View of How to Recognize Polynomials in Higher Order Sobolev Spaces

HOW TO RECOGNIZE POLYNOMIALS IN HIGHER ORDER SOBOLEV SPACES

BOGDAN BOJARSKI, LIZAVETA IHNATSYEVA and JUHA KINNUNEN^∗

Abstract

This paper extends characterizations of Sobolev spaces by Bourgain, Brézis, and Mironescu to the higher order case. As a byproduct, we obtain an integral condition for the Taylor remainder term, which implies that the function is a polynomial. Similar questions are also considered in the context of Whitney jets.

1. Introduction

In this paper we study a new characterization of the higher order Sobolev spacesW^m,p()which is based on J. Bourgain, H. Brézis, and P. Mironescu’s approach [5] (see also [7]). They showed that a functionf ∈L^p()belongs to the first order Sobolev spaceW^1,p(), 1< p <∞, on a smooth bounded domain⊂^Rnif and only if

(1) lim inf

ε→0

|f (x)−f (y)|^p

|x−y|^p ρ_ε(|x−y|) dx dy <∞, whereρ_ε, withε >0, are radial mollifiers. Moreover,

lim inf

ε→0

|f (x)−f (y)|^p

|x−y|^p ρ_ε(|x−y|) dx dy =c

|∇f|^pdx,

where the constantcdepends only onpandn. Forp = 1 this gives a char- acterization of the space of bounded variation BV(). See also [6], [8], [11], [16], [17], [18], [19], [22] and [21] for related results.

We extend the results of [5] and [7] to the higher order case. To characterize the Sobolev spacesW^m,p(), 1< p <∞, we use the condition

(2) lim inf

ε→0

|R^m−1f (x, y)|^p

|x−y|^mp ρ_ε(|x−y|) dx dy <∞,

∗The research is supported by the Academy of Finland, the first author is also partially supported by Polish Ministry of Science grant no. N N201 397837 (years 2009–2012).

Received 28 March 2011.

whereR^m−1f is a Taylor(m−1)-remainder off, generalizing (1). Forp=1 condition (2) describes the space BV^m()of integrable functions whose weak derivatives of ordermare signed Radon measures with finite total variation.

Condition (2) is a priori weaker than the pointwise condition (3) |R^m−1f (x, y)| ≤ |x−y|^m(a_f(x)+a_f(y)), a_f ∈L^p(Rⁿ), characterizing Sobolev classW^m,p(Rⁿ)as in [1] (see also (27) below).

Another variant of extension of the results of [5] and [7] to the higher order case has been introduced in [4], where the characterization ofW^m,p(), 1 <

p <∞, (BV^m()forp =1) is formulated in terms of them-th differences.

According to this result, a functionf ∈L^p()belongs toW^m,p()if (4) lim inf

ε→0

^m

j=0

(−1)^j m

j

f

(m−j ) m x+ j

my

^pρ_ε(|x−y|)

|x−y|^mp dx dy <∞. For smooth functionsf ∈C^m+1the equivalence of the integrands in (2) and (4) moduloO(|x−y|^m+1)is well known. The results of the present paper and of [4] essentially show that both integrands are equivalent in their averaged asymptotic behaviour, for ε → 0, in the ε-neighbourhood of the diagonal = {x=y}in the Cartesian product×.

In close connection with these characterizations, H. Brézis [7] considered conditions under which a measurable functionf defined on a connected open setis a constant. See also [20]. In particular, he showed that if

|f (x)−f (y)|

|x−y|ⁿ⁺¹ dx dy <∞,

thenf is a constant function. We extend this result to the higher order case and show that the condition

|R^m−1f (x, y)|

|x−y|^n+m dx dy <∞

implies that the functionf, with locally integrable weak derivatives up to order m−1, is a polynomial of degree at mostm−1.

Condition (2) applies to Whitney jets as well. Recall that H. Whitney in [24] gave a method to define differentiable functions on closed subsets ofRⁿ. His approach can be adopted to different kind of smoothness conditions. In particular, for an(m−1)-jetF on a subset⊂^Rn, defined as a collection of

functions{f_α : |α| ≤m−1},f_α ∈L^p(), we can study the meaning of (2) using the formal Taylor remainder of orderm−1 of the jetF. The formalism of Taylor-Whitney jets identifies in a natural way virtual derivatives, jets, with Sobolev derivatives.

In Section 4 we show that a jetF on an open setsatisfying (2) for some special case of mollifiers is locally a jet of aW^m,p-function. The proof uses an approximation procedure from [1] where the Sobolev spaces are described in terms of pointwise inequalities (3).

Notice also that a certain version of condition (2) for the jetF on Ahlfors s-regular subsetsS ⊂ ^Rn,n−1 < s ≤ n, charaterizes the Lipschitz spaces Lip(m, p,∞, S)studied in [13]. IfSsupports theq-Poncaré inequality, 1≤ q <∞, then the first order space Lip(1, p,∞, S)coincides with the Hajłasz- Sobolev spaceW^1,p(S)forp > q(see e.g. [25]).

2. Characterization of Sobolev spaces

Our notation is standard. For a multi-indexα = (α1, . . . , αn),αi ≥0, and a pointx=(x1, . . . , x_n)∈^Rn, we denote by

x^α =x₁^α1x₂^α2· · ·x_n^αn the monomial of degree

|α| = n

i=1

αi.

In the same way

D^αf = ∂^|α|f

∂x₁^α1. . . ∂xn^αn

is a (weak) partial derivative of order |α|. We also use the convention that D⁰f =f. Moreover, let∇^mfbe a vector with the componentsD^αf,|α| =m.

Letbe an open set inRⁿ, 1≤p <∞,ma positive integer. The Sobolev spaceW^m,p()consists of all functionsu ∈ L^p() such that for all multi- indexαwith|α| ≤mthe weak derivativeD^αuexists and belongs toL^p().

We use the conventionW^0,p() = L^p(). The Sobolev spaceW^m,p()is equipped with the norm

uW^m,p()=

|α|≤m

|D^αu|^pdx 1/p

.

For the properties of Sobolev functions, see [15].

We write

T_y^mf (x)=

|α|≤m

D^αf (y)(x−y)^α α!

and

R^mf (x, y)=f (x)−T_y^mf (x)

for the Taylor polynomial of ordermand the Taylor remainder of orderm, re- spectively. Iff ∈C^m(Rⁿ)the Taylor formula can be expressed in the following form, see [26] p. 126,

(5) f (x)−T_y^m−1f (x)

=m

|α|=m

1 0

(1−t )^m−1D^αf[(1−t )y+t x)]dt

(x−y)^α α! . Note that we use the same notation for formal Taylor polynomials and remain- ders if we have only weak derivatives.

We define a family of functionsρ_ε∈L¹_loc(0,∞),ε >0, such thatρ_ε≥0, _∞

0

ρ_ε(r)rⁿ⁻¹dr =1 and

εlim→0

_∞

δ

ρε(r)rⁿ⁻¹dr =0 for every δ >0.

These properties are rather standard in the construction of radial mollifiers related to approximations of unity.

First we prove a useful result for smooth functions.

Lemma2.1.Letbe an open set inRⁿ,1≤p <∞,ma positive integer andg∈C₀^m+1(Rⁿ). Then

(6) lim

ε→0

|R^m−1g(x, y)|^p

|x−y|^mp ρε(|x−y|) dx dy

=

∂B(0,1)

|α|=m

D^αg(x) α! e^α

^pde dx.

Proof. Sinceg ∈C₀^m+1(Rⁿ)by Taylor’s formula, we have

|R^m−1g(x+h, x)| ≤

|α|=m

D^αg(x) α! h^α

+c|h|^m+1

from which we conclude that

|R^m−1g(x+h, x)|^p≤(1+θ )

|α|=m

D^αg(x) α! h^α

^p+c_θ|h|^(m+1)p for everyθ >0,x ∈^Rnandh∈^Rn.

We multiply the last inequality byρ_ε(|h|)/|h|^mpand integrate over the set S= {(x, h)∈(suppg∩)×^Rn:x+h∈}

∪ {(x, h)∈(\suppg)×^Rn:x+h∈(suppg∩)}. We have

(7)

S

|R^m−1g(x+h, x)|^p

|h|^mp ρ_ε(|h|) dh dx

≤(1+θ )

Rⁿ

ρ_ε(|h|)

|h|^mp

|α|=m

D^αg(x) α! h^α

^pdh dx +2c_θ|suppg|

Rⁿ

|h|^pρ_ε(|h|) dh.

By the properties of the mollifiersρ_ε, it follows that

(8) lim

ε→0

Rⁿ

|h|^pρ_ε(|h|) dh=0.

Note also, that

Rⁿ

ρ_ε(|h|)

|h|^mp

|α|=m

D^αg(x) α! h^α

^pdh

= _∞

0

rⁿ⁻¹ρε(r) dr

∂B(0,1)

|α|=m

D^αg(x) α! e^α

^pde.

Thus, passing to the limit in (7), first withε →0, then withθ →0, and changing variables in the integral on the left hand side, we arrive at

(9) lim sup

ε→0

|R^m−1g(x, y)|^p

|x−y|^mp ρ_ε(|x−y|) dx dy

≤

∂B(0,1)

|α|=m

D^αg(x) α! e^α

^pde dx.

This concludes the first part of the proof of (6).

Next we show the inequality to the other direction. IfKis a compact subset of, then for anyx∈Kand|h| ≤d, where

d =min{1,dist(K, ∂)/2}, we have

R^m−1g(x+h, x)−

|α|=m

D^αg(x) α! h^α

≤c_K|h|^m+1. Hence

|α|=m

D^αg(x) α! h^α

^p ≤(1+θ )|R^m−1g(x+h, x)|^p+c_θ,K|h|^(m+1)p

for everyθ >0 and consequently (10)

K

B(0,d)

ρ_ε(|h|)

|h|^mp

|α|=m

D^αg(x) α! h^α

^pdh dx

≤(1+θ )

K

B(0,d)

|R^m−1g(x+h, x)|^p

|h|^mp ρ_ε(|h|) dh dx +c_θ,K|K|

B(0,d)

|h|^pρ_ε(|h|) dh.

Passing to the limit asε→0 in (10), and taking into account (8), we have

K

∂B(0,1)

|α|=m

D^αg(x) α! e^α

^pde dx

≤(1+θ )lim inf

ε→0

K

B(0,d)

|R^m−1g(x+h, x)|^p

|h|^mp ρ_ε(|h|) dh dx

≤(1+θ )lim inf

ε→0

|R^m−1g(x, y)|^p

|x−y|^mp ρε(|x−y|) dx dy.

Since the last estimate holds for everyθ >0 and every compact setK ⊂, we have

(11)

∂B(0,1)

|α|=m

D^αg(x) α! e^α

^pde dx

≤lim inf

ε→0

|R^m−1g(x, y)|^p

|x−y|^mp ρ_ε(|x−y|) dx dy.

Combining this with (9) we arrive at (6).

The following theorem is an analog of Theorem 2 in [7] (see also [5]) for higher order Sobolev spaces.

Theorem2.2.Letbe an open set inRⁿ,1< p <∞andmbe a positive integer. Iff ∈W^m−1,p()satisfies

(12) c_f =lim inf

ε→0

|R^m−1f (x, y)|^p

|x−y|^mp ρ_ε(|x−y|) dx dy <∞, thenf ∈W^m,p().

Proof. Assume that andδ <dist(, ∂). Letη∈C₀^∞(Rⁿ)be a nonnegative radial function such that

Rⁿ

η(x) dx=1

and suppη⊂B(0,1). Consider the regularizationf_δ=f ∗η_δoff,η_δ(x)= δ⁻ⁿη(x/δ). For everyf ∈L¹_loc()(extended by zero toRⁿ\) the function f_δis smooth in, and iff has a weak derivativeD^αf in, then

D^α(f_δ)=D^αf ∗η_δ (see e.g. [15]). Thus, for everyx, y∈we have R^m−1fδ(x, y)=fδ(x)−

|α|≤m−1

D^α(fδ)(y)(x−y)^α α!

=

B(0,δ)

f (x−z)−

|α|≤m−1

D^αf (y−z)(x−y)^α α!

η_δ(z) dz

=

B(0,δ)

R^m−1f (x−z, y−z)η_δ(z) dz.

By Jensen’s inequality, it is easy to see that (12) implies (13) lim inf

ε→0

|R^m−1f_δ(x, y)|^p

|x−y|^mp ρε(|x−y|) dx dy≤cf. Next by applying (11) tog=f_δwe get

(14)

∂B(0,1)

|α|=m

D^αf_δ(x) α! e^α

^pde dx≤c_f.

Now let e ∈ ∂B(0,1)and denote by Ethe vector with the components E_α =(1/α!)e^α₁¹· · ·e_n^αn,|α| =m. It is easy to see that

v =

∂B(0,1)|v·E|^pde 1/p

, 1≤p <∞,

is a norm on a linear space of all vectors v = (v_α)_|α|=m. Obviously, it is nonnegative andv =0 if and only ifv=0. The triangle inequality follows from the Minkowski inequality. Since·is equivalent to the Euclidean norm, for any vector∇^mg(x)we have

(15) |∇^mg(x)|^p ≈

∂B(0,1)

|∇^mg(x)·E|^pde

and (14) implies that (16)

|∇^mf_δ(x)|^pdx≤c

for every andδ < dist(, ∂), with the constantcdepending only onc_f,n,mandp.

Sincep > 1 the weak compactness and a diagonal argument show that f ∈W^m,p().

The following statement follows immediately from Theorem 2.2.

Corollary2.3.Let be an open set inRⁿ, 1 < p < ∞, ma positive integer. Iff ∈L^p()satisfies

(17) lim inf

ε→0

|Rⁱ⁻¹f (x, y)|^p

|x−y|^ip ρ_ε(|x−y|) dx dy <∞, for everyi=1, . . . , m, thenf ∈W^m,p().

Here the remaindersRⁱ⁻¹f are defined recursively starting from i = 1.

Thus, if (17) holds fori = 1 by Theorem 2.2 the weak derivatives D^αf,

|α| = 1, exist andRⁱ⁻¹f are defined for i = 2 and this procedure can be continued recursively. The recursion may seem somewhat awckward. A more direct interpretation is possible in terms of Whitney jets, see Section 4.

Remark 2.4. Note that if is a W^m,p-extension domain, i.e. there is a bounded linear operator

E :W^m,p()→W^m,p(Rⁿ)

such thatEf|=f for everyf ∈W^m,p(), then also the converse statement in Theorem 2.2 is true. Indeed, iff ∈W^m,p(Rⁿ), then

(18)

Rⁿ

|R^m−1f (x+h, x)|^pdx ≤c|h|^mp

Rⁿ

|∇^mf (x)|^pdx

for everyh∈^Rn. For smooth functions inequality (18) follows from Taylor’s formula (5), thus we have

R^m−1f (x+h, x)=m

|α|=m

h^α α!

1 0

(1−t )^m−1D^αf (x+t h) dt.

And an integration over the whole space gives

Rⁿ

|R^m−1f (x+h, x)|^pdx

≤c|h|^mp

Rⁿ

1 0

(1−t )^(m−1)p|∇^mf (x+t h)|^pdt dx

≤c|h|^mp 1

0

Rⁿ

|∇^mf (x+t h)|^pdx dt

≤c|h|^mp

Rⁿ

|∇^mf (x)|^pdx.

Since smooth functions are dense in the Sobolev space inequality (18) holds for everyf ∈W^m,p(Rⁿ).

Now letf ∈W^m,p()and denote byfits extension toRⁿ. Since

Rⁿ

ρ_ε(|h|) dh=ω_{n−1 ∞}

0

ρ_ε(r)rⁿ⁻¹dr =ω_n−1,

whereω_n−1is the(n−1)-dimensional surface measure of the unit ball inRⁿ, by (18) we have

(19)

|R^m−1f (x, y)|^p

|x−y|^mp ρ_ε(|x−y|) dy dx

≤

Rⁿ

Rⁿ

|R^m−1f (x, y) |^p

|x−y|^mp ρ_ε(|x−y|) dy dx

≤c

Rⁿ

|∇^mf (x) |^pdx

≤cfW^m,p(Rⁿ)≤cfW^m,p().

More precisely, the following result is true.

Theorem 2.5. Assume that is a W^m,p-extension domain, let m be a positive integer,1< p <∞and letf ∈W^m,p(). Then

(20) lim

ε→0

|R^m−1f (x, y)|^p

|x−y|^mp ρ_ε(|x−y|) dx dy

=

∂B(0,1)

|α|=m

D^αf (x) α! e^α

^pde dx.

Proof. Sinceis an extension domain, any functionf ∈ W^m,p()can be approximated by functions f_n ∈ C₀^m+1(Rⁿ) in W^m,p()-norm. Hence, Lemma 2.1 implies the validity of (6) for everyf ∈ W^m,p(). Indeed, by (15) for the right hand side of (6) we have

∂B(0,1)

|α|=m

D^αf (x) α! e^α

^pde 1/p

−

∂B(0,1)

|α|=m

D^αfn(x) α! e^α

^pde 1/p

= | ∇^mf (x) − ∇^mfn(x) | ≤ ∇^m(f−fn)(x) ≤c|∇^m(f −fn)(x)|. To justify the limit of the left hand side of (6) we can apply (19). Thus, equality (6) is true for anyf ∈W^m,p().

Remark2.6. If=^Rnand

(21) ρ_ε(r)=

⎧⎨

⎩

(n+mp)r^mp

ε^n+mp , r < ε,

0, r ≥ε,

then (12) can be written as (22) lim inf

ε→0

1 ε^mp

Rⁿ

B(y,ε)

|f (x)−T_y^m−1f (x)|^pdx dy <∞.

Here the integral sign with a bar denotes the integral average.

We point out that condition (22) is closely related to Calderón’s character- ization of Sobolev spaces in [9] (see also [12]). To this end, let 1< p < ∞ andmbe a positive integer. Forf ∈L^p()we define a maximal function as (23) N(f, y)=sup

ε>0

1 ε^m

B(y,ε)

|f (x)−P (x, y)|^pdx 1/p

,

if there exists a polynomialP (x, y)inx, of degree at mostm−1, such that the expression on the right hand side of (23) is finite. If no such polynomial exists,

we setN(f, y) = ∞. Then a functionf ∈ L^p(Rⁿ)belongs toW^m,p(Rⁿ)if and only ifN(f,·)∈L^p(Rⁿ).

Now consider the analog of Theorem 2.2 forp= 1. Recall that the space BV() is defined as the space of functions in L¹(), whose weak deriv- atives are Radon measures with finite total variation. Denote by BV^m(), m=2,3, . . ., the set of functions inL¹(), whose derivatives of ordermare finite Radon measures.

Observe, that by the Riesz representation theorem, a functionf ∈L¹() belongs to BV^m()if and only if there existsc >0 such that

f D^αϕ dx

≤cϕ∞

for everyϕ ∈C₀^∞()and every multi-indexαwith|α| =m.

Theorem2.7.Let be an open set inRⁿ andm be a positive integer. If f ∈W^m−1,1()satisfies

(24) lim inf

ε→0

|R^m−1f (x, y)|

|x−y|^m ρ_ε(|x−y|) dx dy <∞, thenf ∈BV^m().

Proof. The proof is the same as for Theorem 2.2, except for the fact that (16) implies that all the derivatives of ordermof functionf are measures with finite total variation.

Remark2.8. As observed in [4], ifis a smooth bounded domain, then BV^m()can be characterized as the set of functionsf ∈W^m−1,1()such that D^αf ∈ BV() for every multi-indexα with|α| = m−1. The equivalence follows from the Sobolev embedding theorem, see e.g. [15].

Using this characterization it is not difficult to see that Theorem 2.7 gives a necessary and sufficient condition for a function to be in BV^m(), when is a bounded smooth domain inRⁿ. This can be seen as in Remark 2.4.

3. A criterion for a function to be a polynomial

The next result is a higher order version of Theorem 1 in [7].

Theorem3.1.Let be a connected open set inRⁿ,1 ≤ p < ∞ andm a positive integer. Assume thatf ∈ L¹_loc()has weak derivativesD^αf with

|α| ≤m−1inand lim inf

ε→0

|R^m−1f (x, y)|^p

|x−y|^mp ρε(|x−y|) dx dy =0.

Thenf is a polynomial of degree at mostm−1a.e. on.

Proof. Suppose first that p > 1. Note that if is a bounded smooth domain in Theorem 2.2 it is enough to assume thatf ∈ L¹_loc() has weak derivativesD^αf with|α| ≤ m−1 in. Thus, applying Theorem 2.2 (see also Theorem 2.5) to a ballB ⊂ we can conclude thatf ∈W^m,p(B)and

∇^mfL^p(B)=0. This implies thatf is a polynomial of degree at mostm−1 a.e. inB(see e.g. generalized Poincaré inequality in [15]) and the claim follows from the assumption thatis connected.

Let thenp = 1. In this case we apply Theorem 2.7 to a ballB ⊂ and conclude that f ∈ BV^m(B)and the total variation |∇^mf|(B)of the vector valued measure∇^mf equal to zero. This implies that there is a polynomialP of degree at mostm−1 such thatf = P a.e. inB (see e.g. Lemma 12 in [4] for more details). Again the claim follows from the assumption thatis connected.

The next result is a higher order generalization of Proposition 1 in [7].

Corollary3.2. Letbe a connected open set inRⁿ,1≤p <∞andm a positive integer. Suppose thatf ∈L¹_loc()has weak derivativesD^αf with

|α| ≤m−1inand (25)

|R^m−1f (x, y)|^p

|x−y|^mp+n dx dy <∞. Thenf is a polynomial of degree at mostm−1a.e. on.

Proof. By choosing

ρε(r)=

εr^−n+ε, r <1,

0, r ≥1,

we have (26) lim inf

ε→0 ε

|R^m−1f (x, y)|^p

|x−y|^mp+n−ε dx dy

≤lim inf

ε→0 ε

|R^m−1f (x, y)|^p

|x−y|^mp+n dx dy =0.

The claim follows from Theorem 3.1.

Again there is an interpretation of the previous result in terms of Whit- ney jets. Indeed, it is possible to state the corollary for Whitney jets without referring to lower order derivatives, see Remark 4.2.

4. Whitney jets

In this section we show that a jet of functions, whose formal Taylor remainder satisfies (22), or its counterpart on a subdomain, can be identified with the jet of weak derivatives of a Sobolev function.

First we recall terminology related to the Whitney jet theory. Assume that is an open set inRⁿand letmbe a positive integer. Anm-jetF ∈J^m()is a collection

{f_α :|α| ≤m}

of functions. The m-jets define the formal Taylor polynomials in x ∈ ^Rn (centered aty∈)

T_y^kF (x)=

|α|≤k

f_α(y)(x−y)^α α! , withk ≤m, and

T_y,j^k−|j|F (x)=

|j+α|≤k

f_j+α(y)(x−y)^α α! , with|j| ≤k≤m. The formal Taylor remainders are defined to be

R^kF (x, y)=f0(x)−T_y^kF (x) and

R_j^k−|j|F (x, y)=f_j(x)−T_y,j^k−|j|F (x), wherex, y∈Kand|j| ≤k ≤m.

LetQbe a fixed cube inRⁿ, an(m−1)-jetF ∈J^m−1(Q), F = {fj :|j| ≤m−1},

wheref_j ∈L^p(Q), is said to be an(m−1)-jet inQwith variable Lipchitz coefficients, denoted byF ∈VLC(m, p, Q), if the pointwise inequality (27) |R^m−1F (x, y)| ≤ |x−y|^m(a_Q(x)+a_Q(y)),

wherex, y∈Q, holds for some functiona_Q=a_Q(F )∈L^p(Q).

The(m−1)-jet spaces VLC(m, p, Q)have been studied in [1] and it has been shown that VLC(m, p, Q)regarded as a Banach space and equipped with the norm

F =max{fjL^p(Q):|j| ≤m−1} +infaQL^p(Q),

can be identified with the classical Sobolev space W^m,p(Q). The fact that inequality (27) holds for a functionf ∈W^m,p(Q)has been proved before in [3].

Let us consider an(m−1)-jetF ∈J^m−1()of locally integrable functions on an open setinRⁿ with the property that its formal Taylor remainder of orderm−1 satisfies the condition

(28) aF = lim

ε→0

1 ε^n+mp

{(x,y)∈:|x−y|<ε}|R^m−1F (x, y)|^pdx dy <∞. This is a special case of condition (12) with mollifiers (21).

Note that pointwise estimate (27) forx, y∈implies (28). Indeed,

|R^m−1F (x, y)|^pχ_{|x−y|<ε}dx dy

≤

|x−y|^mpχ_{|x−y|<ε}(a(x)+a(y))^pdx dy

≤c

|x−y|^mpχ_{|x−y|<ε}a^p(x) dx dy

and we have 1 ε^n+mp

{(x,y)∈:|x−y|<ε}|R^m−1F (x, y)|^pdx dy

≤c

a^p(x)

|x−y|^mp

ε^n+mp χ_{|x−y|<ε}dy dx

≤cω_n−1

n a^pL^p().

At the same time (28) is a sufficient condition for a jet to be identified with a Sobolev function. More precisely, the following theorem holds true.

Theorem4.1.Letbe an open set inRⁿ,1< p <∞andmbe a positive integer. Assume that an(m−1)-jet

F = {f_α :|α| ≤m−1},

wheref_α ∈L^p(), satisfies condition(28). Then for every there is a functionf ∈W^m,p()such that

fα| =D^αf|, |α| ≤m−1,

and ∇^mfL^p()≤c aF.

Remark4.2. Corollary 3.2 reads for Whitney jets as follows. Letbe a connected open set inRⁿ. Suppose that

|R^m−1F (x, y)|^p

|x−y|^mp+n dx dy <∞. Thenf is a polynomial of degree at mostm−1 a.e. on.

To prove the theorem we use the sketch of the proof of Theorem 9.1 from [1]. First we show that the next statement is true.

Lemma4.3.Letbe an open set inRⁿ,1< p <∞andmbe a positive integer. Suppose that(m−1)-jet F satisfies condition (28). Then for every we have

(29) lim

ε→0

1 ε^(m−|j|)p

B(x,ε)

|R_j^m−1−|j|F (x, y)|^pdy dx≤c a_F,

whenever|j| ≤m−1.

Proof. Letand 0< ε <dist(, ∂). Fixx, y∈,|x−y|< ε.

Using Taylor algebra arguments, we have (30)

R_j^m−1−|j|F (x, y)=D^j_z[R^m−1F (z, x)−R^m−1F (z, y)]z=x

=D^j_zP (z;x, y)_z=x,

whereP (z;x, y)is a polynomial inzof order at mostm−1.

Sinceε < dist(, ∂)the setS = B(x, ε)∩B(y, ε) ⊂ .It is easy to see that

c|S| ≥ |B(x, ε)| = |B(y, ε)|

for some constantcwhich is independent ofε. By Markov’s inequality [10]

applied to the subsetSof the ballB(x, ε), we obtain

|D_z^jP (z;x, y)|z=x ≤ c(n) ε^|j|

S

|P (x;x, y)|^pdx 1/p

. Thus, from (30), we have

(31)

|R_j^m−1−|j|F (x, y)| ≤ c(n) ε^|j|

S

|R^m−1F (x, x)|^pdx 1/p

+

S

|R^m−1F (x, y)|^pdx 1/p

= c(n) ε^|i|

I (x)^1/p+I (y)^1/p .

It is easy to see that

B(x,ε)

I (x) dy dx≤c

B(x,ε)

B(x,ε)

|R^m−1F (x, x)|^pdxdy dx

≤c

{(x,x)∈:|x−x|<ε}|R^m−1F (x, x)|^pdxdx.

On the other hand, we obtain a similar estimate for

B(x,ε)

I (y) dy dx,

which together with (31) proves the claim.

Remark4.4. Since (29) implies 1

ε^n+(m−|j|)p

{(x,y)∈:|x−y|<ε}|R_j^m−1−|j|F (x, y)|^pdx dy ≤c aF, asε → 0, |j| ≤ m−1, using the terminology of Jonsson and Wallin (see e.g. [13]) we can formulate Lemma 4.3 in the following way: If(m−1)-jetF satisfies condition (28) then for every we haveF ∈Lip(m, p,∞, ).

Proof of Theorem4.1. Letbe an open set such that. Decom- poseinto dyadic cubes. More precisely, letMkdenote a net with mesh 2^−k inRⁿi.e.Mkis a division ofRⁿinto equally large closed cubes with side lenghts 2^−k, obtained by slicingRⁿwith hyperplanes orthogonal to the coordinate axis.

Set

_k=

x∈ : dist(x, ^c) >√

n2^−k+1 and

_l =

x∈:√

n2^−l+1<dist(x, ^c)≤√

n2^−l+2 , forl =k+1, . . .. Define

Fk⁰=^∞

l=k

{Q∈Ml :Q∩_l = ∅}

and denote byFk the collection of maximal cubes ofFk⁰ (see e.g. [23] for details on the Whitney decomposition).

Then, for eachkthe collection of cubesFk = {Q^k_i}i∈Ik satisfies the condi- tions:

(i) =

i∈IkQ^k_i;