On the Morse-Sard property and level sets of $W^{n,1}$ Sobolev functions on ${\mathbb R}^n$

0.21 MB·English

by  Jean Bourgain

ON THE MORSE–SARD PROPERTY AND LEVEL
SETS OF Wn,1 SOBOLEV FUNCTIONS ON Rn
Jean Bourgain, Mikhail V. Korobkov∗ and Jan Kristensen†
2
1
0 Abstract
2
n WeestablishLuzinN andMorse–SardpropertiesforfunctionsfromtheSobolevspace
a Wn,1(Rn). Using theseresults weprovethat almostalllevelsets arefinitedisjoint unions
J
ofC1-smoothcompactmanifoldsofdimensionn−1. Theseresultsremainvalidalsowithin
6
the larger space of functions of bounded variation BV (Rn). For the proofs we establish
n

] and use some new results on Luzin–type approximation of Sobolev and BV–functions by
P
Ck–functions, wheretheexceptional setshavesmallHausdorffcontent.
A
. Keywords: BV andWn,1–functions,LuzinN–property,Morse–Sardproperty,levelsets,
h n
t approximation bysmoothfunctions.
a
m
[
Introduction

1
v
6 The starting point of the paper is the following classical result (see also [11] for more general
1
expositions):
4
1
1. Theorem (Morse-Sard, 1942, [15], [18]). Letf: Rn → Rm beaCk–smoothmappingwith
0
k ≥ max(n−m+1,1).Then
2
1 Lm(f(Z )) = 0, (1)
f
:
v
where Lm denotesthem-dimensionalLebesguemeasureand Z denotesthesetofcritical
i f
X pointsoff: Z = {x ∈ Rn : rank∇f(x) < m}.
f
r
a
The order of smoothness in the assumptions of this theorem is sharp on the scale Cj (see
[21]). However, some analogs of the Morse–Sard theorem remain valid for functions lack-
ing the required smoothness in the classical theorem. Although (1) may be no longer valid
then, Dubovitski˘ı[9] obtained some results on the structure of level sets in the case of reduced
smoothness(also see[3]).
∗TheauthorwassupportedbytheRussianFoundationforBasicResearch(projectNo.11-01-00819-a)
†Work supported by the EPSRC Science and Innovation award to the Oxford Centre for Nonlinear PDE
(EP/E035027/1).
1
AnotherdirectionoftheresearchwasthegeneralizationoftheMorse–Sardtheoremtofunc-
tions in more refined scales of spaces, and especially in Ho¨lder and Sobolev spaces (for exam-
ples, see [2, 3, 7, 12, 16]). In particular, De Pascale ([7], see also [12]) proved that (1) holds
when f ∈ Wk,p(Rn,Rm) with p > n, k ≥ max(n − m + 1,2). Note that in this case v is
C1–smooth by virtue of the Sobolev imbedding theorem, and so the critical set is defined as
usual.
Forahistoricalreviewfortheplanecasen = 2,m = 1seeforinstance[5]. Wementiononly
the paper [17] where it was proved that (1) holds for Lipschitz functions f of class BV (R2),
2
where BV (R2) is the space of functions f ∈ L1(R2) such that all its partial (distributional)
2
derivativesofthesecond orderareR-valued Radon measureson R2.
In this paper we consider the case of R-valued Sobolev functions v ∈ Wn,1(Rn). It
is known (see, e.g., [8]) that such a function admits a continuous representative which is
(Fre´chet–)differentiable H1–almost everywhere. The critical set Z is defined as the set of
v
pointsx,wherev isdifferentiablewithtotaldifferentialv′(x) = 0. Asourmainresultweprove
thatL1(v(Z )) = 0 (seeTheorem 4.1).
v
Also we show that for any v ∈ Wn,1(Rn) and ε > 0 there exists δ > 0 such that for all
subsetsE ⊂ Rn withH1 (E) < δ wehaveL1(v(E)) < ε,whereH1 istheHausdorffcontent.
∞ ∞
In particular, it follows that L1(v(E)) = 0 whenever H1(E) = 0 (see Theorem 2.1). So the
image of the exceptional “bad” set, where the differential is not defined, has zero Lebesgue
measure. This ties nicely with our definition of the critical set and our version of the Morse–
Sard result.
Finally, using these results we prove that almost all level sets of Wn,1–functions defined
on Rn, are finite disjointunions of C1–smoothcompact manifolds of dimensionn−1 without
boundary(seeTheorem 5.3).
The proofof the last result relies in turn on new Luzin–typeapproximationresults for Wl,1
Sobolev functions by Ck–functions, k ≤ l, where the exceptional sets are of small Hausdorff
content (see Theorem 3.1). The Lp analogs of such results are well-known when p > 1, see,
e.g., [4], [23], [20], where Bessel and Riesz capacities are used instead of Hausdorff content.
In fact, the exceptional set can be precisely characterized in terms of the Bessel and Riesz
capacitiesin thecases f ∈ Wl,p(Rn),p > 1.
We extend our results also to the space BV (Rn) consisting of functions v ∈ L1(Rn) for
n
which all partial (distributional)derivativesof the n-th order are R-valued Radon measures on
Rn (see Section 6).
Fortheplanecasen = 2 theseresultswere obtainedin[5].
As main tools, we use the deep results of [14] on advanced versions of Sobolev imbedding
theorems (see Theorem 1.3), of [1] on Choquet integrals of Hardy-Littlewood maximal func-
tions with respect to Hausdorff content (see Theorem 1.5), and of [22] on the entropy estimate
of near-critical values of differentiable functions (see Theorem 1.6). The key step in the proof
oftheassertionoftheMorse–Sard Theoremis containedinLemma4.2.
2
1 Preliminaries
By an n-dimensional interval we mean a closed cube in Rn whose sides are parallel to the
coordinate axes: I = [a ,a +l]× ... , ×[a ,a +l]. Furthermore we write ℓ(I) = l for its
1 1 n n
sidelength.
We denoteby Ln(F) theouterLebesguemeasure ofa set F ⊂ Rn. Denote by Hk, Hk the
∞
k–dimensionalHausdorffmeasure,Hausdorffcontent,respectively: foranyF ⊂ Rn,Hk(F) =
limHk(F) = sup Hk(F),whereforeach 0 < α ≤ ∞,
α α>0 α
αց0
∞ ∞
Hk(F) = inf (diamF )k : diamF ≤ α, F ⊂ F .
α i i i
i=1 i=1
(cid:8)X [ (cid:9)
Itis wellknownthatHn(F) ∼ Hn (F) ∼ Ln(F)forsetsF ⊂ Rn.
∞
To simplifythenotation,wewritekfk instead ofkfk , etc.
L1 L1(Rn)
The space Wk,1(Rn) is as usual defined as consisting of those functions f ∈ L1(Ω) whose
distributional partial derivatives of order l ≤ k belong to L1(Rn) (for detailed definitions and
differentiability properties of such functions see, e.g., [10], [23], [8]). Denote by ∇lf the
vector-valuedfunctionconsistingofall l-th orderpartialderivativesoff. Weusethenorm
kfk = kfk +k∇fk +···+k∇kfk .
Wk,1 L1 L1 L1
When working with Sobolev functions we always assume that the precise representatives
arechosen. Ifw ∈ L1 (Ω), thenthepreciserepresentativew∗ isdefined by
loc
lim− w(z)dz, ifthelimitexistsandis finite,
w∗(x) = r→0 (2)
 ZB(x,r)
0 otherwise,

wherethedashedintegralas usualdenotestheintegralmean,
1
− w(z)dz = w(z)dz,
Ln(B(x,r))
ZB(x,r) ZB(x,r)
and B(x,r) = {y : |y −x| < r} is the open ball of radius r centered at x. We omit the asterix
andwritesimplyw forthepreciserepresentative.
The following well-known assertion follows immediately from the definition of Sobolev
spaces.
Lemma 1.1. Letf ∈ Wl,1(Rn). Thenforanyε > 0 thereexistfunctionsf ∈ C∞(Rn),
0 0
f ∈ Wl,1(Rn),suchthatf = f +f and kf k < ε.
1 0 1 1 Wl,1
We need a version of the Sobolev Embedding Theorem that gives inclusions in Lebesgue
spaces with respect to general positive measures. Very general and precise statements are
known,butwerestrict attentiontothefollowingclassofmeasures:
3
Definition1.2. LetµbeapositivemeasureonRn. Wesaythatµhasproperty(∗−l)forsome
l ≤ n, if
µ(I) ≤ (ℓ(I))(n−l) (3)
foranyn-dimensionalintervalI ⊂ Rn.
Theorem 1.3 (see[14], §1.4.3). Iff ∈ Wl,1(Rn)andµhasproperty(∗−l),then
|f|dµ ≤ Ck∇lfk , (4)
L1
Z
whereC doesnotdependonµ, f.
Fora functionu ∈ L1(I), I ⊂ Rn, define thepolynomialP [u] ofdegreeat mostk by the
I,k
followingrule:
yα(u(y)−P [u](y)) dy = 0 (5)
I,k
ZI
foranymulti-indexα = (α ,...,α ) oflength|α| = α +···+α ≤ k.
1 n 1 n
Wewilloften usethefollowingsimpletechnicalassertion.
Lemma 1.4. Suppose v ∈ Wn,1(Rn). Then v isacontinuousfunctionandfor any k =
0,...,n−1andforanyn-dimensionalintervalI ⊂ Rntheestimate
k∇k+1vk
sup|v(y)−P [v](y)| ≤ C L1(I) +k∇nvk (6)
I,k ℓ(I)n−k−1 L1(I)
y∈I (cid:18) (cid:19)
holds,whereC dependsonnonly.Moreover,thefunctionv (y) = v(y)−P [v](y),y ∈ I,
I,k I,k
canbeextendedfromI tothewholeofRnsuchthatv ∈ Wn,1(Rn)and
I,k
k∇nv k ≤ C R(I,k), (7)
I,k L1(Rn) 0
whereC alsodependsonn onlyandR(I,k) denotestherighthandsideoftheestimates(6)
0
(inbrackets).
Proof. The existence of a continuous representative for v follows from Remark 2 of §1.4.5
in [14]. Because of coordinate invariance it is sufficient to prove the estimate (6)–(7) for the
case when I is a unit cube: I = [0,1]n. By results of [14, §1.1.15] for any u ∈ Wn,1(I) the
estimates
sup|u(y)| ≤ ckuk ≤ c kP [u]k +k∇k+1uk +k∇nuk , (8)
Wn,1(I) I,k L1(I) L1(I) L1(I)
y∈I
(cid:0) (cid:1)
hold, where c = c(n,k) is a constant. Taking u(y) = v(y) − P [v](y), the first term on the
I,k
right–hand side of (8) vanishes and so the inequality (8) turns to the estimates (6)–(7) (here
we used also the following fact: every function u ∈ Wn,1(I) can be extended to a function
u ∈ Wn,1(Rn) suchthat theestimatek∇nuk ≤ ckuk holds,see[14,§1.1.15]).
L1(Rn) Wn,1(I)
4
Thefollowingtworesultsare crucial forourproof.
Theorem 1.5 ([1]). Iff ∈ Wk,1(Rn),k = 1,...,n−1,then
∞
Hn−k({x ∈ Rn : Mf(x) ≥ λ})dλ ≤ C |∇kf(y)|dy,
∞
Z0 RZn
whereC dependsonn,konlyand
Mf(x) = sup r−n |f(y)|dy
r>0
Z
B(x,r)
istheusualHardy-Littlewoodmaximalfunctionoff.
Theorem 1.6([22]). ForA ⊂ Rmandε > 0let Ent(ε,A)denotetheminimalnumberofballs
ofradiusεcoveringA.ThenforanypolynomialP : Rn → Rofdegreeatmostk,foreachball
B ⊂ Rnofradiusr > 0,andanynumberε > 0theestimate
Ent(εr,{P(x) : x ∈ B, |∇P(x)| ≤ ε}) ≤ C
∗
holds,whereC dependsonn,konly.
∗
To applyTheorem1.5, weneed alsothefollowingsimpleestimateand itscorollary.
Lemma 1.7 (see Lemma 2 in [8]). Let u ∈ W1,1(Rn). Thenforanyball B(z,r) ⊂ Rn,
B(z,r) ∋ x,theestimate
u(x)−− u(y)dy ≤ Cr(M∇u)(x)
(cid:12) ZB(z,r) (cid:12)
(cid:12) (cid:12)
(cid:12) (cid:12)
holds,whereC dependsonnonlyandM∇uisaHardy–Littlewoodmaximalfunctionof∇u.
(cid:12) (cid:12)
Corollary 1.8. Let u ∈ W1,1(Rn). Then for any ball B ⊂ Rn of a radius r > 0 and for any
numberε > 0theestimate
diam({u(x) : x ∈ B, (M∇u)(x) ≤ ε}) ≤ C εr
∗∗
holds,where C is a constantdependingon nonly.
∗∗
Wewillusethefollowingk-order analogofLemma1.7.
Lemma 1.9 (see Lemma 2 in [8]). Letu ∈ Wk,1(Rn),k ≤ n. Thenforanyn-dimensional
intervalI ⊂ Rn,x ∈ I,andforanym = 0,1,...,k −1theestimate
∇mu(x)−∇mP [u](x) ≤ Cℓ(I)k−m(M∇ku)(x) (9)
I,k−1
holds,wherethecon(cid:12)stantC dependsonn,konl(cid:12)y.
(cid:12) (cid:12)
5
2 On images of sets of small Hausdorff contents
Themain resultofthissectionisthefollowingLuzinN–property forWn,1–functions:
Theorem 2.1. Letv ∈ Wn,1(Rn). Thenforeachε > 0 thereexistsδ > 0 suchthatforany
setE ⊂ Rn if H1 (E) < δ,thenH1(v(E)) < ε. Inparticular,H1(v(E)) = 0 whenever
∞
H1(E) = 0.
Fortheplanecase, n = 2, Theorem 2.1was obtainedinthepaper[5].
For the remainder of this section we fix a function v ∈ Wn,1(Rn). To prove Theorem 2.1, we
need somepreliminarylemmasthat weturnto next.
Byadyadicintervalweunderstandanintervaloftheform[k1 , k1+1]×···×[kn, kn+1],where
2m 2m 2m 2m
k ,mareintegers. Thefollowingassertionisstraightforward,andhenceweomititsproofhere.
i
Lemma 2.2. Foranyn-dimensionalintervalI ⊂ Rn thereexistdyadicintervalsQ ,...,Q
1 2n
suchthatI ⊂ Q ∪···∪Q andℓ(Q ) = ··· = ℓ(Q ) ≤ 2ℓ(I).
1 2n 1 2n
Let {I } be a family of n-dimensional dyadic intervals. We say that the family {I } is
α α∈A α
k-regular,ifforanyn-dimensionaldyadicintervalQtheestimate
ℓ(Q)k ≥ ℓ(I )k (10)
α
α:XIα⊂Q
holds.
Thenexttwoassertionsarethemultidimensionalanalogsofthecorrespondingplaneresults
fromthepaper[5].
Lemma 2.3. Letk = 1,...,nandletI beafamilyofn-dimensionaldyadicintervals.Then
α
thereexistsak-regularfamilyJ ofn-dimensionaldyadicintervalssuchthat∪ I ⊂ ∪ J and
β α α β β
ℓ(J )k ≤ ℓ(I )k.
β α
β α
X X
Proof. Define
F = J : J ⊂ Rn dyadicinterval; ℓ(I )k ≥ ℓ(J)k .
α
( )
IXα⊂J
Thus I ∈ F for each α. Denote by F∗ = {J } the collection of maximal elements of F.
α β
Clearly
I ⊂ J , (11)
α β
α β
[ [
andsincedyadicintervalsare eitherdisjointorcontainedinoneanother,the{J } aremutually
β
disjoint. It followsthat
ℓ(J )k ≤ ℓ(I )k ≤ ℓ(I )k. (12)
β α α
Xβ Xβ IαX⊂Jβ Xα
6
Observealsothat foranydyadicintervalQ ⊂ Rn,
ℓ(J )k ≤ ℓ(Q)k. (13)
β
JXβ⊂Q
We used here that if J ⊂ Q for some β, then either J = Q (and then the estimate (13) is
β β
immediate),orQ 6∈ F (becauseJ ismaximal),and henceinthelastcase
β
ℓ(J )k ≤ ℓ(I )k < ℓ(Q)k.
β α
JXβ⊂Q IXα⊂Q
Lemma 2.4. Letk = 0,...,n − 1. Thenforeachε > 0 thereexistsδ = δ(ε,v,k) > 0
suchthatforany(k +1)-regularfamilyI ⊂ Rn ofn-dimensionaldyadicintervalswehaveif
α
ℓ(I )k+1 < δ,then R(I ,k) < ε.
α α α α
PProof. Fixε > 0andlePtI ⊂ Rn bea(k+1)-regularfamilyofn-dimensionaldyadicintervals
α
with ℓ(I )k+1 < δ, where δ > 0 will be specified below. By virtue of Lemma 1.1 we can
α α
find a decomposition v = v0 + v1, where k∇jv0kL∞ ≤ K = K(ε,v) for all j = 0,1,...,n
P
and
k∇nv k < ε. (14)
1 L1
Assumethat
ℓ(I )k+1 < δ < 1 ε. (15)
α K+1
α
X
Definethemeasureµ by
1
µ = 1 Ln, (16)
ℓ(I )n−k−1 Iα
α !
α
X
where1 denotestheindicatorfunctionoftheset I .
Iα α
Claim. 1 µhas property (∗−(k +1)).
2n+k+2
Indeed, writeforadyadicintervalQ
ℓ(Q)n
µ(Q) = ℓ(I )k+1 + ≤ 2ℓ(Q)k+1,
α ℓ(I )n−k−1
α
IXα⊂Q QX⊂Iα
where we invoked (10) and the fact that Q ⊂ I for at most one α. Then for any interval I we
α
havetheestimateµ(I) ≤ 2n+k+2ℓ(I)k+1 (seeLemma2.2). Thisprovestheclaim.
Now return to the estimate of R(I ,k). In addition to (15) we now decrease δ > 0
α α
furthersuchthat
P
k∇nvk < ε/2. (17)
L1(Iα)
α
X
7
By definition of R(I,k) (see Lemma 1.4) and properties (14), (4) (applied to f = ∇k+1v ,
1
l = n−k −1), wehave
1
R(I ,k) = k∇nvk + |∇k+1v|
α L1(Iα) ℓ(I )n−k−1
α α α α ZIα
X X X
1
≤ ε/2+ K ε+ |∇k+1v |
K+1 ℓ(I )n−k−1 1
α α ZIα
X
= C′ε+C |∇k+1v |dµ ≤ C′′ε.
1
Z
Sinceε > 0was arbitrary, theproofofLemma2.4 iscomplete.
NowtheassertionofTheorem2.1followsimmediatelyfromtheabovelemmaswithk = 0.
Further, ifweapplytheabovelemmaswithk = n−1,weobtain
Corollary 2.5. Let v ∈ Wn,1(Rn). Then for each ε > 0 there exists δ > 0 such that for
any family {I } of disjoint n-dimensional intervals we have that if ℓ(I )n < δ, then
α α α
k∇nv k < ε,where thefunctionv wasdefined inLemma 1.4.
α Iα,n−1 L1(Rn) Iα,n−1 P
P
3 On approximation of Wl,1 Sobolev functions
Theorem 3.1. Letk,l ∈ {1,...,n},k ≤ l. Thenforanyf ∈ Wl,1(Rn) andforeachε > 0
thereexistanopensetU ⊂ Rn andafunctiong ∈ Ck(Rn) suchthatHn−l+k(U) < ε and
∞
f ≡ g,∇mf ≡ ∇mgonRn \U form = 1,...,k.
The proof of Theorem 3.1 is based on the results of [1], [8] and on the classical Whitney
ExtensionTheorem:
Theorem3.2. Letk ∈ Nandletf = f ,f beafinitefamilyoffunctionsdefinedontheclosed
0 α
setE ⊂ Rn,whereαrangesoverallmulti-indicesα = (α ,...,α )with|α| = α +···+α ≤
1 n 1 n
k.Forx,y ∈ E andamulti-indexα,|α| ≤ k,put
1
T (x;y) = f (x)·(y −x)β,
α α+β
β!
|β|≤k−|α|
X
R (x;y) = f (y)−T (x;y).
α α α
Supposethatthereexistsafunctionω: [0,+∞) → [0,+∞)suchthatω(t) → 0ast ց 0and
foreachmulti-indexα,|α| ≤ k,andforallx,y ∈ E theestimate
|R (x;y)| ≤ ω(|x−y|)|x−y|k−|α| (18)
α
holds.Thenthereexistsafunctiong ∈ Ck(Rn)suchthatf ≡ g,f ≡ ∂αg onE for|α| ≤ k.
α
8
ProofofTheorem 3.1. Let the assumptions of Theorem 3.1 be fulfilled. For the case k = l the
assertionoftheTheorem iswell-known(see, e.g.,[13], [4], or[23]).
Now fix k < l. Then the gradients ∇mf(x), m ≤ k, are well-defined for all x ∈ Rn \A ,
k
where Hn−l+k(A ) = 0 (see [8]). For a multi-index α with |α| ≤ k denote by T (f,x;y) the
k α
Taylorpolynomialoforderat mostk −|α|forthefunction∂αf withthecenterat x:
1
T (f,x;y) = ∂α+βf(x)·(y −x)β.
α
β!
|β|≤k−|α|
X
ByvirtueoftheWhitneyExtensionTheorem3.2,wefinishtheproofofTheorem3.1bycheck-
ing that for each multi-indexα with |α| ≤ k the corresponding Taylor remainder term satisfies
the estimate ∂αf(y) − T (f,x;y) = o(|x − y|k−|α|) uniformly for x,y ∈ Rn \ U, where
α
Hn−l+k(U) is small.
∞
Takeasequencef ∈ C∞(Rn) suchthat
i 0
k∇lf −∇lfk < 4−i.
i L1
Denotef˜ = f −f . Put
i i
B = {x ∈ Rn : (M∇kf˜)(x) > 2i},
i i
∞
G = A ∪ B .
i k j
!
j=i
[
Thenby Theorem1.5 wehave
Hn−l+k(B ) ≤ c2−i,
∞ i
andconsequently,
Hn−l+k(G ) < C2−i.
∞ i
By construction,
|∇kf˜(x)| ≤ 2−j (19)
j
forallx ∈ Rn\G andallj ≥ i. Foramulti-indexαwith|α| ≤ k−1denotebyT (f,x;y)
i α,k−1
theTaylorpolynomialoforderk −1−|α|forthefunction∂αf withthecenter atx:
1
T (f,x;y) = ∂α+βf(x)·(y −x)β.
α,k−1
β!
|β|≤k−1−|α|
X
In ournotation,
1
T (f,x;y) = T (f,x;y)+ ∂α+βf(x)·(y −x)β.
α α,k−1
β!
|β|=k−|α|
X
We start by estimating the remainder term ∂αf˜(y)−T (f˜,x;y) for a multi-index α with
j α,k−1 j
|α| ≤ k − 1. Fix x,y ∈ Rn \ G , j ≥ i, and an n-dimensional interval I such that x,y ∈ I,
i
|x−y| ∼ ℓ(I). By constructionand Lemma1.9,
∂αf˜(y)−∂αP [f˜](y) ≤ Cℓ(I)k−|α|(M∇kf˜)(y) ≤ C|x−y|k−|α|2−j.
j I,k−1 j j
(cid:12) (cid:12)
(cid:12) (cid:12) 9
Forthesamereasonswefind forany multi-indexβ with|β| ≤ k −1−|α|that
∂α+βf˜(x)−∂α+βP [f˜](x) ≤ Cℓ(I)k−|α|−|β|(M∇kf˜)(x) ≤ C|x−y|k−|α|−|β|2−j.
j I,k−1 j j
Co(cid:12)nsequently, (cid:12)
(cid:12) (cid:12)
|∂αf˜(y)−T (f˜,x;y)| ≤ ∂αf˜(y)−∂αP [f˜](y) (20)
j α,k−1 j j I,k−1 j
(cid:12)+ ∂αPI,k−1[f˜j](y)−Tα,k−1(cid:12)(f˜j,x;y)
(cid:12) (cid:12)
≤ C|x−y|k−|α|2−j
(cid:12) (cid:12)
(cid:12) (cid:12)
1
+ ∂α+βf˜(x)−∂α+βP [f˜](x) ·(y −x)β
j I,k−1 j
β!
|β|≤k−1−|α|
X (cid:12)(cid:0) (cid:1) (cid:12)
≤ C |x−y|k−|α|2(cid:12)−j. (cid:12)
1
Finallyfromthelastestimateand (19)wehave
|∂αf(y)−T (f,x;y)| ≤ |∂αf˜(y)−T (f˜,x;y)|+|∂αf (y)−T (f ,x;y)|
α j α j j α j
≤ |∂αf˜(y)−T (f˜,x;y)|+|∇kf˜(x)|·|x−y|k−|α|
j α,k−1 j j
+ω (|x−y|)·|x−y|k−|α|
fj
≤ C′|x−y|k−|α|2−j +ω (|x−y|)·|x−y|k−|α|
fj
= C′2−j +ω (|x−y|) ·|x−y|k−|α|,
fj
where ω (r) → 0 as r → 0 (the la(cid:0)tter holds since f ∈ C(cid:1)∞(Rn)). We emphasize that the last
fj j 0
inequalityisvalidforall j ≥ i andx,y ∈ Rn \G . Takean open set U ⊃ G suchthat
i i i
Hn−l+k(U ) < C2−i.
∞ i
PutE = Rn \U . Then byconstruction
i i
|∂αf(y)−T (f,x;y)| ≤ C′2−j +ω (|x−y|) ·|x−y|k−|α| (21)
α fj
for all j ≥ i, |α| ≤ k, and x,y ∈ E . Th(cid:0)en the assumptions of(cid:1)the Whitney Extension Theo-
i
rem3.2 arefulfilled, and hencetheproofofTheorem 3.1iscomplete.
Remark3.3. UsingtheextensionformulaandthemethodsfromtheproofofTheorem6.2(see
Section 6 below; this approach was originally introduced in [4]), one can prove that for k < l
the function g from the assertion of Theorem 3.1 can be constructed such that in addition the
estimatekf −gk < ε holds.
Wk+1,1
4 Morse–Sard theorem in Wn,1(Rn)
Recall that if v ∈ Wn,1(Rn) and k = 1,...,n, then ∇kv(x) is well-defined for Hk-almost
all x ∈ Rn (see [8]). In particular, v is differentiable (in the classical Fre´chet sense) and the
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