Whitney’s jets for Sobolev functions
We present two fundamental facts of the jet theory for Sobolev spaces $W^{m, p}$. One of them is that the formal differentiation of $k$-jets theory is compatible with the pointwise definition of Sobolev $(m - 1)$-jet spaces on regular subsets of Euclidean spaces $R^n$. The second result describes t...
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Ukrains’kyi Matematychnyi Zhurnal| _version_ | 1860509376166494208 |
|---|---|
| author | Bojarski, B. Боярський, Б. |
| author_facet | Bojarski, B. Боярський, Б. |
| author_sort | Bojarski, B. |
| baseUrl_str | https://umj.imath.kiev.ua/index.php/umj/oai |
| collection | OJS |
| datestamp_date | 2020-03-18T19:51:00Z |
| description | We present two fundamental facts of the jet theory for Sobolev spaces $W^{m, p}$. One of them is that the formal differentiation of $k$-jets theory is compatible with the pointwise definition of Sobolev $(m - 1)$-jet
spaces on regular subsets of Euclidean spaces $R^n$.
The second result describes the Sobolev embedding operator of Sobolev jet spaces increasing the order of integrability of Sobolev functions up to the critical Sobolev exponent. |
| first_indexed | 2026-03-24T02:40:07Z |
| format | Article |
| fulltext |
UDC 517.5
B. Bojarski (Warszawa, Poland)
WHITNEY’S JETS FOR SOBOLEV FUNCTIONS
STRUMENI U}TNI DLQ FUNKCIJ SOBOLEVA
We present two fundamental facts of the jet theory for Sobolev spaces W m p, . One of them is that the
formal differentiation of k-jets theory is compatible with the pointwise definition of Sobolev ( )m − 1 -jet
spaces on regular subsets of Euclidean spaces R
n . The second result describes the Sobolev embedding
operator of Sobolev jet spaces increasing the order of integrability of Sobolev functions up to the critical
Sobolev exponent.
Vstanovleno dva fundamental\nyx fakty teori] strumeniv dlq prostoriv Soboleva W m p, . Per-
ßyj iz nyx polqha[ v tomu, wo formal\ne dyferencigvannq k-strumeniv [ sumisnym z potoçko-
vym vyznaçennqm sobolevs\kyx prostoriv ( )m − 1 -strumenq na rehulqrnyx pidmnoΩynax evkli-
dovyx prostoriv R
n . Druhyj rezul\tat opysu[ sobolevs\ki operatory vkladennq sobolevs\kyx
prostoriv strumeniv, wo pokrawugt\ porqdok sumovnosti sobolevs\kyx funkcij aΩ do krytyç-
noho pokaznyka.
1. Introduction. We present here two main theorems — Theorem 3.1 and 3.2 below
— as the fundamental facts of the jet theory for Sobolev spaces W m p, . The first one
expresses the basic fact that the formal differentiation D
j : J U J Uk k j( ) ( )→ − of k-
jets theory is compatible with the pointwise definition of Sobolev ( )m −1 -jet spaces
VLC ( m, p, U ) on regular subsets U of Euclidean spaces Rn .
Classically and in any theory describing the Sobolev spaces W Um p, ( ) [1 – 3] by
any process of consecutive differentiations D
α
f ∈ L
p
, e. g. for α < m Theo-
rem 3.1 is essentially part of the definition. However with the spaces VLC ( m, p, U )
defined by the pointwise inequality (1.1) the deduction of the inequalities (3.2) from
(3.1) or (1.1) is far from obvious and quite nontrivial.
Theorem 3.2 describes the Sobolev embedding operator of Sobolev jet spaces in-
creasing the order of integrability of Sobolev functions up to the critical Sobolev expo-
nent.
The case VLC ( 1, p, U ) is identical with the Hajłasz space M p1, [4, 5] which has
been extended by P. Hajłasz for the general case of M X dp1, ( , , )µ spaces on general
measure metric spaces ( X, d, µ ) satisfying the doubling condition. The Haj łasz theo-
rem is one of the cornerstones of the analysis on general metric spaces (fractals etc.).
Theorems 3.1 and 3.2 play a crucial role in incorporating the Sobolev space theory
into the general framework of Whitney type theory for classical function spaces, with
the full exploitation, in a rather deep way, of the classical Taylor approximating for-
mulas.
The classes VLC ( m, p, U ) are described in the form of a pointwise inequality
R F y xm −1 ( , ) ≤ b x y x y m( , ) − , ( x, y ) ∈ U × U, (1.1)
describing the behavior of the ( )m −1 -order Taylor remainder R F x ym −1 ( , ) of the
Sobolev jet F near the diagonal ∆ = { x, x } , ∆ ⊂ U × U , in terms of the distance
x y− of the point ( x, y ) in the cartesian product U × U from the diagonal ∆. We
call (1.1) the S
m
RC condition — the Sobolev remainder condition — describing the
behavior of the ( )m −1 -order Taylor remainder R F y xm −1 ( , ) of the Sobolev ( )m −1 -
jet F.
© B. BOJARSKI, 2007
ISSN 1027-3190. Ukr. mat. Ωurn., 2007, t. 59, # 3 345
346 B. BOJARSKI
In the literature only the “splitted” case
b ( x, y ) ≡ a ( x ) + a ( y ) for some a ∈ L
p
( U ) , p > 1,
was considered [4 – 6]. Our approach allows also to consider somewhat more general
case, not necessarily “splitted”.
As methodological novelty let us mention the general idea of averaging pointwise
inequalities and systematic use of local fractional maximal and sharp maximal functi-
ons. Markov inequality is crucial also, we refer to [7, 8].
In the seminal papers [9, 10] of H. Whitney the space of continuously differentiable
functions C
m
( U ) on a subset U of R
n was described by their m-jets. Recall that
the space J
m
( U ) of m-jets F — also called Whitney fields on U — is defined as a
collection of functions, also called components (or coefficients) of the jet,
F = f xj ( ){ }, x ∈ U, j ≤ m,
indexed by multiindices j = ( , , )j jn1 … , jl ≥ 0, Σl jl = j , from a linear class A of
some usual function space of analysis. This can be the class of measurable, bounded,
continuous, Lebesgue integrable L Up
loc( , )µ , µ — some regular Borel measure on U,
p ≥ 1, etc. The class A is required to admit the multiplication by polynomials in
x.∈ R
n (restricted to U ).
The jets F ⊂ J
m
( U ) define the formal Taylor polynomials in x ∈ R
n (centered at
y ∈ U )
T F y xj
k j− ( , ) = T F xy j
k j
, ( )− = f y
x y
lj l
l
l j k
+
+ ≤
−∑ ( )
( )
!
,
k ≤ m, l ≥ 0, l ≤ k j− , (1.2)
T F y xk ( , ) = T F xy
k ( ) = f y
x y
ll
l
l k
( )
( )
!
−
≤
∑
(also called Taylor fields) and the formal Taylor remainders R F y xj
k j− ( , ) defined by
the formulas
f xj ( ) ≡ T F x T F y xy j
k j
j
k j
, ( ) ( , )− −+ , R F y xk ( , ) ≡ R F y xk
0 ( , ),
( y, x ) ∈ U × U, j ≤ m .
On the jet spaces J
m
( U ) there are three basic natural operations:
(I) The formal jet differential
D
i : J U J Um m i( ) ( )→ −
defined by the formula
{ }Di
kF = fi k+ for k ≤ m .
Obviously Di F ≡ 0 for i > m .
(II) The reduction C
l, l ∈ N ,
C
l : J U J Um m l( ) ( )→ − , l ≤ m ,
( )Cl
kF = fk , k ≤ m – l .
ISSN 1027-3190. Ukr. mat. Ωurn., 2007, t. 59, # 3
WHITNEY’S JETS FOR SOBOLEV FUNCTIONS 347
(III) The restriction or trace operation. For a subset Σ of U the trace TrS :
J
m
( U ) → J
m
( Σ ) is defined componentwise
{ }F kΣ = { }fk Σ .
The restriction operation is unconditionally meaningful and well understood for jet
spaces with continuous coefficients or when Σ is an open subset of an open domain U
in Rn . For general Sobolev type jet spaces, e. g. VLC-spaces, defined below, the re-
striction operation is defined and understood only under very special conditions on the
set Σ. Classically Σ should be a smooth submanifold, typically a hyperplane, of U ⊂
⊂ R
n
. In this context for Sobolev spaces the d-sets Σ of U are an important class of
admissible subsets (see [11]).
It is convenient to distinguish between the polynomial (or space) variables x ∈ R
n
and the field variables y ∈ U of the Taylor fields T F xy j
k j
, ( )− . Thus in the definition
of m-jets in J Um( ) the components fj of the jet F are to be considered as (local)
functions of the field variables.
The Taylor fields (1.2) T
k
F ( y, x ) considered as functions on the product U × R
n
are polynomials in x ∈ R
n with coefficients in A . Moreover,
T F xy j
k j
, ( )− = D T F xx
j
y
k ( ), T F xy
k ( ) ≡ T F xy
k
, ( )0 ,
where Dx
i are the standard differential operators acting on polynomials in x ∈ R
n.
Though the Taylor remainders R F y xj
k j− ( , ) are defined on the product U × U as
functions of the field variables x, y and are in the class A only, their difference
R F y x R F z xj
k j
j
k j− −−( , ) ( , ) ≡ T F x T F xz j
k j
y j
k j
, ,( ) ( )− −−
is a polynomial in x for arbitrary y, z ∈ U × U.
H. Whitney considered the case when U = K is a compact subset K of R
n and
A the ring of continuous functions on K, A = C ( K ) . By definition an m-jet F ⊂
⊂ J
m
( K ) is in C
m
( K ) if
F = G K = D gm
K
(1.3)
is the restriction to K of an m-jet G = { }:D g j mj ≤ of a function g ∈ C
m
( R
n
) .
Numerous natural and delicate questions related with the above definition and the in-
verse extension operators Ext : C
m
( K ) → C
m
( R
n
) have been discussed in detail in
the vast literature of the subject [9 – 13].
A necessary and sufficient condition for (1.3) was formulated and proved by
H. Whitney in the form of the famous Whitney’s remainder condition WRC
m
( K ) on
the behavior of the Taylor remainders R F y xj
m j− ( , ) near the diagonal ∆ ⊂ K × K :
R F y xj
m j− ( , ) = o x y m j−( )− , j ≤ m,
when ( y, x ) → ( x0 , x0 ) uniformly on K, x0 ∈ K .
Actually it is enough to require
R F y xm ( , ) = o x y m−( ), j = 0
(see [7, 14]).
ISSN 1027-3190. Ukr. mat. Ωurn., 2007, t. 59, # 3
348 B. BOJARSKI
In the present paper U will be mainly an open subset of R
n
, possibly U = R
n or
a model (closed) cube Q ⊂ R
n. Functions in the Sobolev space W Qm p, ( ), p > 1,
will be described by their ( )m −1 -jets. The corresponding precise definition is formu-
lated below as jet spaces VLC.
Remark. Most of our concepts and results are meaningful and hold when U is
considered to be a regular subset of R
n, e. g. bounded John domain or extension do-
main for Sobolev spaces.
The notation throughout this paper is either standard in the literature on Sobolev
spaces or self-explanatory. As a general reference for the Sobolev spaces W Gm p, ( ) ,
G ⊂ R
n, we propose W. P. Ziemer’s monograph [15]. Since this paper is a continua-
tion of [4, 6, 7, 16], our notation and the basic definitions are consistent with those in
the quoted papers. For convenience we recall that for a multiindex α = ( α1 , … , αn ) ,
D
α is the classical or distributional differential operator, α = Σi αi . For a locally
integrable function g and a measurable subset E ⊂ R
n, E will be its Lebesgue
measure and −∫ g
E
= gE = E g
E
− ∫1 the average value of the function g over E .
B ( a, r ) will denote the open ball in R
n of radius r centered at a. The letter Q will
be used either for our fixed model open cube, where all our function spaces “live” or
for a generic subcube of “small” diameter δ ( Q ) or sidelength s ( Q ), containing all
the points x , y , z of R
n which appear in the considered pointwise inequality.
Usually s ( Q ) ≈ x y− , x, y, z ∈ Q , x ≠ y ≠ z, 2 Q ⊂ B ( x, r ) , r ≈ 4 x y− ,
where 2Q is the notation for the cube concentric with Q and of the doubled diameter.
Given two quantities A and B, we write A ≈ B if C1 A ≤ B ≤ C2 A for some
positive constants C1 and C2 . In the case of the Lipschitz moduli aQ ( x ) or the local
Hardy – Littlewood maximal functions MQ g ( x ) , the subscript Q indicates the local
character: x ∈ Q , s ( Q ) ≈ 2 x y− , of the pointwise estimates and the concepts used:
local maximal functions, local Riesz potentials, local Lipschitz moduli, etc. All our
pointwise estimates are essentially local, i.e., the important information, conveyed by
the pointwise estimates of the Taylor remainders R F x yj
k ( , ), is relevant only when
x y− → 0 or x y− < δ, i.e., in an open neighborhood of the diagonal { ( x, x ) }
of Q × Q for sufficiently small δ .
Finally we explicitly state that the constants in our formulas can change values in
the same string of estimates. Writing C ( n, m, p ) we emphasize that the constant de-
pends on the indicated parameters only.
2. Fractional maximal function. The local fractional maximal function of a
locally integrable function f : R
n → [ – ∞, ∞ ] is defined by
M f xα ( ) = M f xRα, ( ) = −
< <
∫sup ( )
( , )0 r R B x r
r f y dyα , 0 ≤ α ≤ n .
For α = 0 we obtain the local Hardy – Littlewood maximal function. The function
M fRα, is closely related to the Riesz potential ( R = ∞ ) and its localized form
I f xRα, ( ) =
f y
x y
dyn
x y R
( )
− −
− ≤
∫ α .
Generally the Riesz potential I fα majorizes the maximal function M fα
M f xα ( ) ≤ C I f x1 α ( ) ,
ISSN 1027-3190. Ukr. mat. Ωurn., 2007, t. 59, # 3
WHITNEY’S JETS FOR SOBOLEV FUNCTIONS 349
but for 1 < p < ∞
I f Lpα ≤ C M f Lp2 α
by a known result of Muckenhoupt – Wheeden [17] (with the corresponding inequali-
ties for local versions). The constants C1 , C2 are universal and depend on n, p on-
ly and possibly on R in some controlled way. For 0 < α p < n the Hardy – Little-
wood – Pólya theorem on fractional integration gives
I f Lqα ≤ C n p f Lp( , , )α ,
1
q
=
1
p n
− α
. (2.1)
It follows that the local fractional operators M Rα, also increase the integrability expo-
nent
M Rα, : L Qp
loc( ) → L Qp
loc
α ( ), pα =
np
n p− α
> p . (2.2)
The fractional maximal operator M Rα, is meaningful on arbitrary measure metric
space ( X, d, µ ) . If the Borel measure µ on X is an s-measure for some real s > 0
in the sense that
C rs
1 ≤ µ ( ( , ))B x r ≤ C rs
2 , (2.3)
then (2.2) holds if the L
p
spaces are understood as L Xp( , )µ . The real number s in
(2.3) is then interpreted as the Hausdorff dimension, dimH ( X, d, µ ) = s, of the regu-
lar measure metric space ( X, d, µ ) . The mapping property (2.2) then holds with the
Euclidean dimension n replaced by the Hausdorff dimension s satisfying the conditi-
on
α p < s. (2.4)
In this paper our main concern is naturally the case of Sobolev spaces in the Eucli-
dean spaces R
n
: the explicit expressions for the remainders R
m
F ( x, y ) are meaning-
ful only in this case. The above remarks on Sobolev spaces on measure metric spaces
are included only to see the broader perspective of the special cases of the theory ( m =
= 1, d-sets in R
n
, d < n [11], etc. ) . In this context it is important to notice that the
proof [17] of the inequality (2.2) is presented independently of the Hardy – Littlewood
– Pólya – Sobolev inequalities (2.1) and thus the inequalities (2.2) antecede the inequa-
lities (2.1).
The proof of the Hardy – Littlewood type result for the fractional operators M fα
for measure metric spaces satisfying the doubling condition
µ ( B ( x, 2r )) ≤ C µ ( B ( x, r )) (2.5)
should be presented in connection with this aspect of [17], rather than the traditional
reference to the classical Hardy – Littlewood – Pólya theorem [2, 3].
The result of P. Hajłasz [5] is somehow more subtle since it does not use the full
strength of the doubling condition (2.3) – (2.5).
The local fractional sharp maximal function of a locally integrable function f is de-
fined by
f xRβ,
# ( ) = − −
< <
− ∫sup ( , )
( , )0 r R
B x r
B x r
r f f dxβ , β ≥ 0.
For R = ∞ we simply write f xβ
# ( ). The case β = 0 is the Feffermann – Stein sharp
maximal function and instead of f xR0,
# ( ) we write f xR
# ( ) for the local version of the
Feffermann – Stein operator.
ISSN 1027-3190. Ukr. mat. Ωurn., 2007, t. 59, # 3
350 B. BOJARSKI
It has been understood during the last few decades that the fractional maximal func-
tions are a convenient way to describe various subtle properties of Sobolev function
spaces. They are also intimately connected with the pointwise inequalities characteri-
zation of these spaces [6, 16, 18 – 21], etc.
Let us briefly recall the case of first order Sobolev spaces W Up1, ( ). A. P. Calde-
rón characterized the W p n1, ( )R spaces by the pointwise condition f ∈ L
p
, and
f x1
# ( ) ∈ L
p
( R
n
) [19]. They are also characterized by the pointwise inequality
f x f y( ) ( )− ≤ a x a y x y( ) ( )+( ) − , x, y ∈ U ,
with a ∈ L
p
( U ) , p > 1, and the closely related averaged Poincaré inequality [4, 22]
− −∫ f x f dyB x r
B x r
( ) ( , )
( , )
≤ −∫Cr g y dy
B x r
( )
( , )
(2.6)
with g estimated by
g ( y ) ≤ a ( y ) + MQ a ( y )
for some local maximal function MQ a of a ∈ L
p
( g ∈ L
p
) , Q — cube of diameter
Q ≤ 4 x y− , containing x and y.
Maximizing the both sides of (2.6) we obtain [18]
f xR
# ( ) ≤ CM g xR1, ( )
which implies, by (2.2), that f xR
# ( ) ∈ Lp1 and consequently M fR and f ∈ Lp1 .
This is the Sobolev imbedding of W Up1, ( ) into L Up1
∗
( ) . This procedure is meaning-
ful for the case when U is replaced by a measure metric space ( X, d, µ ) with the
doubling measure µ, and the Sobolev imbedding holds with the geometric dimension
n replaced by the Hausdorff dimension s (for p < s ) [5] (see also the proof of
Proposition 4.1 below).
The described properties of the fractional maximal operator M R1, for measure me-
tric spaces lead then to an alternative proof of a weaker version of the Hajłasz theorem
(Hajłasz in [5] does not use the full strength of the doubling condition). For us it will
be important to understand the integrability properties of the function b ( x, y ) =
= ( )( ) ( )a x a y x y+ − on the product space U × U. They are controlled by the itera-
ted local maximal functions or iterated Steklov means
−∫
B x r( , )
−
∫
B y r
b x y dx dy
( , )
( , ) ≤ −∫
B x r
RM b y dy
( , )
( )1 ≤ M M b xR R
2 1( )( )
or (2.7)
−∫
B y r( , )
−
∫
B x r
b x y dy dx
( , )
( , ) ≤ −∫
B y r
RM b x dx
( , )
( )1 ≤ M M b yR R
2 1( )( ),
where M b yR
1 ( ) = −
<
∫sup ( , )
( , )r R B y r
b x y dx and M b xR
2 ( ) = −
<
∫sup ( , )
( , )r R B x r
b x y dy.
The representation b ( x, y ) = ( )( ) ( )a x a y x y+ − implies that each of the iterated
M MR R
2 1 maximal functions is controlled by either M M aR R( ),1 or M M aR R1, ( ) eval-
uated at x and y, respectively (see also the proof of Proposition 4.2 below). Thus by
(2.2) we have the following proposition.
Proposition 2.1. The iterated Steklov means (2.7) of the function b ( x, y ) =
= ( )( ) ( )a x a y x y+ − with a ∈ L
p
are in Lp1 .
ISSN 1027-3190. Ukr. mat. Ωurn., 2007, t. 59, # 3
WHITNEY’S JETS FOR SOBOLEV FUNCTIONS 351
3. The classes VLC (((( m, p, Q )))) . The ( )m −1 -jet F ∈ J Qm−1( ), F = { :f ii ≤
≤ m − 1}, fi ∈ L
p
( Q ) is said to be a ( )m −1 -jet in Q with variable Lipschitz
coefficient, F ∈ VLC ( m, p, Q ) for short, if for some function aQ = aQ ( F ) ∈ L
p
( Q )
the pointwise inequality holds
R F x ym −1 ( , ) ≤ x y a x a ym
Q Q− +[ ]( ) ( ) , x, y ∈ Q. (3.1)
Intimately related with (3.1) is the a priori stronger condition requiring the inequalities
R F x yi
m i− −1 ( , ) ≤ x y a x a ym i
Q
i
Q
i− +− [ ]( ) ( ) , aQ
i ∈ L
p
( Q ) , (3.2)
to be satisfied for all i ≤ m – 1 ( )R F R Fk k
0 ≡ .
Functional coefficients a xQ( ) , a xQ
i ( ) in (3.1) and (3.2) will be called the Lip-
schitz moduli of the jet F. The Lipschitz moduli in (3.1) and (3.2) are obviously not
defined uniquely. The equivalence relation aQ ≈ bQ in the sense of the pointwise
inequality C1 aQ ( x ) ≤ bQ ( x ) ≤ C2 aQ ( x ) , C1 , C2 — positive constants, preserves all
essential information contained in (3.1) and (3.2).
The concepts of the Taylor algebra recalled above give a convenient tool to for-
mulate the pointwise inequalities satisfied by the ( )m −1 -jet F = { :f D fα
α α≡ ≤
≤ m − 1} of a function f in the Sobolev class W m n
loc
, ( )1
R .
Indeed, the pointwise inequalities from [4] (Theorem 2) and [6] (formulas (3.5),
(3.6)) may be stated as the following proposition.
Proposition 3.1. If F = { }:D f mα α ≤ − 1 , f ∈ W m n
loc
, ( )1
R , then
R F x ym −1 ( , ) ≤ C x y a x a ym
Q Q− +( )( ) ( ) , aQ ≡ aQ
0 (3.3)
and
R F x yi
m i− −1 ( , ) ≤ C x y a x a ym i
Q
i
Q
i− +− ( )( ) ( ) , i ≤ m – 1. (3.4)
The local Lipschitz moduli aQ , aQ
i in the right-hand side of (3.3) and (3.4) in
view of [4, 6, 7, 16] can be chosen as dominated by the local Hardy – Littlewood maxi-
mal function M f xQ
m( )( )∇ of the highest gradient ∇m f of f. Also the constants
C in (3.3) and (3.4) are universal and depend on n and m only. Q is a model cube,
containing x and y of the diam Q ≈ δ
, δ ≥ x y− ( usually δ ≤ 5 x y− ) .
The jet spaces VLC ( m, p, Q ) can be supplied with Banach space norms in a natu-
ral way. For instance, the ( m – 1 ) -jet space VLC ( m, p, Q ) can be given the norm
F = max , inf( )( ) ( )
f i m ai L Q Q L Q
p p≤ − +1
where the inf is taken over all admissible moduli in (3.3) ( or aQ
i
L Qi k p ( )≤∑ for
the space VLC
##
( , , )m p Q defined by the inequalities (3.2) ) .
For simplicity we use the same notation VLC ( m, p, Q ) for spaces of jets or spaces
of functions generating the jets. The actual object is defined by the context. The aris-
ing normed spaces are complete Banach spaces.
The space VLC ( 0, p, Q ) is identified with jet space J
0
( Q ) ≡ L
p
( Q ) .
The space VLC ( 1, p, Q ) is the space M
1, p
( Q ) of P. Hajłasz whose immediate
natural generalizations to the case of measure metric spaces ( X, d, µ ) were studied by
P. Hajłasz [5] in an original and profound way.
ISSN 1027-3190. Ukr. mat. Ωurn., 2007, t. 59, # 3
352 B. BOJARSKI
For m = 1 and the jet space J
0
( Q ) = L
p
( Q ) , R
0
f ( x, y ) ≡ f ( x ) – f ( y ) . Inequa-
lity (3.3) has the form
f x f y( ) ( )− ≤ x y a x a y− +( )( ) ( ) , a ∈ L
p
( Q ) .
This makes sense for the general measure metric space ( X, d, µ ) as
f x f y( ) ( )− ≤ d x y a x a y( , ) ( ) ( )[ ]+ , a ∈ L
p
( X, dµ ) ,
f X d( , , )µ = f aL X Lp p( ) inf+
and produces the Banach space M X dp1, ( , , )µ .
The jet spaces VLC ( m, p, Q ) as Banach spaces have been identified with the clas-
sical Sobolev spaces W Qm p, ( ), Q — a cube in R
n. This is Theorem 9.1 in [16],
which we recall here referring for the detailed proof to [16].
Theorem. Let F be a ( m – 1 ) -jet, F = { }:f mα α ≤ − 1 ,
F ∈ VLC ( m, p, Q ) , p > 1, m ≥ 1.
Then in any cube Q0 � Q there exists a function f in the Sobolev class W Qm p, ( )0
such that the ( m – 1 ) -jet J fm−1 ∈ J Qm−1
0( ) coincides with F Q0
or
D f
Q
α
0
= f Qα 0
for α ≤ m – 1
and the extended m -jet, F̃ = { }:D f mα α ≤ ∈ J Qm( )0 , satisfies the inequality
˜
( )
F
L Qp
0
≤ C F m p QVLC( , , )
or
D F
L Qp
α ˜
( )0
≤ C F m p QVLC( , , ) for α = m.
Moreover, for any α + β, α β+ ≤ m, α ≤ m, β ≤ m,
D fα β+ = D D fα β( ), fα β+ = D fα
β, β ≤ m – 1, (3.5)
with the differential operators Dα , Dβ understood in the generalized Sobolev, or
distributional, sense.
Remark. The component fα , α = m, of the extended m -jet F̃ is obtained
also from fα , α = m – 1, by approximate differentiation [7, 8].
Corollary 3.1. The reduced jets C1( )F , F ⊂ VLC ( m, p, Q ) are in VLC ( m –
– 1, p1, Q ) where p1 is the Sobolev conjugate of p, 1 1/ p = 1 1/ /p n− .
This is the simplest of the Sobolev imbedding theorems for the spaces W Qm p, ( ),
for the case mp < n.
However, our general idea in this paper is to study the properties of the jet spaces
VLC independently of the identification with the classical Sobolev spaces W Qm p, ( ).
We state now and prove the two main theorems of this paper — Theorems 3.1
and 3.2.
Theorem 3.1. The formal differential operator
D
i : J U J Um m i( ) ( )→ −
acts as a bounded operator
D
i : VLC VLC( , , ) ( , , )m p Q m i p Q→ − .
ISSN 1027-3190. Ukr. mat. Ωurn., 2007, t. 59, # 3
WHITNEY’S JETS FOR SOBOLEV FUNCTIONS 353
Proof. We have
R Fm i i− −1( )D ≡ R Fi
m i− −1
with F ∈ VLC ( m, p, Q ) satisfying the pointwise inequality (3.1). We need to assess
that the inequality (3.2) holds. This we formulate as Lemma 3.1 below, which is the
general case of the main Lemma 8.1 in [16]. We enclose the proof from [16], because
the formulation here is simpler and explicitly drops the unnecessary assumption of qua-
sisuperharmonicity.
Lemma 3.1. If
R F x ym −1 ( , ) ≤ x y a x a ym
Q Q− +[ ]( ) ( ) , x, y ∈ Q,
(3.6)
diam Q = δ
, δ ≈ 2 x y−
,
then
R F y zi
m i− −1 ( , ) ≤ z y a y a zm i
Q
i
Q
i− +−
′ ′[ ]( ) ( ) , y, z ∈ Q ′,
δ ′ = diam Q ′, Q ′ ⊂ Q, δ ′ ≈ 2 y z− ≈
1
2
δ
,
with aQ
i
′ ≈ aQ or
a yQ
i
′( ) ≤ C n m a y M a yi Q Q Q( , ) ( ) ( )( )[ ]+ ′ , y ∈ Q ′,
for some constant Ci depending on n and m only.
Proof. From the Taylor algebra
R F y zi
m i− −1 ( , ) = D R F x y R F x zx
i m m
x y[ ]( , ) ( , )− −
=−1 1 ≡ D P x y zx
i
x y( ; , ) = (3.7)
with
P x y z( ; , ) ≡ R F x y R F x zm m− −−1 1( , ) ( , ) (3.8)
which is a polynomial in x of order ≤ m – 1, with coefficients obtained by restriction
to Q , of the coefficients of the local Whitney jet F . These are functions from
L Q1( )′δ . For y, z fixed, y ≠ z, consider the balls B ( y, r ) and B ( z, r ) , r =
= y z− and the spherical segment
S ≡ Sr = B ( y, r ) ∩ B ( z, r ) , Sr = Sr ( y, z ) .
By elementary geometry
B z r( , ) = B y r( , ) ≤ σ Sr
with 1 < σ < σ ( n ) independent of r.
By Markov’s inequality, [7, 8, 16, 23], applied to the subset S of the ball B ( y, r ) ,
r = y z− , we obtain
D P x y zx
i
x y
( ; , )
=
≤
C n
r i
( ) − ′ ′∫ P x y z dx
S
( ; , ) . (3.9)
The integrand P x y z( ; , )′ for x ′ ∈ S is estimated in view of (3.8) and (3.6) as
P x y z( ; , )′ ≤ R F x y R F x zm m− −′ + ′1 1( , ) ( , ) ≤
≤ [ ] [ ]( ) ( ) ( ) ( )a x a y x y a x a z x zQ Q
m
Q Q
m
′ ′ ′ ′′ + ′ − + ′ + ′ − .
For x ′ ∈ Sr we have
′ −x y ≤ y z− and ′ −x z ≤ y z−
ISSN 1027-3190. Ukr. mat. Ωurn., 2007, t. 59, # 3
354 B. BOJARSKI
and the integral in the right-hand side of (3.9) splits into the sum I r I r1 2( ) ( )+ with
I r1( ) ≡
C
r
ri
m
− ′ + ′′ ′∫ [ ]( ) ( )a x a y dxQ Q
S
,
I r2( ) ≡
C
r
ri
m − ′ + ′′ ′∫ [ ]( ) ( )a x a z dxQ Q
S
.
In I r1( ) we consider S as a subset of the ball B ( z, r ) and in I r2( ) we consider
it as a subset of B ( y, r ) and we get
I r1( ) ≤ Cr C M a z a ym i
Q Q Q
−
′ ′+( )( )( ) ( )1 ≤ CC r M a z a ym i
Q Q Q1
−
′ ′+[ ]( )( ) ( )
(3.10)
and
I r2( ) ≤ CC r C a z M a ym i
Q Q Q2 1
−
′ ′+[ ]( ) ( )
this ends the proof of the lemma with the explicit estimates for the constants
C n mi ( , ) = max( , )CC CC1 2
Theorem 3.2. The reduction operators Cl
act as a bounded operators ( pm < n )
Cl : VLC ( m, p, Q ) → VLC ( m – l, pl
, Q )
with pl
defined by the Sobolev relation
1
pl
=
1
p
l
n
− for l = 1, … , m.
For m = 1 Theorem 3.2 reduces to Theorem 3.3 or the case of the spaces
W Qp1, ( ) = M Qp1, ( ) treated by P. Hajłasz [5].
It is convenient for us to formulate the case m = 1 in the form of a separate theo-
rem, which we state here in the general Hajłasz form for general measure metric spaces
M X dp1, ( , , )µ .
Theorem 3.3 (P. Hajłasz [5]). If µ is a doubling measure with the doubling
constant C, and s = dimH ( X, d, µ ) the Hausdorff dimension ( C = 2
s
) , then
M Xp1, ( ) O L Xp∗
( ) ,
1
p∗ =
1 1
p s
− , p < s,
and
M Xp1, ( ) ⊂ Lip ( α, X ) , α = 1 − s
p
, p > s.
Proof. [5] or Proposition 4.2 below.
Proof of Theorem 3.2. Let F = { , , }f fm0 1… − be in VLC ( m, p, Q ) . Then
D Fm−1 = fm−1 and by Theorem 3.1 fm−1 is a (vector valued) function in
VLC ( 1, p, Q ) . By the quoted Theorem 3.3,
fm Lp−1 1 ≤ C F m p QVLC( , , ) .
We have
R Fm−1 ≡ R F f y
x y
m
m
m
m
−
−
−
+ −
−
2 1
1
1
1
( ) ( )
( )
( )!
C
and we see that the ( m – 2 ) -jet C1F satisfies the pointwise inequality
ISSN 1027-3190. Ukr. mat. Ωurn., 2007, t. 59, # 3
WHITNEY’S JETS FOR SOBOLEV FUNCTIONS 355
R F x ym−2 1( )( , )C ≤ C x y f y a F x a F y x ym
m Q Q− + +( ) −( )−
−
1
1( ) ( )( ) ( )( )
(3.11)
with f ym−1( ) ∈ L Qp1 ( ), aQ ( F ) ∈ L
p
( Q ) .
Symmetrizing the right-hand side of (3.11) by adding the term f xm−1( ) and ap-
plying the iterated maximal function operation as in Proposition 2.1 and Lemma 3.1 we
deduce from (3.11) the pointwise inequality
R F x ym−2 1( )( , )C ≤ C x y b x b ym
Q Q− +[ ]−1 ( ) ( )
with b xQ( ) controlled by
b xQ( ) ≤ C f x M M a F xm Q− + ( )( )1 1( ) ( ) ( )( ) ∈ L Qp1 ( ).
Since Cl may be represented as a composition of C1, Cl = ( )C1 l and the formulas
1/ pk = 1 11/ /p nk− − , k = 1, … , l, also respect composition, we see that the descri-
bed procedure ends the proof.
4. The averaging of the pointwise inequalities. Averaging of the pointwise in-
equalities as a method of generating new inequalities out of known ones has been sys-
tematically applied probably for the first time in [16], see also [8]. The proof of Lem-
ma 8.1 in [16], in some slightly modified form reproduced also above in the proof of
our Lemma 3.1, gives a general scheme of the averaging process and is in principle ap-
plicable to the pointwise inequalities in function spaces ( X, d, µ ) satisfying the doub-
ling condition for the measure µ .
In this context it seems proper now to formulate a following general proposition for
the simplest case of the pointwise inequalities characterizing the Sobolev space
W Qp1, ( ) , p > 1, or Hajłasz – Sobolev spaces M X dp1, ( , , )µ .
Proposition 4.1. Let a real valued function f satisfy the pointwise estimate
f x f y( ) ( )− ≤
≤ x y a x a y x y a x a y x y a x a y− + + − + + − + + …[ ]( ) ( ) ( ) ( ) ( ) ( )( ) ( )1 1
2
2 2 (4.1)
for some finite sequence of nonnegative valued functions
a ( x ) ≡ a0 ( x ) , a1 , a2 , … , ai ∈ Lp i− ,
1
p i−
=
1
p
i
n
+ , i = 1, 2, … ,
then
f x f y( ) ( )− ≤ C x y a x a y− +[ ]˜( ) ˜( ) (4.2)
with
˜( )a x ≤ C M a x M M a x M M a xi i( )( ) ( )( ) ( )( )0 0 1 1 0+ + … + + … , (4.3)
where Mi is the local fractional maximal function of order i, M f0 ≡ M fQ .
Proof. Integrating inequality (4.1) over a ball B ( x0 , r ) with respect to x and y
we obtain
− −∫ f y f dyB
B
( ) ≤ −
∫Cr a dx
B
+ −∫r a dx
B
1 + … + −
−
−∫r a dxk
k
B
1
1 . (4.4)
Hence, for r ≤ RQ we conclude for some R > 0 the inequality
− −∫ f y f dyB
B
( ) ≤ f xRQ
# ( ) ≤
≤ C M a M a M aR R k R kQ Q Q
( ), , ,1 2 1 1+ + … + − (4.5)
ISSN 1027-3190. Ukr. mat. Ωurn., 2007, t. 59, # 3
356 B. BOJARSKI
for the local fractional maximal functions M a xi R i, ( )−1 and the local sharp maximal
function fR
# and the inequality for the local fractional sharp maximal function f R1,
#
− −∫1
r
f y f dyB
B x r
( )
( , )
≤ f xR1,
# ( ) ≤ C M a x M a M aR R k R k( )( ) , ,+ + … + − −1 1 1 1 . (4.6)
In view of (2.2) all terms in the right-hand side of (4.5) and (4.6) are in Lp
loc
1 and Lp
loc
respectively.
Thus (4.5) implies fR
# ∈ Lp
loc
1 , hence f ∈ Lp1 — which is one of the conclusions
of the Hajłasz theorem for generalized M p1, pointwise inequalities (4.1). In an analo-
gous way (4.6) implies f xR1,
# ( ) ∈ Lp
loc and we recognize the A. Calderón [19] conditi-
on characterizing the Sobolev classes W Qp1, ( ) .
Application of the telescoping argument at the Lebesgue points of the function f to
the generalized Poincaré inequality (4.4), probably first used by L. Hedberg in [24],
allows to recover the pointwise inequality (4.2) in the same way as in the proof of
Theorem 3.2 [22] (see also [25]). This finishes the proof of the proposition.
By the imbedding theorem (2.2) for fractional maximal functions all terms in the
right-hand side of (4.3) are in the space L
p
( Q ) .
Closely related with Proposition 4.1 is following proposition.
Proposition 4.2. Assume that a function f ∈ L
p
( Q ) satisfies instead of (4.2) the
inequality
f x f y( ) ( )− ≤ b x y x y( , ) − , x, y ∈ Q .
Then for a small concentric cube Q0 ⊂ Q ( e.g. Q ⊂ 4 Q0 where 4 Q0 is the
concentric 4-times extended sube Q0 ) ,
f x f y( ) ( )− ≤ −
∫Cr
B x r( , )
b x z dz( , ) + −
∫
B y r
b z y dz
( , )
( , )
with some constant C = C ( n ) .
Proof. Let x, y ∈ Q0 and r = x y− . Then the balls B ( x, r ) and B ( y, r ) are
both contained in Q and so is the spherical segment S = B ( x, r ) ∩ B ( y, r ) . The ob-
vious inequality
f x f y( ) ( )− ≤ f x f z f z f y( ) ( ) ( ) ( )− + − (4.7)
averaged over z ∈ S gives
f x f y( ) ( )− ≤
S
b x z x z b z y z y dz∫ − + −( )( , ) ( , ) ≤
≤ −
∫Cr
B x r( , )
b x z dz( , ) + −
∫
B y r
b z y dz
( , )
( , ) (4.8)
for some constant C depending on n only.
For b ( x, y ) symmetric: b ( x, y ) = b ( y, x ) , what we can always assume, (4.8) has
the “splitted form”, for b ( x, y ) “splitted”, b ( x, y ) = a ( x ) + a ( y ) , the Steklov means
−∫ b x z dz
B x r
( , )
( , )
have the form a x M aQ( ) + and we recognize the terms in the right-
hand side of (3.10) in our proof of Lemma 3.1. If b ( x, y ) does not appear in a
symmetric form, say b ( x, y ) = a ( x ) + a ( y ) + c x x y( ) − , the second integral in (4.8)
will produce the term of the form M c z1 ( ) of a fractional maximal function of the
ISSN 1027-3190. Ukr. mat. Ωurn., 2007, t. 59, # 3
WHITNEY’S JETS FOR SOBOLEV FUNCTIONS 357
coefficient c ( x ) . The described averaging process can be “iterated” and applied to the
pointwise inequalities (4.8). It leads then to the iterated fractional maximal functions
appearing in (2.7) in the sketch of the proof of Proposition 2.1.
Proposition 4.2 and its proof generalize to the higher order Taylor remainder esti-
mates
R F y xm−1 ( , ) ≤ b x y x y m( , ) − .
Instead of the obvious inequality (4.7) the proof requires somewhat more sophisticated
formula (3.7) and the Markov inequality (3.9) for i = 0.
The pointwise inequality (4.2) characterizes the behavior of the zero-order remain-
der term R f x y0 ( , ) ≡ f x f y( ) ( )− in the neighborhood of the diagonal ∆ of the carte-
sian product Q × Q. Inequality (4.1) can be viewed as a more subtle description of the
asymptotic of R f x y0 ( , ) in the neighborhood of the diagonal in terms of powers of the
distance x y− of the point ( x, y ) ∈ Q × Q from the diagonal ∆, dist (( x, y ), ∆ ) ∼
∼ x y− .
Proposition 4.1 states that the a priori more general asymptotics (4.1) can be always
reduced to the “simplest” case (4.2).
Actually the averaging process described in Proposition 4.1 can be continued one
step further reducing (4.2) to the pointwise inequality
f x f y( ) ( )− ≤ C x y a x a y− +0( )˜̃( ) ˜̃( ) ≡ C a x a y( )˜̃( ) ˜̃( )+
with the estimate
˜̃( )a x ≤ C M M a x M M a x M M a xi i( ) ( ) ( )( ) ( ) ( ) ( ) ( ) ( )1 2 1 1+ + … +( )+
with all the terms locally in L Qp1 ( ).
Thus the extended averaging process gives essentially the Sobolev and Sobolev –
Hajłasz embedding theorem for p < n ( p < s = dimH ( X, d, µ ) — for measure me-
tric spaces ) .
Analogous asymptotic Sm RC conditions can be formulated for m > 1 and Pro-
position 4.1 generalizes to this context also.
When complemented by the recent extension results of Hajłasz – Koskela – Tuomi-
nen [26] or P. Shvartsman [27], the poinwise theory approach to Sobolev spaces
W Qm p, ( ) , p > 1, as presented in [5, 7, 16] and outlined above for Sobolev jet spaces
VLC ( m, Q, p ) extends to regular ( n-Ahlfors regular ) open subsets of R
n. In this
way the Sobolev jet space theory seems to be brought to a satisfactory shape, according
to the expectations expressed in the final comments of §10 of [16]. It can be expected
that the averaging process described above when applied with respect to a d-measure
µ supported by a d-set Σ ⊂ Q ( Σ ⊂ R
n
) [11, 28] will be useful for fully understan-
ding from the pointwise point of view the case of fractional Sobolev – Besov spaces on
n-Ahlfors regular subsets of R
n (see also [21], wich almost fulfils the task) and the
case of Besov spaces or traces of Sobolev spaces on d-subsets Σ of R
n.
This problem will be, hopefully, the subject of a subsequent publication in coopera-
tion with P. Hajłasz and P. Strzelecki.
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Received 10.11.2006
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|
| id | umjimathkievua-article-3311 |
| institution | Ukrains’kyi Matematychnyi Zhurnal |
| keywords_txt_mv | keywords |
| language | English |
| last_indexed | 2026-03-24T02:40:07Z |
| publishDate | 2007 |
| publisher | Institute of Mathematics, NAS of Ukraine |
| record_format | ojs |
| resource_txt_mv | umjimathkievua/be/6efae394f44873ffe851e952b2fdc3be.pdf |
| spelling | umjimathkievua-article-33112020-03-18T19:51:00Z Whitney’s jets for Sobolev functions Струмені Уїтні для функцій Соболева Bojarski, B. Боярський, Б. We present two fundamental facts of the jet theory for Sobolev spaces $W^{m, p}$. One of them is that the formal differentiation of $k$-jets theory is compatible with the pointwise definition of Sobolev $(m - 1)$-jet spaces on regular subsets of Euclidean spaces $R^n$. The second result describes the Sobolev embedding operator of Sobolev jet spaces increasing the order of integrability of Sobolev functions up to the critical Sobolev exponent. Встановлено два фундаментальних факти теорії струменів для просторів Соболева $W^{m, p}$. Перший із них полягає в тому, що формальне диференціювання $k$-струменів є сумісним з поточко-вим визначенням соболевських просторів $(m - 1)$-струменя на регулярних підмножинах евклідових просторів $R^n$. Другий результат описує соболевські оператори вкладення соболевських просторів струменів, що покращують порядок сумовності соболевських функцій аж до критичного показника. Institute of Mathematics, NAS of Ukraine 2007-03-25 Article Article application/pdf https://umj.imath.kiev.ua/index.php/umj/article/view/3311 Ukrains’kyi Matematychnyi Zhurnal; Vol. 59 No. 3 (2007); 345–358 Український математичний журнал; Том 59 № 3 (2007); 345–358 1027-3190 en https://umj.imath.kiev.ua/index.php/umj/article/view/3311/3367 https://umj.imath.kiev.ua/index.php/umj/article/view/3311/3368 Copyright (c) 2007 Bojarski B. |
| spellingShingle | Bojarski, B. Боярський, Б. Whitney’s jets for Sobolev functions |
| title | Whitney’s jets for Sobolev functions |
| title_alt | Струмені Уїтні для функцій Соболева |
| title_full | Whitney’s jets for Sobolev functions |
| title_fullStr | Whitney’s jets for Sobolev functions |
| title_full_unstemmed | Whitney’s jets for Sobolev functions |
| title_short | Whitney’s jets for Sobolev functions |
| title_sort | whitney’s jets for sobolev functions |
| url | https://umj.imath.kiev.ua/index.php/umj/article/view/3311 |
| work_keys_str_mv | AT bojarskib whitneysjetsforsobolevfunctions AT boârsʹkijb whitneysjetsforsobolevfunctions AT bojarskib strumeníuítnídlâfunkcíjsoboleva AT boârsʹkijb strumeníuítnídlâfunkcíjsoboleva |