Electronic Journal of Differential Equations, Vol. 2002(2002), No. 66, pp. 1–17.
ISSN: 1072-6691. URL: ¡http://ejde.math.swt.edu¿ or ¡http://ejde.math.unt.edu¿
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An embedding theorem for Campanato spaces
∗
Azzeddine El Baraka
Abstract
The purpose of this paper is to give a Sobolev type embedding theon
rem for the spaces Lλ,s
p,q (R ). The homogeneous versions of these spaces
contain well known spaces such as the Bounded Mean Oscillation spaces
(BMO) and the Campanato spaces L2,λ . Our result extends some injections obtained by Campanato [3, 4], Strichartz [11], and Stein and
Zygmund [10].
1
Introduction and statement of results
The main goal of this work is to give a Sobolev type embedding theorem for the
n
n
λ,s
appropriate scaled functions in Lλ,s
p,q (R ) whose homogeneous version L̇p,q (R )
contains some well known spaces as special cases: John and Nirenberg space
BMO (Bounded Mean Oscillation) and more generally Campanato spaces L2,λ
modulo polynomials [5].
s
(Rn ) spaces coIt is well known that the homogeneous Triebel-Lizorkin Ḟp,q
incide with BMO modulo polynomials for some values of p, q and s. Namely,
0
s
BM O = Ḟ∞,2
[12, chap. 5] and thus I s (BM O) = Ḟ∞,2
, where I s = F −1 (|·|−s F)
is the Riesz potential operator. Strichartz [11] discussed the connexion between
I s (BM O) and the homogeneous Besov space Λ̇s∞,q and proved the following
injections (Theorem 3.4.)
.s
Λ̇s∞,2 ⊆ I s (BM O) ⊆ Λ∞,∞
Let us mention that others classical embeddings have been obtained respectively
by Stein and Zygmund [10] and Campanato [3, 4]; they proved that
Hpn/p ,→ BM O
λ−n
and Lp,λ ∼
=C p
if n < λ < n + p
n/p
here Hp is the Bessel potential space and Lp,λ is the Campanato space (cf.
Definition 1.5). In this paper we extend these injections to a more general frame
n
(Theorem 1.8), by giving some embedding results between the spaces Lλ,s
p,q (R )
(which include I s (BM O) and I s (L2,λ ) as special cases), Triebel-Lizorkin spaces
∗ Mathematics Subject Classifications: 46E35.
Key words: Sobolev embeddings, BMO, Campanato spaces.
c
2002
Southwest Texas State University.
Submitted April 15, 2002. Published July 11, 2002.
1
2
An embedding theorem for Campanato spaces
EJDE–2002/66
n/p
s
s
Fp,q
(Rn ) (containing Hp as particular case) and Besov-Peetre spaces Bp,q
(Rn ),
n
and on the other side between the same spaces Lλ,s
p,q (R ) and Hölder-Zygmund
s
n
ones C (R ). Such embeddings shed some light on duals of the closure of
Schwartz space S(Rn ) in BMO and in Campanato spaces L2,λ (Corollary 3.1).
The present article is organised as follows: first, we give the definition of the
n
λ,s
n
Lλ,s
p,q (R ) spaces and their homogeneous counterparts L̇p,q (R ) which invoke a
Littlewood-Paley partition of unity and some expressions with balls (or cubes)
of Rn , and we state the main result (Theorem 1.8). Next, we proceed to a
discussion of some properties of these spaces: we give the connexion between
n
the nonhomogeneous spaces Lλ,s
p,q (R ) and the homogeneous ones, and we specify
the differential dimension of the dotted spaces which depends on p, q, s, and λ.
Finally, we prove Theorem 1.8 and Corollary 3.1.
To define these spaces, we will need a Littlewood-Paley partition of unity.
Denote x ∈ Rn and ξ its dual variable. Consider the cutoff function ϕ ∈
C0∞ (Rn ), ϕ ≥ 0 and ϕ equal to 1 for |ξ| ≤ 1, 0 for |ξ| ≥ 2. Let θ(ξ) = ϕ(ξ)−ϕ(2ξ)
which satisfies supp θ ⊂ { 21 ≤ |ξ| ≤ 2}. For j ∈ Z we set
˙ j u = θ(2−j Dx )u
∆
and
˙ j u if j ≥ 1
∆0 u = ϕ(Dx )u, ∆j u = ∆
P
Remark 1.1 We have ϕ(ξ) + k≥1 θ(2−k ξ) = 1 for all ξ ∈ Rn , thus the nonhomogeneous partition of u ∈ S 0 (Rn ) is given by the formula
X
u=
∆k u
k≥0
Set f (ξ) = k∈Z θ(2−k ξ); for each fixed ξ 6= 0, f (ξ) contains at most two nonvanishing terms, we deduce f (ξ) = 1 for any |ξ| ≥ 2. Note that f (2−j ξ) = f (ξ)
for all j ∈ Z, and choose j more and more large, to obtain f (ξ) = 1 for any
ξ 6= 0. Then, for u ∈ S 0 (Rn ), with 0 ∈
/ supp Fu, the homogeneous partition of
u is given by the formula
X
˙ ku
u=
∆
P
k∈Z
n
Now, let us define the nonhomogeneous space Lλ,s
p,q (R ).
Definition 1.2 Let s ∈ R, λ ≥ 0, 1 ≤ p < +∞ and 1 ≤ q < +∞. The space
n
0
n
Lλ,s
p,q (R ) denotes the set of all tempered distributions u ∈ S (R ) such that
kukLλ,s
n =
p,q (R )
n
sup
B
o1/q
X
1
q
jqs
2
k∆
uk
< +∞
p (B)
j
L
|B|λ/n
+
(1.1)
j≥J
where J + = max (J, 0), |B| is the measure of B and the supremum is taken
over all J ∈ Z and all balls B of Rn with radius 2−J . When p = q, the space
n
p,λ,s
Lλ,s
(Rn ).
p,p (R ) will be denoted L
EJDE–2002/66
Azzeddine El Baraka
3
n
Note that the space Lλ,s
p,q (R ) equipped with the norm (1.1) is a Banach
space.
n
To give the homogeneous counterpart of the space Lλ,s
p,q (R ), we recall the
notations of [12, chap. 5]. Let
Z(Rn ) = ϕ ∈ S(Rn ); (Dα Fϕ)(0) = 0 for every multi-index α .
The space Z(Rn ) is considered as a subspace of S(Rn ) with the induced topology,
and Z 0 (Rn ) its topological dual. We may identify Z 0 (Rn ) with S 0 (Rn )/P(Rn ),
where P(Rn ) is the set of all polynomials of Rn with complex coefficients i.e.
Z 0 (Rn ) is interpreted as S 0 (Rn ) modulo polynomials.
Definition 1.3 Let s ∈ R, λ ≥ 0, 1 ≤ p < +∞ and 1 ≤ q < +∞. The dotted
n
0
n
space L̇λ,s
p,q (R ) denotes the set of all u ∈ Z (R ) such that
n
=
sup
kukL̇λ,s
n
p,q (R )
B
o1/q
1 X jqs ˙
q
2
k
∆
uk
< +∞
j
Lp (B)
|B|λ/n j≥J
(1.2)
where the supremum is taken over all J ∈ Z and all balls B of Rn with radius
2−J .
n
p,λ,s
If p = q, the space L̇λ,s
(Rn ). If P is a polynomial
p,p (R ) will be denoted L̇
of P(Rn ) and u ∈ S 0 (Rn ), it follows immediatly that
ku + P kL̇λ,s
n = kukL̇λ,s (Rn )
p,q (R )
p,q
n
This shows that the norm (1.2) is well defined. Further, the space L̇λ,s
p,q (R )
equipped with this norm is a Banach space.
Remark 1.4 The supremum given in expressions (1.1) and (1.2) may be taken
over all J ∈ Z and all cubes B of Rn of length side 2−J .
Now we define Campanato spaces modulo polynomials L̇p,λ (Rn ).
Definition 1.5 Let λ ≥ 0 and 1 ≤ p < +∞.
(i) The space Lp,λ (Rn ) denotes the set of (classes of) functions u ∈ Lploc (Rn )
such that
kukLp,λ (Rn )
= sup
B
1
|B|λ/n
Z
1/p
|u − mB u| dx
< +∞
p
B
1
where mB u = |B|
u(y)dy is the mean value of u and the supremum is
B
taken over all balls B of Rn .
R
Note that Lp,λ (Rn ) is a Banach space modulo constants equal to {0} if λ > n+p.
4
An embedding theorem for Campanato spaces
EJDE–2002/66
(ii) The space L̇p,λ (Rn ) denotes the set of functions u ∈ Z 0 (Rn ) such that
kukL̇p,λ (Rn ) < +∞, where
kukL̇p,λ (Rn )

 inf P ∈P ku + P kLp,λ (Rn )
=
+∞

if u ∈ L1loc (Rn )
if u ∈ Z 0 (Rn ) but
not locally integrable
the infimum is taken over the set P of all polynomials of Rn with complex
coefficients.
Remark 1.6 (i) Modulo polynomials, the space L̇p,λ (Rn ) is the Campanato
space Lp,λ (Rn ). For 0 ≤ λ < n + p, L̇p,λ (Rn ) can be defined as the set of
all equivalence classes modulo P of elements of Lp,λ (Rn ), equipped with the
following norm kU kL̇p,λ (Rn ) = kukLp,λ (Rn ) where u is the unique (modulo a
constant) element of Lp,λ (Rn ) belonging to the class U .
(ii) Using a result of John and Nirenberg [9] it is classical that if λ = n,
then for every 1 ≤ p < +∞, Lp,n (Rn ) coincides with the space BM O (Rn ) of
all functions u ∈ L1loc (Rn ) such that there exists a constant C > 0 satisfying
the following inequality
Z
1
|u − mB u|dx ≤ C
|B| B
. 2,λ
for all balls B of Rn . We show in [5] that in general L̇2,λ,0 (Rn ) ≡ L (Rn )
provided 0 ≤ λ < n + 2; therefore the characterization of Littlewood-Paley type
of Campanato spaces L̇2,λ (Rn ) is given. This result is not true in general for
p 6= 2, nevertheless if p ≥ 2 and 0 ≤ λ < n + p we show that L̇p,λ (Rn ) is
continuously embedded in L̇p,λ,0 (Rn ).
Now, recall the definition of some function spaces.
Definition 1.7 Let s ∈ R, 1 ≤ p < +∞ and 1 ≤ q < +∞.
s
(i) We denote Fp,q
(Rn ) the set of all tempered distributions u ∈ S 0 (Rn ) such
that
Z X
p/q 1/p
s (Rn ) =
kukFp,q
2jqs |∆j u|q
dx
< +∞
Rn
j≥0
s
(ii) The space B∞,q
(Rn ) denotes the set of all tempered distributions u ∈
0
n
S (R ) such that
nX
o1/q
s
kukB∞,q
2kqs k∆k ukqL∞ (Rn )
< +∞
(Rn ) =
k≥0
(iii) We denote C s (Rn ) the set of all tempered distributions u ∈ S 0 (Rn ) such
that
kukC s (Rn ) = sup 2js k∆j ukL∞ (Rn ) < +∞
j≥0
EJDE–2002/66
Azzeddine El Baraka
5
.
(iv) If we replace in (i), (ii) and (iii) S 0 (Rn ) by Z 0 (Rn ), ∆j by ∆j and j ≥ 0
by j ∈ Z, we obtain respectively the definition of the homogeneous spaces
s
s
Ḟp,q
(Rn ), Ḃ∞,q
(Rn ) and Ċ s (Rn ).
Here is the main result of this paper.
Theorem 1.8 Let s ∈ R, 1 ≤ p < +∞, 1 ≤ q < +∞ and λ ≥ 0. We have the
following continuous embeddings
λ
λ,s+ n
p−q
Lp,q
s+ n
(Rn ) ,→ C s (Rn )
λ
λ,s+ n
p−q
Fp,q p (Rn ) ,→ Lp,q
s+ n
Fp,q p (Rn ) ,→ Lp,λ,s−
s− n + λ
q
B∞,qp
λ−n
p
(Rn ) provided q ≥ p
(Rn ) provided p ≥ q.
n
(Rn ) ,→ Lλ,s
p,q (R ) provided λ ≥ n
q
p
and finally if λ ≥ n then
X
λ−n
2jq(s+ q ) |∆j u|q ∈ L∞ (Rn ) implies u ∈ Lq,λ,s (Rn )
j≥0
We have also the same continuous embeddings if we replace B, C, F and L respectively by the dotted spaces Ḃ, Ċ, Ḟ and L̇.
Remark 1.9 (i) In the case s = 0, S. Campanato [3, 4] showed that if n < λ <
λ−n
n + p we have Lp,λ ∼
= C p and Lp,n+p = Lip (cf. [8] too).
(ii) If λ = n and p = q = 2, we obtain the theorem 3.4. of R. S. Strichartz
· s
·s
[11], namely B ∞,2 (Rn ) ,→ I s (BM O) ,→ C (Rn ), where I s is the Riesz potential operator and I s (BM O) = L̇2,n,s (Rn ), cf. [5], BMO is defined modulo
polynomials. We note that Theorem 1.8 generalise Theorem 2.1 of [6].
(iii) Choosing s = 0, p = q = 2 and λ = n in the third embedding, we find
a result due to E. M. Stein and A. Zygmund [10]
n
n
2
Ḣ 2 = Ḟ2,2
,→ L̇2,n,0 (Rn ) = BM O modulo polynomials
2
Properties of the spaces
We start with some helpful lemmas needed in the further considerations.
Lemma 2.1 Let 1 ≤ p < +∞, and A a real < 0. If (ajν )j,ν is a sequence of
positive real numbers satisfying (ajν )j ∈ lp for all ν ≥ 1, then there exists a
constant C > 0 such that
p
X X
X p
2νA ajν ≤ C sup
ajν .
j≥1 ν≥1
ν≥1
j≥1
6
An embedding theorem for Campanato spaces
Proof Let
1
p
+
1
q
EJDE–2002/66
= 1. Using Hölder’s inequality we get
X X
2νA ajν
p
p
ν
ν
2 2 A (2 2 A ajν )
≤
X X
≤
XX
j≥1 ν≥1
j≥1 ν≥1
j≥1
≤ C
ν
2 2 Aq
ν≥1
XX
2
p/q X
ν
2 2 Ap apjν
ν≥1
ν
2 Ap
apjν
X
apjν ≤ C sup
j≥1 ν≥1
≤ C
X
ν
2 2 Ap
ν≥1
ν≥1
j≥1
X
apjν
j≥1
˙ l is uniformly bounded on Lp (Rn ). On the other hand, on
It is known that ∆
p
n
L (B), B ⊂ R , we have the following result. We denote αB, α > 0, the ball
with the same center x0 as B and of radius αr, r is the radius of B.
Lemma 2.2 For each integer M > 0, there exists a constant CM > 0, such
that for any J ∈ Z, x0 ∈ Rn , for any ball B centered at x0 and of radius 2−J ,
for any l ∈ Z and u ∈ Lp (Rn ) , 1 ≤ p ≤ +∞,
n
o
X
kAl ukLp (B) ≤ CM kukLp (2B) +
2−(l+ν)M kukLp (Fν )
ν≥−J+1
holds with Fν = x ∈ Rn ; 2ν ≤ |x − x0 | ≤ 2ν+1 and Al denotes any product of
∆l , Sl and the dotted operators.
˙ l . Remark that
Proof We give the proof for Al = ∆
X
u = χ2B u +
χFν u
ν≥−J+1
where χΩ is the characteristic function of the set Ω. Thus
X
˙ l ukLp (B) ≤ k∆
˙ l χ2B ukLp (B) +
˙ l χF ukLp (B)
k∆
k∆
ν
ν≥−J+1
≤ CkukLp (2B) +
X
˙ l χF ukLp (B) ,
k∆
ν
ν≥−J+1
where C is a constant independent of l and B. Now
Z
˙ l χF u(x) = 2nl
∆
u(y)F −1 θ(2l (x − y))dy
ν
Fν
For x ∈ B and y ∈ Fν , we have |x − y| ∼ |x0 − y| ∼ 2ν . Since θ ∈ S (Rn ), for
every integer N , there exists CN > 0 such that
|F −1 θ 2l (x − y) | ≤ CN 2−(l+ν)N
EJDE–2002/66
Azzeddine El Baraka
7
We deduce
˙ l χF u(x)|
|∆
ν
≤ CN 2nl 2−(l+ν)N
Z
|u(y)|dy
Fν
1
≤ CN 2nl 2−(l+ν)N kukLp (Fν ) |Fν |1− p
n
≤ CN 2−(l+ν)(N −n) 2−ν p kukLp (Fν )
It follows
n
˙ l χF u(x)kLp (B) ≤ CN 2−(l+ν)(N −n) 2− p (ν+J) kukLp (F )
k∆
ν
ν
Therefore
˙ l ukLp (B) ≤ CkukLp (2B) + CN
k∆
X
2−(l+ν)(N −n) kukLp (Fν )
ν≥−J+1
Put M = N − n > 0, and the proof is complete.
♦
Remark 2.3 Note that
|F −1 θ 2l (x − y) | ≤ kF −1 θkL∞ (Rn )
Thus we may improve the statement of the last lemma by replacing the term
2−(l+ν)M by inf (1, 2−(l+ν)M ).
Here is a classical lemma [1].
Lemma 2.4 Let λ > 0, 1 ≤ p ≤ q ≤ +∞. For any u ∈ Lp (Rn ) with supp Fu ⊂
{|ξ| ≤ λ} and for any multi-index α, there exists Cα > 0 such that
1
1
kDxα ukLq (Rn ) ≤ Cα λ|α|+n( p − q ) kukLp (Rn )
Lemma 2.5 Let R be a real > 1; there exists C > 0 such that for any λ > 0
and 1 ≤ p ≤ +∞, for any u ∈ Lp (Rn ) with supp Fu ⊂ {λ ≤ |ξ| ≤ Rλ} and for
any k ∈ N we have
kukLp (Rn ) ≤ C k λ−k sup kDxα ukLp (Rn )
|α|=k
Proof The above lemma gives that Dxα u ∈ Lp (Rn ) for all multi-index α. Let
ψ ∈ C0∞ (Rn ), ψ(ξ) = 1 on a neighbourhood of the annulus {1 ≤ |ξ| ≤ R} and
ξ
ψ(ξ) = 0 on a neighbourhood of 0. Setting ψλ (ξ) = ψ( λξ ) and ψλ,j = |ξ|j2 ψλ we
Pn
obtain j=1 ξj ψλ,j (ξ) = 1 on a neighbourhood of supp Fu. We deduce
u=
n
X
j=1
Fλ,j ∗ Dxj u
where Fλ,j = F −1 ψλ,j ∈ S(Rn )
(2.2)
8
An embedding theorem for Campanato spaces
EJDE–2002/66
We have Fλ,j (x) = λn−1 F1,j (λx), kFλ,j kL1 = λ−1 kF1,j kL1 and using (2.2) we
deduce lemma 2.5 for k = 1. For the general case we do an induction on k > 1.
♦
The following lemma yields the homogeneous decomposition modulo polynomials of a distribution u ∈ S 0 (Rn ) without taking in account the condition
0∈
/ supp Fu that we have met in remark 1.1.
Lemma 2.6 For any u ∈ S 0 (Rn ) we have the decomposition
X
˙ j u in Z 0 (Rn ).
u=
∆
j∈Z
P+∞ ˙
0
n
Proof The series j=k ∆
j u converges in S (R ) for all k ∈ Z. It suffices to
n
show that for any ψ ∈ Z(R ),
lim
k→−∞
+∞
DX
˙ j u, ψ̌
∆
j=k
E
S 0 ×S
D
E
= u, ψ̌
S 0 ×S
where ψ̌(x) = ψ(−x) for x ∈ Rn . Now
+∞
X
˙ ju − u =
∆
j=k
0
X
˙ j u − ∆0 u = ϕ(2−k Dx )u
∆
j=k
which is a smooth function. Thus
(2π)n
+∞
DX
˙ j u − u, ψ̌
∆
j=k
E
S 0 ×S
= Fu, ϕ(2−k .)Fψ S 0 ×S
Since Fu ∈ S 0 (Rn ), there exists a constant C > 0 and an integer N such that
+∞
DX
E
˙ j u − u, ψ̌
∆
(2π)n j=k
≤ C
S 0 ×S
sup |ξ α Dξβ (ϕ(2−k ξ)Fψ(ξ))|
sup
|α|+|β|≤N ξ∈Rn
≤ C0
sup
sup
|α|+|β|≤N
|ξ|≤2k+1
X
2k(|α|−|β−γ|) |(Dξβ−γ ϕ)(2−k ξ)||Dξγ Fψ(ξ)|
γ≤β
By the assumption ψ ∈ Z(Rn ) and Taylor’s formula, for all nonnegative large
integer M
+∞
DX
E
˙ j u − u, ψ̌
(2π)n ∆
j=k
S 0 ×S
≤ C2k(M −N ) −→ 0
k→−∞
The proof is complete.
n
λ,s
n
Now we state the connexion between Lλ,s
p,q (R ) and L̇p,q (R ).
♦
EJDE–2002/66
Azzeddine El Baraka
9
Lemma 2.7 Let 1 ≤ p, q < +∞, λ ≥ 0 and s ∈ R.
n
p
n
(i) If the class of u modulo P belongs to L̇λ,s
p,q (R ) and if ∆0 u ∈ L (R ), then
λ,s
n
u ∈ Lp,q (R ).
n
λ,s
n
(ii) Lp (Rn ) ∩ L̇λ,s
p,q (R ) ⊂ Lp,q (R ) with the same meaning as (i).
n
λ,s
n
(iii) Lλ,s
p,q (R ) ⊂ L̇p,q (R ) provided s > 0.
Remark 2.8 It follows that if s > 0 then
n
p
n
λ,s
n
Lp (Rn ) ∩ L̇λ,s
p,q (R ) = L (R ) ∩ Lp,q (R ).
n
0
n
Proof of Lemma 2.7 (i) If u ∈ L̇λ,s
p,q (R ), then u ∈ S (R ) and there exists
n
a constant M > 0 such that for any ball B of R with radius 2−J with J ∈ Z,
1 X lsq ·
2 k∆l ukqLp (B) ≤ M < +∞ .
|B|λ/n l≥J
(2.3)
Let J ∈ Z and B be a ball of Rn with radius 2−J . If J ≥ 1, then inequality
(2.3) gives
1 X lsq
2 k∆l ukqLp (B) ≤ M
λ
n
|B| l≥J +
If J ≤ 0, we have
X
1
1
2lsq k∆l ukqLp (B) =
k∆0 ukqLp (B) +
λ/n
|B|
|B|λ/n
+
l≥J
1 X lsq
2 k∆l ukqLp (B) ≤ C2Jλ k∆0 ukqLp (B) + M
|B|λ/n l≥1
≤ Ck∆0 ukqLp (Rn ) + M
Since ∆0 u ∈ Lp (Rn ), we obtain
1
|B|
λ
n
X
2lsq k∆l ukqLp (B) ≤ C 0 M
l≥J +
n
Hence u ∈ Lλ,s
p,q (R ).
(ii) follows from (i).
n
(iii) Let u ∈ Lλ,s
p,q (R ). There exists a constant M > 0 such that for any ball B
of Rn with radius 2−J and J ∈ Z,
X
1
2lsq k∆l ukqLp (B) ≤ M < +∞ .
λ/n
|B|
+
l≥J
10
An embedding theorem for Campanato spaces
EJDE–2002/66
We have to show (2.3). Let J ∈ Z and B a ball of Rn of radius 2−J . If J ≥ 1,
then J + = J and (2.3) is valid. If J ≤ 0 (i.e. J + = 0). We have
0
1 X lsq ·
1 X lsq ·
q
2
k
∆
uk
=
2 k∆l ukqLp (B) +
l
Lp (B)
|B|λ/n l≥J
|B|λ/n l=J
0
1 X lsq ·
1 X lsq ·
q
2
k
∆
uk
≤
2 k∆l ukqLp (B) + M
l
Lp (B)
|B|λ/n l≥1
|B|λ/n l=J
(2.4)
Using the nonhomogeneous decomposition of u we have for l ≤ 0
X
˙ l u = ∆0 ∆
˙ l u + ∆1 ∆
˙ lu
˙ lu =
∆
∆k ∆
k≥0
and then
˙ l ukLp (B) ≤ k∆0 ∆
˙ l ukLp (B) + k∆1 ∆
˙ l ukLp (B) .
k∆
(2.5)
Lemma 2.2 gives for M large,
q
−(l−J)M q
˙ l ∆0 ukq p
k∆
[
L (B) ≤ CM {k∆0 ukLp (2B) + 2
X
2−νM k∆0 ukLp (Bν−J ) ]q }
ν≥1
ν+1
where Bν−J = 2
B. Thus
0
1 X lsq ·
2 k∆l ∆0 ukqLp (B)
|B|λ/n l=J
≤
0
0
q
X
C
1 X lsq X −νM
q
lsq
p
k∆
uk
2
+
2
2
k∆
uk
0
0
L (Bν−J )
Lp (2B)
|B|λ/n
|B|λ/n l=J
ν≥1
l=J
Since s > 0,
P0
lsq
l=J 2
≤C
0
1 X lsq ·
C
2 k∆l ∆0 ukqLp (B) ≤
k∆0 ukqLp (2B)
|B|λ/n l=J
|B|λ/n
X
q
λ
1
p
+C
2ν(−M + q )
k∆
uk
0
L (Bν−J )
λ
|Bν−J | nq
ν≥1
≤ CM + C sup
ν
1
|Bν−J |λ/n
(2.6)
k∆0 ukqLp (Bν−J ) ≤ CM
In the same way we have
0
1 X lsq ·
2 k∆l ∆1 ukqLp (B) ≤ CM
|B|λ/n l=J
From (2.4), (2.5), (2.6) and (2.7) it follows
1 X lsq ·
2 k∆l ukqLp (B) ≤ CM
|B|λ/n l≥J
(2.7)
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Azzeddine El Baraka
11
n
and then u ∈ L̇λ,s
p,q (R ).
n
Now we give the differential dimension of L̇λ,s
p,q (R ).
♦
n
Proposition 2.9 For any u ∈ L̇λ,s
p,q (R ) and any t > 0, we have
d
ku(t−1 ·)kL̇λ,s
n ≈ t kukL̇λ,s (Rn )
p,q (R )
p,q
where d =
n
p
λ
q
−
− s.
Proof Let t = 2N , N ∈ Z and set v(x) = u( 2xN ). From
·
·
∆l v(x) = (∆l+N u)(
x
)
2N
we get
·
n
·
k∆l vkLp (B) = 2N p k∆l+N ukLp (2−N B)
where B is a ball of Rn with radius 2−J , J ∈ Z. Then we have
kvkq λ,s
L̇p,q
(Rn )
≈ sup 2Jλ
B
X
·
n
2lsq 2N p q k∆l+N ukqLp (2−N B)
l≥J
n
B
λ
(n
p − q −s)qN
sup 2Jλ
≈2
B
·
X
≈ 2N p q 2−N λ 2−N sq sup 2(J+N )λ
2(l+N )sq k∆l+N ukqLp (2−N B)
l+N ≥J+N
X
·
2lsq k∆l ukqLp (B)
l≥J
Lemma 2.10 Let 1 ≤ p ≤ p0 < +∞, 1 ≤ q 0 ≤ q < +∞ and s ∈ R. Further, let
λ and µ ≥ 0 such that pn0 − qµ0 ≥ np − λq . Then we have the continuous embedding
µ,s+ pn0 − qµ0
Lp0 ,q0
λ
λ,s+ n
p−q
(Rn ) ,→ Lp,q
(Rn ) .
We also have the same result for the dotted spaces L̇.
µ,s− µ
λ,s− λ
In particular, if p = p0 and q = q 0 then Lp,q q (Rn ) ,→α Lp,q q (Rn ) holds for
all µ ≤ λ. Furthermore if p = p0 = q = q 0 we get Lp,λ,s+ p (Rn ) ,→ Lp,λ+α,s (Rn )
for all α ≥ 0 and λ ≥ 0.
µ,s+
n
0
−
µ
0
Proof Let u ∈ Lp0 ,q0 p q (Rn ). Let B be a ball of Rn with radius 2−J , J ∈ Z.
Since p ≤ p0 and qq0 ≥ 1 we obtain
1
|B|
λ
n
≤
X
n
λ
2l(s+ p − q )q k∆l ukqLp (B)
l≥J +
1
λ
X
|B| n l≥J +
q
1
n
λ
−1
2l(s+ p − q )q k∆l ukLp0 (B) |B| p p0
12
An embedding theorem for Campanato spaces
≤
≤
1
λ
X |B| n l≥J +
1 X
λ
|B| n
n
λ
q/q0
|B|
q/q0
q
0
0
2l(s+ p − q )q k∆l ukqLp0 (B)
2
0
λ
l(s+ n
p − q )q
0
k∆l ukqLp0 (B)
EJDE–2002/66
q
|B|
1
1
p − p0
1
1
p − p0
l≥J +
Now |B| ∼ 2−nJ , l ≥ J + ≥ J and the assumption pn0 − qµ0 ≥ np − λq yield
n 1 X
o1/q
n
λ
2l(s+ p − q )q k∆l ukqLp (B)
λ
|B| n l≥J +
1 X
1/q0
µ
0
1
0
n
λ
−1
− λ
≤
2l(s+ p − q )q k∆l ukqLp0 (B)
|B| p p0 |B| nq0 nq
µ
|B| n l≥J +
1 X n µ
1/q0
0
λ
0
0
n
λ
J ( − )−( n
p−q) q
≤ C
2l(s+ p − q )q k∆l ukqLp0 (B)
2 p0 q0
µ
|B| n l≥J +
1 X
1/q0
0
l(s+ pn0 − qµ0 )q 0
≤ C
2
k∆l ukqLp0 (B)
µ
|B| n l≥J +
♦
The proof of lemma 2.10 is complete.
Lemma 2.11 The
n
Lλ,s
p,q (R ) and from
derivation Dxα is a bounded
λ,s+|α|
n
L̇p,q
(Rn ) to L̇λ,s
p,q (R ).
operator from
λ,s+|α|
Lp,q
n
(R ) to
Proof Let |α| = 1. We have
kDx ukq λ,s
Lp,q
(Rn )
= kDx
X
∆l ukq λ,s
l≥0
= sup
B
Lp,q (Rn )
j+1
X
X
1
jsq
Dx ∆j ∆l ukqLp (B)
2
k
|B|λ/n
+
l=j−1
j≥J
X
X
1
≤ C sup
2jsq
kDx ∆j ∆l ukqLp (B)
λ/n
B |B|
+
l∼j
j≥J
(in the case of the dotted spaces we use lemma 2.6). Let B a ball of Rn of radius
2−J , J ∈ Z. Set fl = ∆l u. Note that
X
fl = χ2B fl +
χFν fl
ν≥−J+1
and use lemma 2.4 to get
kDx ∆j fl kLp (B) ≤kDx ∆j χ2B fl kLp (B) +
X
kDx ∆j χFν fl kLp (B)
ν≥−J+1
≤C2j k∆j χ2B fl kLp (Rn ) +
X
kDx ∆j χFν fl kLp (B)
ν≥−J+1
≤C2j kfl kLp (2B) +
X
ν≥−J+1
kDx ∆j χFν fl kLp (B)
(2.8)
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Azzeddine El Baraka
13
On the other hand,
Dx ∆j χFν fl (x) = 2
(n+1)j
Z
Fν
fl (y) ψ 2j (x − y) dy
where ψ = Dx F −1 θ. Therefore, if x ∈ B and y ∈ Fν , ν ≥ −J + 1, then
|x − y| ∼ |x0 − y| ∼ 2ν . Since ψ ∈ S(Rn ), for each integer N there exists a
constant CN such that
Z
j(n+1−N ) −νN
|Dx ∆j χFν fl (x) | ≤ CN 2
2
|fl (y)| dy
Fν
j(n+1−N ) −νN
≤ CN 2
2
1
kfl kLp (Fν ) |Fν |1− p
We deduce
kDx ∆j χFν fl kLp (B) ≤ CN 2j(n+1−N ) |B|1/p 2ν (−N + q − p +n)
λ
n
1
λ
|Fν | nq
kfl kLp (Fν )
Thus
X
kDx ∆j χFν fl kLp (B)
ν≥−J+1
n
2ν (−N + q − p +n)
λ
X
≤ CN 2j(n+1−N ) 2−J p
n
λ
X
λ
|Fν | nq
ν≥−J+1
≤ C2j 2(j−J)(n−N ) 2−J q
kfl kLp (Fν )
2ν (−N + q − p +n)
λ
n
kfl kLp (Fν−J )
λ
|Fν−J | nq
ν≥1
Choose N large and use inequalities (2.8), (2.9) and lemma 2.1 to deduce
X
X
1
2jsq
kDx ∆j ∆l ukqLp (B)
λ/n
|B|
+
l∼j
j≥J
≤ C
X
1
2j(s+1)q kfj kqLp (2B)
λ/n
|B|
+
j≥J
+CN
X nX
j≥J +
≤ C
1
|3B|λ/n
0
+CN
sup
ν≥1
2ν (−N + q − p +n)
λ
n
|Fν−J |
ν≥1
X
2j(s+1)
λ
nq
kfj kLp (Fν−J )
2j(s+1)q k∆j ukqLp (2B)
j≥(J−2)+
1
|Fν−J |λ/n
X
2j(s+1)q k∆j ukqLp (Fν−J )
j≥(J−ν−1)+
Finally,
kDx ukq λ,s
Lp,q (Rn )
≤ Ckukq λ,s+1
which completes the proof of lemma 2.11.
Lp,q
(Rn )
oq
(2.9)
14
3
An embedding theorem for Campanato spaces
EJDE–2002/66
Proof of theorem 1.8
λ,s+ n − λ
For the first embedding, let u ∈ Lp,q p q (Rn ). Let j ≥ 1 and x ∈ Rn be fixed.
Let φ ∈ S (Rn ) with Fφ = 1 on 21 ≤ |ξ| ≤ 2.
From F∆j u(ξ) = F(∆j u)(ξ)Fφ(2−j ξ) we deduce
Z
∆j u (x) = 2nj
∆j u(y)φ(2j (x − y))dy
(3.1)
Rn
(in the case j = 0, we assume Fφ = 1 on |ξ| ≤ 2). We decompose Rn into
disjunctive cubes with the side-length 2−j ,
Qν = y ∈ Rn : ky − x − 2−j νk∞ ≤ 2−j−1 ,
ν ∈ Zn , and we set aν = supky−νk∞ ≤ 21 |φ(y)| which is a rapidly decreasing
sequence. Then, we have
XZ
|∆j u (x) | ≤ 2nj
|∆j u(y)φ(2j (x − y))|dy
Qν
ν
≤ 2nj
Z
X
aν
≤ 2nj
X
aν k∆j ukLp (Qν ) |Qν |1− p
|∆j u(y)|dy
Qν
ν
1
ν
nj
≤ C2
X
≤ C2nj
X
n
λ
aν 2−j(s+ p − q ) kuk
λ,s+ n − λ
Lp,q p q
ν
λ
−j(s+ n
p−q)
aν 2
λ,s+ n − λ
p
q
Lp,q
λ
kuk
ν
≤ C2−js kuk
(Rn )
λ
1
|Qν | nq |Qν |1− p
λ,s+ n − λ
Lp,q p q
1
2−jn( nq +1− p )
(Rn )
.
(Rn )
The first embedding is proved.
s+ n
Now let u ∈ Fp,q p (Rn ) and B a ball of Rn centered at x0 with radius 2−J ,
J ∈ Z. The assumption pq ≥ 1 gives
1
λ
|B| nq
nX
2
1
|B|
1
|B|
λ
nq
λ
nq
Z
B
j≥J +
=
≤
λ
jq(s+ n
p−q)
q/p o
1/q
|∆j u| dx
p
Z
n
λ
2jp(s+ p − q ) |∆j u|p dx
B
C
nZ
≤ C2J q
n
λ
X
n
λ
≤ C
Rn
X
n
λ
j≥J +
2jq(s+ p − q ) |∆j u|q
Rn j≥J +
nZ
j≥J
k 2jp(s+ p − q ) |∆j u|p
Rn
nZ
λ
1/p
pq
+
l
j≥J +
o1/p
k pq dx
l
p/q o1/p
2Jλ 2jq(s+ p − q ) |∆j u|q
p/q o1/p
≤ Ckuk
s+ n
Fp,q p (Rn )
.
EJDE–2002/66
Azzeddine El Baraka
15
Therefore the second embedding yields true.
For the third embedding, use pq ≥ 1 to obtain
1
|B|
λ
n
Z X
2jp(s−
λ−n
p )
|∆j u|p dx =
B j≥J +
Z X
1
|B|
λ
n
2jq(s−
λ−n
p )
|∆j u|q
p/q
dx
2jq(s−
λ−n
p )
|∆j u|q
p/q
dx
B j≥J +
Jλ
≤ C2
Z X
B j≥J +
≤
Z X
q
2Jλ p 2jq(s−
λ−n
p )
|∆j u|q
B j≥J +
≤ C
Z
X
n
2jq(s+ p ) |∆j u|q
Rn j≥J +
p
≤ Ckuk
p/q
p/q
dx
dx
s+ n
Fp,q p (Rn )
Let B be a ball of Rn centered at x0 with radius 2−J for J ∈ Z. Let u ∈
s− n + λ
q
B∞,qp
(Rn ). Then
1
|B|
λ
n
X
2jqs k∆j ukqLp (B)
≤ C2Jλ
j≥J +
X
q
2jqs |B| p k∆j ukqL∞ (B)
j≥J +
≤ C
X
q
2jqs 2J(λ−n p ) k∆j ukqL∞ (B)
j≥J +
≤ C
X
n
λ
2jq(s− p + q ) k∆j ukqL∞ (B)
j≥0
Therefore the fourth injection is proved. For the last assertion,
Z X
X
1
1
q
jqs
2
k∆
uk
≤
C
2jqs |∆j u|q dx
j
Lq (B)
λ/n
|B|λ/n
|B|
B
j≥J +
j≥J +
X
λ
≤ C sup
2jqs |∆j u(x)|q |B|− n +1
x∈Rn
≤ C sup
x∈Rn
j≥J +
X
2jq(s+
λ−n
q )
|∆j u|q
j≥0
♦
and the proof of theorem 1.8 is complete.
◦ λ,s
n
Now we can state a partial result on the topological dual of Lp,q (R ), the
n
closure of Schwartz space S(Rn ) in Lλ,s
p,q (R ).
Corollary 3.1 Let s ∈ R, λ ≥ 0, 1 ≤ p < +∞, 1 ≤ q < +∞, 1 < p0 ≤ +∞
and 1 < q 0 ≤ +∞ with p1 + p10 = 1, 1q + q10 = 1. We have
n
−s− λ
q+p
F1,1
◦ λ,s
−s− λ
(Rn ) ,→ (Lp,q (Rn ))0 ,→ Fp0 ,q0 q (Rn ) provided p ≤ q.
(3.2)
16
An embedding theorem for Campanato spaces
n
−s− λ
p+p
F1,1
◦ p,λ,s
(Rn ) ,→ (L
EJDE–2002/66
−s− λ
(Rn ))0 ,→ Fp0 ,q0 p (Rn ) provided q ≤ p
(3.3)
in particular
n
−λ
2
2
F1,1
(Rn ) ,→
◦ 2,λ
L
0
−λ
(Rn ) ,→ F2,q20 (Rn ) for all q 0 ≥ 2.
We have also the same injections for the dotted spaces.
Proof We prove only (3.3). Applying the first and the third embedding of
theorem 1.8 with s + λ−n
p instead of s we obtain
s+ λ
Fp,q p (Rn ) ,→ Lp,λ,s (Rn ) ,→ C s+
λ−n
p
s+ λ−n
(Rn ) ≡ B∞,∞p (Rn )
hence
s+ λ
◦ p,λ,s
Fp,q p (Rn ) ,→ L
◦ s+
λ−n
p
◦ s+
λ−n
p
(Rn ) ,→ B ∞,∞ (Rn )
s+ λ−n
where B ∞,∞ (Rn ) denotes the closure of S(Rn ) in B∞,∞p (Rn ). Now
0
◦ s+ λ−n
p
−s− λ + n
B ∞,∞ (Rn ) = F1,1 p p (Rn )
yields the result.
♦
Acknowledgement The author wants to express his gratitude to Professors
Pascal Auscher, Gerard Bourdaud and Jacques Camus who read this manuscript
and made many valuable corrections and comments.
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Azzeddine El Baraka
17
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Azzeddine El Baraka
Université Mohamed Ben Abdellah, FST Fes-Saiss
Département de Mathématiques
BP2202, Route Immouzer
Fes 30000 Morocco
e-mail: aelbaraka@yahoo.com