Morse--Sard theorem and Luzin N -property: a new synthesis result for Sobolev spaces
aa r X i v : . [ m a t h . A P ] S e p Morse–Sard theorem and Luzin N -property: a newsynthesis result for Sobolev spaces Adele Ferone, Mikhail V. Korobkov, and Alba RovielloMarch 21, 2019
Abstract
For a regular (in a sense) mapping v : R n → R d we study the following problem: let S be a subset of m -critical a set e Z v,m = { rank ∇ v ≤ m } and the equality H τ ( S ) = 0 (or the inequality H τ ( S ) < ∞ ) holds for some τ > . Does it implythat H σ ( v ( S )) = 0 for some σ = σ ( τ, m ) ? (Here H τ means the τ -dimensionalHausdorff measure.)For the classical classes C k -smooth and C k + α -Holder mappings this problemwas solved in the papers by Bates and Moreira. We solve the problem for Sobolev W kp and fractional Sobolev W k + αp classes as well. Note that we study the Sobolevcase under minimal integrability assumptions p = max(1 , n/k ), i.e., it guarantees ingeneral only the continuity (not everywhere differentiability) of a mapping.In particular, there is an interesting and unexpected analytical phenomena here:if τ = n (i.e., in the case of Morse–Sard theorem), then the value σ ( τ ) is the samefor the Sobolev W kp and for the classical C k -smooth case. But if τ < n , then thevalue σ depends on p also; the value σ for C k case could be obtained as the limitwhen p → ∞ . The similar phenomena holds for Holder continuous C k + α and forthe fractional Sobolev W k + αp classes.The proofs of the most results are based on our previous joint papers withJ. Bourgain and J. Kristensen (2013, 2015). We also crucially use very deepY. Yomdin’s entropy estimates of near critical values for polynomials (based onalgebraic geometry tools). MSC 2010:
Key words:
Holder mappings, Morse–Sard theorem, Dubovitski˘ı–Federer theorems,Sobolev–Lorentz mappings, Bessel potential spaces
The Morse–Sard theorem in its classical form states that the image of the set of criticalpoints of a C n − d +1 smooth mapping v : R n → R d has zero Lebesgue measure in R d .1ore precisely, assuming that n ≥ d , the set of critical points for v is Z v = { x ∈ R n :rank ∇ v ( x ) < d } and the conclusion is that L d ( v ( Z v )) = 0 (1.1)whenever v ∈ C k with k ≥ max(1 , d − m + 1). The theorem was proved by Morse [47]in 1939 for the case d = 1 and subsequently by Sard [52] in 1942 for the general vector–valued case. The celebrated results of Whitney [57] show that the C n − d +1 smoothnessassumption on the mapping v is sharp.Another important item of the real analysis, N -property, means that the image v ( E )has zero measure whenever E has zero measure (see the recent paper [26], where wediscuss the history of the topic).We need some usual notation. Fix a pair of positive parameters τ and σ . A continuousmapping v : R n → R d is said to satisfy ( τ, σ )- N - property, if H σ ( v ( E )) = 0 whenever H τ ( E ) = 0,where H τ means the Hausdorff measure.For a C -smooth mapping v : R n → R d and for an integer number m ∈ Z + denote e Z v,m = { x ∈ R n : rank ∇ v ( x ) ≤ m } . Then for parameters τ, σ > v : R n → R d satisfies ( τ, σ, m )- N - property, if H σ ( v ( E )) = 0 whenever E ⊂ e Z v,m with H τ ( E ) = 0.Further, we say that that a mapping v : R n → R d satisfies strict ( τ, σ, m ) - N -property, if H σ ( v ( E )) = 0 whenever E ⊂ e Z v,m with H τ ( E ) < ∞ .Using this notation, the above classical Morse–Sard theorem means, that every C k -mapping v : R n → R d has strict ( n, d, d − N -property if k ≥ n − d + 1.The starting point for our research is the following recent result for classically smoothcase. Theorem 1.1 (Bates S.M. and Moreira C., 2002 [10, 46]) . Let m ∈ { , . . . , n − } , k ≥ , d ≥ m , ≤ α ≤ , and v ∈ C k,α ( R n , R d ) . Then for any τ ∈ [ m, n ] the mapping v has ( τ, σ, m ) - N -property with σ = m + τ − mk + α . (1.2) We use the symbol e Z , since in our previous papers we denoted Z v,m = { x ∈ R n : rank ∇ v ( x ) < m } .So in the present notation e Z v,m = Z v,m +1 . oreover, this N -property is strict if at least one of the following additional assumptionsis fulfilled:1) τ = n (in particular, it includes the case of the classical Morse–Sard theorem);2) τ > m and α = 0 (that means v ∈ C k );3) τ > m and v ∈ C k,α + ( R n , R d ) . Here we say that a mapping v : R n → R d belongs to the class C k,α for some positiveinteger k and 0 < α ≤ v ∈ C k and there exists a constant L ≥ |∇ k v ( x ) − ∇ k v ( y ) | ≤ L | x − y | α for all x, y ∈ R n .To simplify the notation, let us make the following agreement: for α = 0 we identify C k,α with usual spaces of C k -smooth mappings.Analogously, we say that a mapping v : R n → R d belongs to the class C k,α + for somepositive integer k and 0 < α ≤
1, if there exists a function ω : R + → R + such that ω ( r ) → r → |∇ k v ( x ) − ∇ k v ( y ) | ≤ ω ( r ) · | x − y | α whenever | x − y | < r. (1.3)Note that the assertion of Theorem 1.1 is rather sharp: for example, if its conditions1)–3) are not satisfied, than the corresponding ( τ, σ, m )- N -property is not strict in general,it follows from Whitney’s counterexamples [57], see also [48] for commentaries.Of course, the assertion of Theorem 1.1 includes Morse–Sard theorem and many otherresults on this topic as partial cases; for convenience, we made some historical referencesbelow in Subsection 1.2. The purpose of our paper is to extend this result to the mappingsof Sobolev spaces. In this subsection W kp ( R n , R d ) means the space of Sobolev mappings with all derivativesof order j ≤ k belonging to the Lebesgue space L p .Let k ∈ N , 1 < p < ∞ and 0 ≤ α <
1. One of the most natural type of fractionalSobolev spaces is (Bessel) potential spaces L k + αp .Recall, that a function v : R n → R d belongs to the space L k + αp , if it is a convolution ofa function g ∈ L p ( R n ) with the Bessel kernel G k + α , where \ G k + α ( ξ ) = (1 + 4 π ξ ) − ( k + α ) / .It is well known that for the integer exponents (i.e., when α = 0) one has the identity L kp ( R n ) = W kp ( R n ) if 1 < p < ∞ . As well-known, if ( k + α ) p > n , then functions from the potential space L k + αp ( R n ) arecontinuous by Sobolev Imbedding theorem, but in general the gradient ∇ v is not well-defined everywhere. Thus now for the Sobolev case the m -critical set is defined as e Z v,m = { x ∈ R n : x ∈ A v or x ∈ R n \ A v with rank ∇ v ( x ) ≤ m } . A v means the set of ‘bad’ points at which either the function v is not differentiableor which are not the Lebesgue points for ∇ v . So in the paper we consider these ‘bad’nonregular points automatically as m -critical for any m (such assumption, of course,makes the corresponding ( τ, σ, m )- N -properties more stronger). Theorem 1.2.
Let m ∈ { , . . . , n − } , k ≥ , d ≥ m , ≤ α < , p > , ( k + α ) p > n ,and let v ∈ L k + αp ( R n , R d ) . Denote τ ∗ = n − ( k + α − p . Suppose in addition that τ > m and τ > τ ∗ , then the mapping v has strict ( τ, σ, m ) - N -property with σ = m + p ( τ − m ) τ + ( k + α ) p − n . (1.4) Further, if τ = m > τ ∗ , then v has nonstrict ( τ, m, m ) - N -property. We need to make several remarks here. • First of all, let us note, that the value σ in Theorems 1.1–1.2 coincide for theboundary cases τ = m or τ = n , but they are different for m < τ < n (of course,then σ for Sobolev case is larger). Nevertheless, σ in Theorem 1.1 could be obtainedby taking a limit in (1.4) as p → ∞ ; • Recall, that by approximation results (see, e.g., [54] and [36] ) the set of ‘bad’points A v is rather small, i.e., it has the Hausdorff dimension τ ∗ : H τ ( A v ) = 0 ∀ τ > τ ∗ := n − ( k + α − p if v ∈ L k + αp ( R n ) . (1.5)In particular, A v = ∅ if ( k + α − p > n . • The condition τ > τ ∗ in Theorem 1.2 is essential and sharp: namely, in the paper [26]we constructed a counterexample of a mapping from L k + αp ( R n ) not satisfying the( τ, σ, m )- N -property with τ = τ ∗ = m = σ = 1. • The usual ( τ, σ )- N -properties (without constraints on the gradient, i.e., when m = n ) were studied in our previous paper [26], see also subsection 1.2, Theorems 1.4–1.5. (One has to use these usual N -properties also if the assumptions τ > m and τ > τ ∗ of Theorem 1.2 are not satisfied.)Thus above Theorem 1.2 omits the limiting cases ( k + α ) p = n and τ = τ ∗ . However,it is possible to cover these cases as well using the Lorentz norms. Namely, denote by L k + αp, ( R n , R d ) the space of functions which could be represented as a convolution ofthe Bessel potential G k + α with a function g from the Lorentz space L p, (see the definitionof these spaces in the section 2). In our previous papers we consider the m -critical points and ‘bad’ points A v separately. heorem 1.3. Let m ∈ { , . . . , n − } , k ≥ , d ≥ m , ≤ α < , p ≥ and let v : R n → R d be a mapping for which one of the following cases holds:(i) α = 0 , k ≥ n , and v ∈ W k ( R n , R d ) ;(ii) ≤ α < , p > , ( k + α ) p ≥ n , and v ∈ L k + αp, ( R n , R d ) .Denote τ ∗ = n − ( k + α − p . Suppose in addition that τ > m and τ ≥ τ ∗ , then the mapping v has strict ( τ, σ, m ) - N -property with the same σ defined by (1.4).Further, if τ = m ≥ τ ∗ , then v has the corresponding nonstrict ( τ, m, m ) - N -property. So here the limiting case τ = τ ∗ is included. Some other commentaries: • Recall, that by approximation results (see, e.g., [54] and [36] ) the set of ‘bad’points A v for this Sobolev–Lorentz case has the same Hausdorff dimension τ ∗ = n − ( k + α − p , but it is smaller in a sense, namely: H τ ∗ ( A v ) = 0 if v is from Theorem 1.3 . (1.6)(compare with (1.5) ). In particular, A v = ∅ if ( k + α − p ≥ n . • For the integer exponents (i.e., when α = 0) the Sobolev–Lorentz potential spacehas a more simple and natural description: L kp, ( R n ) = W kp, ( R n ) if 1 < p < ∞ , there by W kp, we denote the subspace of W kp consisting of functions whose derivativesof order k belongs to the Lorentz space L p, (see, e.g., [26]). There are a lot of papers devoted to the Morse–Sard theorem, and the above formulatedresults includes many previous theorems as partial cases. For example, for smooth caseif α = 0, τ = n , then we have σ = m + n − mk , and the assertion of Theorem 1.1 coincides with the classical Federer–Dubovitski˘ı theorem,obtained almost simultaneously by Dubovitski˘ı [22] in 1967 and Federer [25, Theorem3.4.3] in 1969. Of course, it includes the original Morse–Sard theorem as partial case(when k = n − m, σ = m + 1 ).Note also, that Theorem 1.1 was formulated as a Conjecture by A.Norton in [48,page 369] and it includes as partial cases some relative results of other mathematicians:Norton himself (who proved the assertion for the case σ = d , τ = ( k + α )( d − m ) + m ),5.Yomdin [58] (case τ = n , v ∈ C k,α + , see also [13] ), M. Kucera [38] (case τ = n , m = 1,i.e., when the gradient totally vanishes on the critical set), etc.Concerning the Sobolev case, in the pioneering paper by De Pascale [18] the assertionof the initial Morse–Sard theorem (1.1) (i.e., when k = n − d + 1, m = d − σ = d ) wasobtained for the Sobolev classes W kp ( R n , R m ) under additional assumption p > n (in thiscase the classical embedding W kp ( R n , R m ) ֒ → C k − holds, so there are no problems withnondifferentiability points).Some other Morse–Sard type theorems for Sobolev cases were obtained in [13] and [29],these papers mainly concern the Dubovitski˘ı–Fubini type properties for the Morse–Sardtheorem, which will be discussed in the next subsection.In addition to the above mentioned papers there is a growing number of papers on thetopic, including [6, 7, 8, 9, 17, 28, 49, 55, 56].Finally, Theorems 1.2 and 1.3 for the most important case τ = n were obtained in ourprevious paper [27] (see also our preceding articles [15, 16, 30, 35, 36] of the first authorwith J.Bourgain, J.Kristensen, and P. Haj lasz on this topic).The usual ( τ, σ )- N -properties (without constraints on the gradient, i.e., when m = n )were studied in our previous paper [26], where we proved the following two theorems: Theorem 1.4 ([26]) . Let α > , < p < ∞ , αp > n , and v ∈ L αp ( R n , R d ) . Supposethat < τ ≤ n . Then the following assertions hold:( ◦ ) if τ = τ ∗ = n − ( α − p , then v has the ( τ, σ ) - N -property, where the value σ = σ ( τ ) is defined as σ ( τ ) := τ, if τ ≥ τ ∗ := n − ( α − p ; p ταp − n + τ , if 0 < τ < τ ∗ . (1.7) ( ◦◦ ) if α > and τ = τ ∗ > then σ ( τ ) = τ ∗ and the mapping v in general hasno ( τ ∗ , τ ∗ ) - N -property, i.e., it could be H τ ∗ ( v ( E )) > for some E ⊂ R n with H τ ∗ ( E ) = 0 . The similar results were announced in [5], see [26] for our commentaries and otherhistorical remarks on this important case.The above Theorem 1.4 omits the limiting cases αp = n and τ = τ ∗ . As above, it ispossible to cover these cases as well using the Lorentz norms. Theorem 1.5 ([30, 26]) . Let v : R n → R d be a mapping for which one of the followingcases holds:(i) v ∈ W k ( R n , R d ) for some k ∈ N , k ≥ n ;(ii) v ∈ L αp, ( R n , R d ) for some α > , p ∈ (1 , ∞ ) with αp ≥ n .Suppose that < τ ≤ n . Then v is a continuous function satisfying the ( τ, σ ) - N -property,where again the value σ = σ ( τ ) is defined in (1.7) (with α = k and p = 1 for the (i) case). So, in the last theorem the critical case τ = τ ∗ is included .6 .3 The Dubovitski˘ı–Fubini type properties for the Morse–Sardtheorem As it was mentioned by A.Norton [48, page 369], the absence of a Fubini theorem forHausdorff measure makes an obstacle for proofs of some new Morse–Sard type theorems.Nevertheless, in 1957 Dubovitski˘ı proved, that surprisingly some Fubini type propertiesalways hold for the Morse–Sard topic.
Theorem A (Dubovitski˘ı 1957 [21]) . Let n, d, k ∈ N , and let v : R n → R d be a C k –smooth mapping. Then H µ ( Z v ∩ v − ( y )) = 0 for L d -a.a. y ∈ R d , (1.8) where µ = n − d + 1 − k and Z v = { x ∈ R n : rank ∇ v ( x ) < d } . Here and in the following we interpret H β as the counting measure when β ≤
0. Thusfor k ≥ n − d + 1 we have ν ≤
0, and H µ in (1.8) becomes simply the counting measure,so the Dubovitski˘ı theorem contains the Morse–Sard theorem as particular case.It turns out that the similar Fubini type extensions hold for the Theorems 1.1–1.3stated above. Remark 1.1.
The following language below may seem too technical and cumbersome. So,a disinterested reader can omit them; anyway the main results of the article are the abovetheorems 1.2–1.3. Nevertheless, authors consider the following theorems as importantstrengthens of theorems 1.1–1.3, as they allow to realise the idea of Dubovitsky’s approachin general situation, and include all the theorems given in this article as a particular case;moreover, they are new even for the classical smooth cases C k and C k,α .We need some notation. For parameters µ ≥ q ≥ m , τ > v : R n → R d satisfies ( τ, µ, q, m )- N - property, if H µ ( E ∩ v − ( y )) = 0 for H q -almost all y ∈ v ( E ) whenever E ⊂ e Z v,m with H τ ( E ) = 0.(1.9)Recall, that here as above e Z v,m = { x ∈ R n : rank ∇ v ( x ) ≤ m } . Obviously,if µ ≤
0, then the ( τ, µ, q, m )- N -property is equivalent to the ( τ, q, m )- N -property.(1.10)Further, we say that that a mapping v : R n → R d satisfies strict ( τ, µ, q, m ) - N -property, if H µ ( E ∩ v − ( y )) = 0 for H q -almost all y ∈ v ( E ) whenever E ⊂ e Z v,m with H τ ( E ) < ∞ . Theorem 1.6 (Smooth case v ∈ C k,α ( R n , R d )) . Under assumptions of Theorem 1.1 onecan replace the assertion about ( τ, σ, m ) - N -properties by the more strong assertion about ( τ, q, µ, m ) - N -property for any τ ∈ [ m, n ] and q ∈ [ m, σ ] with µ = τ − m − ( k + α )( q − m ) . (1.11)7 urther, if q > m and at least one of the corresponding conditions of Theorem 1.1is fulfilled, then this ( τ, q, µ, m ) - N -property is strict. The similar assertions hold for Sobolev and Sobolev–Lorentz cases (we use the defini-tion from subsection 1.1 for the m -critical set e Z v,m of Sobolev functions). Theorem 1.7 (Sobolev case v ∈ L k,αp ( R n , R d ), ( k + α ) p > n ) . Under assumptions ofTheorem 1.2 one can replace the assertion about strict ( τ, σ, m ) - N -properties by the morestrong assertion about strict ( τ, q, µ, m ) - N -property for any τ > max( τ ∗ , m ) , q ∈ ( m, σ ] with µ = τ − m − ( k + α − np + τp )( q − m ) . (1.12) Further, if q = m , τ > τ ∗ , and τ ≥ m , then v has nonstrict ( τ, m, µ, m ) - N -property with µ = τ − m . Theorem 1.8 (Sobolev–Lorentz case v ∈ L k,αp, ( R n , R d ), kp ≥ n ) . Under assumptions ofTheorem 1.3 one can replace the assertion about strict ( τ, σ, m ) - N -properties by the morestrong assertion about strict ( τ, q, µ, m ) - N -property for any τ ≥ τ ∗ , τ > m , q ∈ ( m, σ ] ,and with the same µ as in (1.12). Further, if q = m and τ ≥ max( m, τ ∗ ) , then v hasnonstrict ( τ, m, µ, m ) - N -property with µ = τ − m . It is easy to see, that in formulation of Theorems 1.6–1.8 if we take q = σ , then µ = 0, where σ is defined in formulation of the corresponding Theorems 1.1–1.3. It means(see (1.10) ), that Theorems 1.6–1.8 include the previous Theorems 1.1–1.3 as particularcase. Remark 1.2 (Some historical remarks) . It is interesting to note that this Dubovitski˘ıTheorem A remained almost unnoticed by West mathematicians for a long time; anotherproof was given in the recent paper Bojarski B. et al. [13], where they proved also a versionof this theorem for Holder classes C k,α + with vanishing condition (1.3). Further, in [29]Haj lasz and Zimmerman replaced the assumption v ∈ C k ( R n , R d ) of Theorem A by theassumption of Sobolev regularity v ∈ W kp ( R n , R d ) with p > n (this is an analog of DePas-cale extension for the Morse-Sard, see subsection 1.2, cf. with our assumptions kp > n or kp ≥ n in theorems 1.2–1.8 ).It is easy to see that Dubovitski˘ı Theorem A is a partial case of Theorem 1.6 of thepresent paper with parameters τ = n , α = 0 and q = m + 1 = d . Note, that the lastassumption (which also used in [13], [29] ) simplifies the proofs very essentially, becauseautomatically one has that the image v ( E ) is H q - σ -finite. But in general in theorems 1.6–1.8 the image v ( E ) may have Hausdorff dimension much large than q for E ⊂ e Z v,m with H τ ( E ) = 0. Nevertheless, the equality H µ ( v − ( y ) ∩ Z v,m ) = 0 is fulfilled for q -almostall y ∈ v ( E ) as required in definition (1.9 )Finally, let us note that the assertions of Theorems 1.6–1.8 for the case τ = n wereproved in our previous paper [27] and in the papers of [30] by Haj lasz, Korobkov, Kris-tensen. 8ithout the gradient constraints, the Dubovitski˘ı–Fubini analogs of Theorems 1.4–1.5were obtained in our previous paper [26]. Theorem 1.9 ([26], Sobolev case) . Let α > , < p < ∞ , αp > n , and v ∈ L αp ( R n , R d ) .Suppose that < τ ≤ n and τ = τ ∗ = n − ( α − p . Then for every q ∈ [0 , σ ] and for anyset E ⊂ R n with H τ ( E ) = 0 the equality H µ ( E ∩ v − ( y )) = 0 for H q -a.a. y ∈ R d (1.13) holds, where µ = τ (cid:0) − qσ (cid:1) and the value σ = σ ( τ, α, p ) is defined in (1.7). The above Theorem 1.9 omits the limiting cases αp = n and τ = τ ∗ . As above, it ispossible to cover these cases as well using the Lorentz norms. Theorem 1.10 ([26], Sobolev–Lorentz case) . Let v : R n → R d be a mapping for whichone of the following cases holds:(i) v ∈ W k ( R n , R d ) for some k ∈ N , k ≥ n ;(ii) v ∈ L αp, ( R n , R d ) for some α > , p ∈ (1 , ∞ ) with αp ≥ n .Suppose that < τ ≤ n . Then for every q ∈ [0 , σ ] and for any set E ⊂ R n with H τ ( E ) = 0 the equality (1.13) holds with the same µ and σ defined in (1.7) (with α = k and p = 1 for the case (i) ). Taking τ ≥ τ ∗ , we obtain, in particular, Corollary 1.1.
Let α > , < p < ∞ , αp > n , and v ∈ L αp ( R n , R d ) . Suppose that < τ ≤ n and τ > τ ∗ = n − ( α − p . Then for every q ∈ [0 , τ ] and for any set E ⊂ R n with H τ ( E ) = 0 the equality H τ − q ( E ∩ v − ( y )) = 0 for H q -a.a. y ∈ R d (1.14) holds. Further, if v ∈ L αp, ( R n , R d ) or if v ∈ W k ( R n , R d ) , then the same assertion holdsunder weaker assumptions αp ≥ n (respectively, k ≥ n ) and τ ≥ τ ∗ . By an n –dimensional interval we mean a closed cube in R n with sides parallel to the coor-dinate axes. If Q is an n –dimensional cubic interval then we write ℓ ( Q ) for its sidelength.For a subset S of R n we write L n ( S ) for its outer Lebesgue measure (sometimes weuse the symbol meas S for the same purpose ). The m –dimensional Hausdorff measure isdenoted by H m and the m –dimensional Hausdorff content by H m ∞ . Recall that for anysubset S of R n we have by definition H m ( S ) = lim t ց H mt ( S ) = sup t> H mt ( S ) , < t ≤ ∞ , H mt ( S ) = inf ( ∞ X i =1 (diam S i ) m : diam S i ≤ t, S ⊂ ∞ [ i =1 S i ) . 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 k f k L p instead of k f k L p ( R n ) , etc.The Sobolev space W kp ( R n , R d ) is as usual defined as consisting of those R d -valuedfunctions f ∈ L p ( R n ) whose distributional partial derivatives of orders l ≤ k belong toL p ( R n ) (for detailed definitions and differentiability properties of such functions see, e.g.,[23], [45], [59], [19]). We use the norm k f k W kp = k f k L p + k∇ f k L p + · · · + k∇ k f k L p , and unless otherwise specified all norms on the spaces R s ( s ∈ N ) will be the usualeuclidean norms.Working with locally integrable functions, we always assume that the precise repre-sentatives are chosen. If w ∈ L , loc (Ω), then the precise representative w ∗ is defined forall x ∈ Ω by w ∗ ( x ) = lim r ց − Z B ( x,r ) w ( z ) d z, if the limit exists and is finite,0 otherwise , where the dashed integral as usual denotes the integral mean, − Z B ( x,r ) w ( z ) d z = 1 L n ( B ( x, r )) Z B ( x,r ) w ( z ) d z, and B ( x, r ) = { y : | y − x | < r } is the open ball of radius r centered at x . Henceforth weomit special notation for the precise representative writing simply w ∗ = w .If k < n , then it is well-known that functions from Sobolev spaces W kp ( R n ) are con-tinuous for p > nk and could be discontinuous for p ≤ p ◦ = nk (see, e.g., [45, 59]). TheSobolev–Lorentz space W kp ◦ , ( R n ) ⊂ W kp ◦ ( R n ) is a refinement of the corresponding Sobolevspace. Among other things functions that are locally in W kp ◦ , on R n are in particular con-tinuous (see, e.g., [35] ).Here we only mentioned the Lorentz space L p, , p ≥
1, and in this case one mayrewrite the norm as (see for instance [42, Proposition 3.6]) k f k L p, = + ∞ Z (cid:2) L n ( { x ∈ R n : | f ( x ) | > t } ) (cid:3) p d t. Of course, we have the inequality k f k L p ≤ k f k L p, . (2.1)10enote by W kp, ( R n ) the space of all functions v ∈ W kp ( R n ) such that in addition theLorentz norm k∇ k v k L p, is finite.By definition put k g k L p, ( E ) := k E · g k L p, , where 1 E is the indicator function of E .We need the following analog of the additivity property for the Lorentz norms: X i k f k p L p, ( Q i ) ≤ k f k p L p, ( ∪ i Q i ) for any family of disjoint cubes Q i (2.2)(see, e.g., [42, Lemma 3.10] or [51] ).For a function f ∈ L , loc ( R n ) we often use the classical Hardy–Littlewood maximalfunction: M f ( x ) = sup r> − Z B ( x,r ) | f ( y ) | d y. Recall that by usual Fubini theorem, if a set E ⊂ R has a zero plane measure, then for H -almost all straight lines L parallel to coordinate axes we have H ( L ∩ E ) = 0. Thenext result could be considered as functional Fubini type theorem. Theorem 2.1 (see Theorem 5.3 in [30]) . Let µ ≥ , q > , and v : R n → R d be acontinuous function. For a set E ⊂ R n define the set function Φ( E ) = inf E ⊂ S j D j X j (cid:0) diam D j (cid:1) µ (cid:2) diam v ( D j ) (cid:3) q , (2.3) where the infimum is taken over all countable families of compact sets { D j } j ∈ N such that E ⊂ S j D j . Then Φ( · ) is a countably subadditive and the implication Φ( E ) = 0 ⇒ (cid:20) H µ (cid:0) E ∩ v − ( y ) (cid:1) = 0 for H q -almost all y ∈ R d (cid:21) holds. In this section we formulate estimates of the above defined set function Φ obtained in [27,Appendix] for subsets of critical set in cubes for different classes of mappings .For all the following four subsections fix m ∈ { , . . . , n − } and d ≥ m . Take alsoa positive parameter q ≥ m and nonnegative µ ≥ The only technical difference is that in [27] we used the notation Z ′ v = { x ∈ R n \ A v : rank ∇ v ( x ) 1, 0 ≤ α ≤ 1, and v ∈ C k,α ( R n , R d ). By definition of the space C k,α , thereexists a constant A ∈ R + such that |∇ k v ( x ) − ∇ k v ( y ) | ≤ A · | x − y | α for all x, y ∈ R n . (3.2) Theorem 3.1 ([27]) . Under above assumptions, for any sufficiently small n -dimensionalinterval Q ⊂ R n the estimate Φ( Q ∩ Z ′ v ) ≤ C A q − m ℓ ( Q ) q + µ +( k + α − q − m ) (3.3) holds, where the constant C depends on n, m, k, α, d only. Fix k ≥ 1, 0 ≤ α < 1, 1 < p < ∞ , and v ∈ L k + αp, ( R n , R d ). In this subsection weconsider the case, when k + α > k + α ) p > n, (3.4)i.e., when v is a continuous function (see, e.g., [35] ), but the gradient ∇ v could bediscontinuous in general (if ( k + α − p < n ). Theorem 3.2 ([27]) . Under above assumptions, there exists a function h ∈ L p ( R n ) (depending on v ) such that the following statements are fulfilled:(i) if ( k + α − p > n , then the gradient ∇ v is continuous and uniformly boundedfunction, and for any sufficiently small n -dimensional interval Q ⊂ R n the estimate Φ( Z ′ v ∩ Q ) ≤ C σ q − m r q + µ +( k + α − − np )( q − m ) (3.5) holds, where r = ℓ ( Q ) , σ = k h k L p ( Q ) . (3.6) and the constant C depends on n, m, k, α, d, p only. ii) if ( k + α − p < n , then under additional assumption q + µ > τ ∗ := n − ( k + α − p (3.7) for any n -dimensional interval Q ⊂ R n the estimate Φ( Z ′ v ∩ Q ) ≤ C (cid:18) σ q r ( k + α − np ) q + µ + σ q − m r q + µ +( k + α − − np )( q − m ) (cid:19) (3.8) holds with the same σ, r . Fix k ≥ 1, 0 ≤ α < 1, 1 < p < ∞ , and v ∈ L k + αp, ( R n , R d ). In this subsection weconsider the case, when k + α > k + α ) p ≥ n, (3.9)i.e., when v is a continuous function (see, e.g., [35] ), but the gradient ∇ v could bediscontinuous in general (if ( k + α − p < n ). Theorem 3.3 ([27]) . Under above assumptions, there exists a function h ∈ L p, ( R n ) (depending on v ) such that the following statements are fulfilled:(i) if ( k + α − p ≥ n , then the gradient ∇ v is continuous and uniformly bounded func-tion, and for any sufficiently small n -dimensional interval Q ⊂ R n the estimate (3.5)holds with r = ℓ ( Q ) , σ = k h k L p, ( Q ) . (3.10) (ii) if ( k + α − p < n , then under additional assumption q + µ ≥ τ ∗ := n − ( k + α − p (3.11) for any n -dimensional interval Q ⊂ R n the estimate (3.8) holds with the same σ, r as in (3.10). Remark 3.1. Formally estimates in Theorem 3.3 are the same as in Theorems 3.2, theonly difference is in the definition of σ (using the Lorentz norm instead of Lebesgue one ).However, Theorem 3.3 is ‘stronger’ in a sense than the previous Theorems 3.2. Namely,there are some important (limiting) cases, which are not covered by Theorem 3.2, but onecould still apply the Theorem 3.3 for these cases. It happens for the following values ofthe parameters:( k + α ) p = n, (3.12)13r ( k + α − p = n, (3.13)or q + µ = τ ∗ . (3.14)It means, that the Lorentz norm is a sharper and more accurate tool here than theLebesgue norm. W k ( R n ) , k ≥ n . In this subsection we consider the limiting case p = 1 for Sobolev spaces W k . It is wellknown that functions from the Sobolev space W k ( R n , R d ) are continuous if k ≥ n, (3.15)so we assume this condition below. Fix k ≥ n and v ∈ W k ( R n , R d ). Theorem 3.4 ([27]) . Under above assumptions, the following statements hold:(i) if k − ≥ n , then the gradient ∇ v is continuous and uniformly bounded function,and for any sufficiently small n -dimensional interval Q ⊂ R n the estimate Φ( Z ′ v ∩ Q ) ≤ C σ q − m r q + µ +( k − − n )( q − m ) (3.16) holds, where again r = ℓ ( Q ) , σ = k∇ k v k L ( Q ) . (3.17) and the constant C depends on n, m, k, d only.(ii) if k = n , then under additional assumption q + µ ≥ for any n -dimensional interval Q ⊂ R n the estimate Φ( Z ′ v ∩ Q ) ≤ C (cid:18) σ q r µ + σ q − m r µ + m (cid:19) , (3.19) holds with the same r, σ , and with C depending on n, m, k, d only. Proofs of the main results We have to prove three theorems 1.6–1.8 (because other two theorems 1.2–1.3 are thepartial cases of Theorems 1.7–1.8 when q = σ and µ = 0 ).For the extremal case τ = n all these three theorems were proved in [30] and [27], sobelow we always assume that0 < τ < n. (4.1)Let us first check the assertions about strict N -properties. Fixed the correspondingparameters m ∈ { , , . . . , n − } , µ ≥ q ∈ ( m, σ ], and a mapping v : R n → R d satisfyingassumptions of one of the Theorems 1.6–1.8. We have to prove that H µ ( E ∩ v − ( y )) = 0 for H q -almost all y ∈ R d whenever E ⊂ e Z v,m with H τ ( E ) < ∞ . (4.2)First of all, we will simplify the situation and eliminate some technical difficultiesassociated with irregular points of mappings from Sobolev classes. Recall, that for theSobolev case the m -critical set is defined as e Z v,m = { x ∈ R n : x ∈ A v or x ∈ R n \ A v with rank ∇ v ( x ) ≤ m } . Here A v means the set of ‘bad’ points at which either the function v is not differentiableor which are not the Lebesgue points for ∇ v . Recall that the set A v is relatively small: H t ( A v ) = 0 ∀ t > τ ∗ := n − ( k + α − p if v ∈ L k + αp ( R n ) (case of Theorem 1.7); (4.3) H τ ∗ ( A v ) = 0 if v is from Theorem 1.8 . (4.4)In particular, A v = ∅ if ( k + α − p > n (respectively, if ( k + α − p ≥ n ).For the case of Theorem 1.7, take t ∈ ( τ ∗ , τ ). Then by Corollary 1.1 we have H t − q ( A v ∩ v − ( y )) = 0 for H q -almost all y ∈ R d . (4.5)By elementary direct calculation, if τ ∗ > 0, then τ ∗ − q < µ = τ − m − (cid:0) k + α − np + τp (cid:1) ( q − m ) . (4.6)Indeed, by definition of τ ∗ = n − ( k + α − p , the last inequality is equivalent to( τ − τ ∗ ) (cid:0) − q − mp (cid:1) > . (4.7)But really by our assumptions q − mp ≤ σ − mp = τ − mτ + ( k + α ) p − n < ττ = 1 , 15o (4.6)–(4.7) is fulfilled. From inequality (4.6) it follows that for t ∈ ( τ ∗ , τ ) sufficientlyclose to τ ∗ we have t − q < µ. From this inequality and (4.5) we obtain H µ ( A v ∩ v − ( y )) = 0 for H q -almost all y ∈ R d , (4.8)so indeed A v is negligible in property (4.2).If v is from Theorem 1.8, then again Corollary 1.1 implies H τ ∗ − q ( A v ∩ v − ( y )) = 0 for H q -almost all y ∈ R d . (4.9)And by the same calculations we obtain τ ∗ − q ≤ µ, (4.10)therefore, the identity (4.8) is fulfilled as well and in any case the ’bad’ set A v is negligiblein property (4.2).It means, that in the required property (4.2) we could replace the set e Z v,m by smaller(regular) set Z v = { x ∈ R n \ A v : rank ∇ v ( x ) ≤ m } . Moreover, since the countable union of the sets of H µ -measure zero has again H µ -measurezero, we could replace the set Z v by the smaller set Z ′ v = { x ∈ R n \ A v : |∇ v ( x ) | ≤ ∇ v ( x ) ≤ m } , i.e., instead of (4.2) we need to check only H µ ( E ∩ v − ( y )) = 0 for H q -almost all y ∈ R d whenever E ⊂ Z ′ v with H τ ( E ) < ∞ . (4.11)Because of Theorem 2.1, for the proof of the last assertion it is sufficient to check, thatΦ( E ) = 0 whenever E ⊂ Z ′ v with H τ ( E ) < ∞ , (4.12)where the set function Φ was defined in Theorem 2.1.In our previous paper [27, Appendix] we obtained the general estimates forΦ( Z ′ v ∩ Q ), here Q is an arbitrary n -dimensional cube, for all considered cases: Holder,Sobolev (including fractional Sobolev), and Sobolev–Lorentz (see their formulation inSection 3 of the present paper). From these estimates and from the Holder inequality therequired property (4.12) follows easily . Really, the present paper and [27] were written in the same time, so we had in mind the purposes ofthe present paper when we formulated and proved the estimates for Φ( Z ′ v ∩ Q ) in [27, Appendix]. τ = n . The present case τ < n is even simpler: indeed, the mostdifficult and subtle part in [27] was to prove the strict ( τ, q, µ, m )- N -property for Holdercase when τ = n , — it requires the application of some generalised Coarea formula, etc.We do not need to touch these difficulties here. The strictness of considered N -propertiesfor the present case 0 < τ < n follows from the following three simple facts: |∇ k v ( x ) − ∇ k v ( y ) | ≤ ω ( r ) · | x − y | α whenever | x − y | < r (4.13)with ω ( r ) → r → v ∈ C k,α + or v ∈ C k (i.e., α = 0 ); X i k h k pL p ( Q i ) → X i ℓ ( Q i ) τ ≤ C, sup i ℓ ( Q i ) → , for any (fixed) function h ∈ L p ( R n ), where Q i is a family of nonoverlapping n -dimensionalcubes; X i k h k pL p, ( Q i ) → X i ℓ ( Q i ) τ ≤ C, sup i ℓ ( Q i ) → , for any (fixed) function h ∈ L p, ( R n ), where again Q i is a family of nonoverlapping n -dimensional cubes (see (2.2) ). 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