
Take any point y ∈ E m−m 0 ∩ B(x, ρ), then by construction y ∈ E j for some index j ≥ m − m 0 , in particular, I E (y) ≥ t j . Suppose j ≥ i + 1. Then (3.11) implies |x − y| α ≤ t i+1 ≤ t j , therefore, by Lemma 3.2 (applying for t = t j ) we have I E (x) ≥ C 2 t j . Thus by (3.12) we obtain j ≤ m + m 2 for some constant m 2 depending on α, n only. Finally, we have j ≤ max(i + 1, m + m 2 ) ≤ max(i + 1, i + m 1 + m 2 ) finishing the proof of the Lemma.
where the symbol i ↔ j indicates that the order of summation was changed. holds. (3.16) where again the symbol m ↔ j means that the order of summation was changed.
Combining Lemma 3.7 with the initial estimate (3.6) gives the validity of the Trace Theorem 1.3.

On Approximation of Sobolev-Lorentz Mappings
Using the established Theorem 1.2 and Adam's estimate from Theorem 2.9 with β = n − (k − l) p, we obtain the following estimates, which are key ingredients in the proof of N -property. ∈ (1, ∞), k, l ∈ {1, . . . , n}, l ≤ k, (k − l) p < n. Then for any function f ∈ W k p,1 (R n ) the estimates

Corollary 3.8 Let p
hold, where β = n − (k − l) p and the constant C depends on n, k, p only.
The main result of this subsection is the following Then for any f ∈ W k p,1 (R n ) and for each ε > 0 there exist an open set U ⊂ R n and a function g ∈ C l (R n ) such that Note that in the analogous theorem for the case of Sobolev mappings f ∈ W k p (R n ), the assertion (i) should be reformulated as follows: denotes the Bessel capacity of the set U (see, e.g., [36,Chap. 3] or [9] ).
Proof of Theorem 3.9 Let the assumptions of the Theorem be fulfilled. By Theorem 2.5 and Corollary 2.7, we can choose the sequence of mappings Then one could repeat almost word by word the proof of [12, Theorem 3.1]. Since there are no essential differences, we omit the detailed calculations here.

On Differentiability Properties of Sobolev-Lorentz Mappings
We start with the following simple technical observation.
holds for x ∈ R n \ U .
Proof The proof of the Lemma follows standard arguments, we reproduce it here for reader's convenience. Fix σ > 0. Let {B α } be a family of disjoint balls B α = B(x α , r α ) such that and sup α r α < δ for some δ > 0, where δ is chosen small enough to guarantee that sup α 1 B α · ∇ k v L p•,1 < 1. Then by Lemma 2.1 we have Since the last term tends to 0 as L n ( α B α ) → 0, and L n ( α B α ) ≤ c δ n−l p • α r l p • α , we get easily that α r l p • α → 0 as δ 0. Using this fact and some standard covering lemmas, we arrive in a routine manner that for a set A σ,δ := x ∈ R n : ∃r ∈ (0, δ] holds for any fixed σ > 0. The rest part of the proof of the lemma is obvious, so we omit it. From the last lemma (for l = 1), Theorem 3.9 (ii) and estimate (2.7) we obtain the following result: Then there exists a Borel set A v ⊂ R n such that H p • (A v ) = 0 and for any x ∈ R n \ A v the function v is differentiable (in the classical Fréchet sense) at x, furthermore, the classical derivative coincides with ∇v(x) (x is a Lebesgue point for ∇v ).
The case k = 1, p • = n is a classical result due to Stein [34] (see also [22]), and for k = n, p • = 1 the result is also proved in [14].
There holds the following extension of Theorem 3.11.
Theorem 3.12 Let k, l ∈ {1, . . . , n}, l ≤ k, and v ∈ W k p • ,1 (R n , R d ). Then there exists a Borel set A v ⊂ R n such that H l p • (A v ) = 0 and for any x ∈ R n \ A v the function v is l-times differentiable (in the classical Fréchet-Peano sense) at x, i.e., where T v,l,x (y) is the Taylor polynomial of order l for v centered at x (which is well defined H l p • -a.e. by Theorem 3.9).
Proof We consider only the case l < n; for l = n, the arguments are similar and becomes even simpler. Below we follow methods of [11, proof of Lemma 5.5] and [12, proof of Theorem 3.1]. By Theorem 3.9 of the present paper, there exists a set A l such that H l p • (A l ) = 0 and the derivatives ∇ j v(x) are well defined for all x ∈ R n \ A l and j = 0, 1, . . . , l. Further, by Lemma 3.10, there exists a sequence of open sets holds for x ∈ R n \ U i . It means that there exists a function ω i : (0, +∞) → (0, +∞) such that ω i (r ) → 0 as r 0 and Take a sequence of mappings v i : Then by estimate (3.18) we have for all x ∈ R n \ G i and all j ≥ i. Moreover, since v j ∈ C ∞ 0 (R n ), there exists constants M j such that |∇ k v j (x)| ≤ M j ∀x ∈ R n , this fact and (3.20) implies We start by estimating the remainder termṽ j (y) − Tṽ j ,l,x (y). Fix y ∈ R n , x ∈ R n \ G i , j ≥ i, and an n-dimensional cubic interval Q such that x, y ∈ Q, |x − y| ∼ (Q). By construction and Lemma 2.10, for any multi-index α with |α| ≤ l we have (2.8), (3.23), (3.24) ≤ C2 − j r l + ω i (r )r l + M j r n + |α|≤l 1 α! ∂ αṽ (3.26) Finally from the last estimate and equality v =ṽ j + v j , we have where ω i (r ) → 0 and ω v j (r ) → 0 as r → 0 (the latter holds since v j ∈ C ∞ 0 (R n ) ). We emphasize that the last inequality is valid for all y ∈ R n , j ≥ i, and x ∈ R n \ G i . Therefore uniformly for all x ∈ R n \ G i . This means, that v is uniformly l-times differentiable (in the classical Fréchet-Peano sense) at every x ∈ R n \ G i . Then the estimate (3.22) finishes the proof.

The Proof of the N-Property
In this subsection, we need to prove the assertion of Theorem Recall that for the case k = 1 this assertion was proved in [22], and for k = n it was proved in [12], so we omit these cases. Our proof here for the remaining cases follows and expands on the ideas from [12]. For the remainder of this section, we fix k ∈ {2, . . . , n − 1}, and a mapping v in W k p • ,1 (R n , R d ). To prove Theorem 3.13, we need some preliminary lemmas that we turn to next.
Applying Corollary 3.8 for the case p = p • = n k , l = 1, we obtain where C depends on n, p • , d only.
By a dyadic interval, we understand a cubic interval of the form where k i , m are integers. The following assertion is straightforward, and hence we omit its proof here. Lemma 3.14 For any n-dimensional cubic interval J ⊂ R n , there exist dyadic intervals Q 1 , . . . , Q 2 n such that J ⊂ Q 1 ∪· · ·∪ Q 2 n and (Q 1 ) = · · · = (Q 2 n ) ≤ 2 (J ).
Let {Q α } α∈A be a family of n-dimensional dyadic intervals. We say that the family {Q α } is regular, if for any n-dimensional dyadic interval Q the estimate holds. Since dyadic intervals are either nonoverlapping or contained in one another, (3.28) implies that any regular family {Q α } must in particular consist of nonoverlapping intervals.
Proof Fix ε ∈ (0, 1) and let {Q α } be a regular family of n-dimensional dyadic intervals satisfying (3.29), where δ > 0 will be specified below. Let us start by checking (3.30). We have Using (2.2), we can rewrite the last estimate as it follows that the integral on the right-hand side of (3.32) tends to zero as L n ( α Q α ) tends to zero. In particular, it will be less than ε if the condition (3.29) is fulfilled with a sufficiently small δ. Thus (3.30) is established for all δ ∈ (0, δ 1 ], where δ 1 = δ 1 (ε, v) > 0.
Next we check (3.31). By virtue of Corollary 2.7, applied coordinate-wise, we can find a decomposition v = v 0 + v 1 , where ∇v 0 L ∞ ≤ K = K (ε, v) and (3.33) Assume that δ ∈ (0, δ 1 ] and α (Q α ) p • < δ < 1 K p• +1 ε. (3.34) Define the measure μ by (3.35) where 1 Q α denotes the indicator function of the set Q α . Claim The estimate sup holds, where the supremum is taken over all n-dimensional cubic intervals. Indeed, write for a dyadic interval Q By regularity of {Q α } the first sum is bounded above by (Q) p • . If the second sum is nonzero, then there must exist an index α such that Q α Q and Q α , Q overlap. But as the intervals {Q α } are nonoverlapping and dyadic, we must then precisely have one such interval Q α and Q α ⊃ Q. But then the first sum is empty and the second sum has only the one term (Q) n / (Q α ) n− p • , hence is at most (Q) p • . Thus the estimate μ(Q) ≤ (Q) p • holds for every dyadic Q. The inequality (3.36) in the case of a general cubic interval J follows from the above dyadic case and Lemma 3.14. The proof of the claim is complete. Now return to (3.31). By properties (3.27), (3.33)" and (3.34) (applied to the mapping v 1 ), we have Since ε > 0 was arbitrary, the proof of Lemma 3.16 is complete. ∞ (E) < δ 0 , then E can be covered by a regular family {Q α } of n-dimensional dyadic intervals with α (Q α ) p • < δ. Remark 3.17 Note that the order of integrability p • is sharp: for example, the Luzin N -property fails in general for continuous mappings v ∈ W 1 n (R n , R n ) (here k = 1, q = p • = n), see, e.g., [26].
Proof By virtue of (2.5) it suffices to prove that (3.40) for the mapping v Q defined in Lemma 2.2, where C = C(n, m, k, d) is a constant. To establish (3.40), it is possible to repeat almost verbatim the proof of Lemma 3.2 in [24]. One must observe the following minor changes: first q • = p • , and next, instead of Corollary 1.8 from [24] one must use Corollary 3.8 established above. Note that in the present situation the calculations simplify since for m = 1 many of terms from [24, proof of Lemma 3.2] disappear.

Corollary 3.19
For any ε > 0 there exists δ > 0 such that for every subset E of R n we have H Proof Let L n (E) ≤ δ, then we can find a family of nonoverlapping n-dimensional dyadic intervals Q α such that E ⊂ α Q α and α n (Q α ) < Cδ. Then in view of This estimate together with Lemma 3.18 allow us to conclude the required smallness of Invoking Dubovitskiȋ-Federer's Theorem (see commentary to the Theorem 1.5 in the Introduction) for the smooth case g ∈ C k (R n , R d ), Theorem 3.9 (iii) (applied to the case l = k ) implies Corollary 3.20 (see, e.g., [13]) There exists a set Z v of n-dimensional Lebesgue measure zero such that From Corollaries 3.19 and 3.20, we conclude that H p • (v(Z v )) = 0, and this ends the proof of Theorem 1.5.