Hindawi Publishing Corporation
Abstract and Applied Analysis
Volume 2012, Article ID 929381, 14 pages
doi:10.1155/2012/929381
Research Article
Necessary and Sufficient Conditions for
Boundedness of Commutators of
the General Fractional Integral Operators on
Weighted Morrey Spaces
Zengyan Si1 and Fayou Zhao2
1
2
School of Mathematics and Information Science, Henan Polytechnic University, Jiaozuo 454000, China
Department of Mathematics, Shanghai University, Shanghai 200444, China
Correspondence should be addressed to Fayou Zhao, zhaofayou2008@yahoo.com.cn
Received 20 March 2012; Revised 5 July 2012; Accepted 20 July 2012
Academic Editor: Giovanni Galdi
Copyright q 2012 Z. Si and F. Zhao. This is an open access article distributed under the Creative
Commons Attribution License, which permits unrestricted use, distribution, and reproduction in
any medium, provided the original work is properly cited.
We prove that b is in Lipβ ω if and only if the commutator b, L−α/2 of the multiplication operator
by b and the general fractional integral operator L−α/2 is bounded from the weighted Morrey space
Lp,k ω to Lq,kq/p ω1−1−α/nq , ω, where 0 < β < 1, 0 < α β < n, 1 < p < n/α β, 1/q 1/p − α β/n, 0 ≤ k < p/q, ωq/p ∈ A1, and rω > 1 − k/p/q − k, and here rω denotes the
critical index of ω for the reverse Hölder condition.
1. Introduction and Main Results
Suppose that L is a linear operator on L2 Rn which generates an analytic semigroup e−tL
with a kernel pt x, y satisfying a Gaussian upper bound, that is,
pt x, y ≤ C e−c|x−y|2 /t
tn/2
1.1
for x, y ∈ Rn and all t > 0. Since we assume only upper bound on heat kernel pt x, y and
no regularity on its space variables, this property 1.1 is satisfied by a class of differential
operator, see 1 for details.
For 0 < α < n, the general fractional integral L−α/2 of the operator L is defined by
L
−α/2
1
fx Γα/2
∞
0
e−tL f
dt
x.
t−α/21
1.2
2
Abstract and Applied Analysis
Note that, if L −Δ is the Laplacian on Rn , then, L−α/2 is the classical fractional integral
Iα which plays important roles in many fields. Let b be a locally integrable function on Rn ,
the commutator of b and L−α/2 is defined by
b, L−α/2 fx bxL−α/2 fx − L−α/2 bf x.
1.3
For the special case of L −Δ, many results have been produced. Paluszyński 2
obtained that b ∈ Lipβ Rn if the commutator b, Iα is bounded from Lp Rn to Lr Rn , where
1 < p < r < ∞, 0 < β < 1 and 1/p − 1/r α β/n with p < n/α β. Shirai 3 proved that
b ∈ Lipβ Rn if and only if the commutator b, Iα is bounded from the classical Morrey spaces
Lp,λ Rn to Lq,λ Rn for 1 < p < q < ∞, 0 < α, 0 < β < 1, and 0 < α β 1/p − 1/qn − λ < n
or Lp,λ Rn to Lq,μ Rn for 1 < p < q < ∞, 0 < α, 0 < β < 1, 0 < α β 1/p − 1/q <
n, 0 < λ < n − α βp, and μ/q λ/p. Wang 4 established some weighted boundedness
of properties of commutator b, Iα on the weighted Morrey spaces Lp,k under appropriated
conditions on the weight ω, where the symbol b belongs to weighted Lipschitz spaces. The
weighted Morrey space was first introduced by Komori and Shirai 5. For the general case,
Wang 6 proved that if b ∈ Lipβ Rn , then the commutator b, L−α/2 is bounded from Lp ωp to Lq ωq , where 0 < β < 1, 0 < α β < n, 1 < p < n/α β, 1/p − 1/q α β/n, and
ω q ∈ A1 .
The purpose of this paper is to give necessary and sufficient conditions for
boundedness of commutators of the general fractional integrals with b ∈ Lipβ ω the
weighted Lipschitz space. Our theorems are the following.
Theorem 1.1. Let 0 < β < 1, 0 < α β < n, 1 < p < n/α β, 1/q 1/p − α β/n, 0 ≤ k <
min{p/q, pβ/n}, and ωq ∈ A1 . Then one has the following.
a If b ∈ Lipβ Rn , then b, L−α/2 is bounded from Lp,k ωp , ωq to Lq,kq/p ωq ;
b If b, L−α/2 is bounded from Lp,k ωp , ωq to Lq,kq/p ωq , then b ∈ Lipβ Rn .
Theorem 1.2. Let 0 < β < 1, 0 < α β < n, 1 < p < n/α β, 1/q 1/p − α β/n, 0 ≤ k <
p/q, ωq/p ∈ A1 , and rω > 1 − k/p/q − k, where rω denotes the critical index of ω for the reverse
Hölder condition. Then one has the following.
a If b ∈ Lipβ ω, then b, L−α/2 is bounded from Lp,k ω to Lq,kq/p ω1−1−α/nq , ω;
b If b, L−α/2 is bounded from Lp,k ω to Lq,kq/p ω1−1−α/nq , ω, then b ∈ Lipβ ω.
Our results not only extend the results of 4 from −Δ to a general operator L, but
also characterize the weighted Lipschitz spaces by means of the boundedness of b, L−α/2 on the weighted Morrey spaces, which extend the results of 4, 6. The basic tool is based on
a modification of sharp maximal function ML# introduced by 7.
Throughout this paper all notation is standard or will be defined as needed. Denote
the Lebesgue measure of B by |B| and the weighted measure of B by ωB, where ωB ωxdx. For a measurable set E, denote by χE the characteristic function of E. For a real
B
number p, 1 < p < ∞, let p be the dual of p such that 1/p 1/p 1. The letter C will be used
for various constants, and may change from one occurrence to another.
Abstract and Applied Analysis
3
2. Some Preliminaries
A nonnegative function ω defined on Rn is called weight if it is locally integral. A weight ω
is said to belong to the Muckenhoupt class Ap Rn for 1 < p < ∞, if there exists a positive
constant C such that
p−1
1
1
−1/p−1
2.1
ωxdx
ωx
dx
≤ C,
|B| B
|B| B
for every ball B ⊂ Rn . The class A1 Rn is defined replacing the above inequality by
1
ωxdx ≤ C essinf ωx.
x∈B
|B| B
2.2
When p ∞, ω ∈ A∞ , if there exist positive constants δ and C such that given a ball B and E
is a measurable subset of B, then
δ
ωE
|E|
≤C
.
ωB
|B|
2.3
A weight function ω belongs to Ap,q for 1 < p < q < ∞ if for every ball B in Rn , there
exists a positive constant C which is independent of B such that
1
|B|
ωxq dx
1/q B
1
|B|
ωx−p dx
1/p
≤ C.
2.4
B
From the definition of Ap,q , we can get that
ω ∈ Ap,q ,
iff ωq ∈ A1q/p .
2.5
Since ωq/p ∈ A1 , then by 2.5, we have ω1/p ∈ Ap,q .
A weight function ω belongs to the reverse Hölder class RHr if there exist two
constants r > 1 and C > 0 such that the following reverse Hölder inequality,
1
|B|
ωxr dx
B
1/r
≤C
1
|B|
ωxdx,
2.6
B
holds for every ball B in Rn .
It is well known that if ω ∈ Ap with 1 ≤ p < ∞, then there exists r > 1 such that
ω ∈ RHr . It follows from Hölder inequality that ω ∈ RHr implies ω ∈ RHs for all 1 < s < r.
Moreover, if ω ∈ RHr , r > 1, then we have ω ∈ RHrε for some > 0. We thus write rw sup{r > 1 : ω ∈ RHr } to denote the critical index of ω for the reverse Hölder condition. For
more details on Muckenhoupt class Ap,q , we refer the reader to 8–10.
Definition 2.1 see 5. Let 1 ≤ p < ∞ and 0 ≤ k < 1. Then for two weights μ and ν, the
weighted Morrey space is defined by
p Lp,k μ, ν f ∈ Lloc μ : f Lp,k μ,ν < ∞ ,
2.7
4
Abstract and Applied Analysis
where
f p,k
sup
L μ,ν
B
1
νBk
fxp μxdx
1/p
2.8
,
B
and the supremum is taken over all balls B in Rn .
If μ ν, then we have the classical Morrey space Lp,k μ with measure μ. When k 0,
then L μ Lp μ is the Lebesgue space with measure μ.
p,0
Definition 2.2 see 11. Let 1 ≤ p < ∞, 0 < β < 1, and ω ∈ A∞ . A locally integral function b is
p
said to be in Lipβ ω if
bLipp ω sup
β
B
1
ωBβ/n
1
ωB
|bx − bB |p ωx1−p dx
1/p
≤ C < ∞,
2.9
B
where bB |B|−1 B bydy and the supremum is taken over all ball B ⊂ Rn . When p 1, we
p
denote Lipβ ω by Lipβ ω.
p
p
Obviously, for the case ω 1, then the Lipβ ω space is the classical Lipβ space.
Remark 2.3. Let ω ∈ A1 , Garcı́a-Cuerva 11 proved that the spaces fLipp ω coincide, and the
β
norms of || · ||Lipp ω are equivalent with respect to different values of provided that 1 ≤ p < ∞.
β
Given a locally integrable function f and β, 0 ≤ β < n, define the fractional maximal
function by
Mβ,r fx sup
x∈B
1
|B|1−βr/n
r
f y dy
1/r
,
2.10
r ≥ 1,
B
when 0 < β < n. If β 0 and r 1, then M0,1 f Mf denotes the usual Hardy-Littlewood
maximal function.
Let ω be a weight. The weighted maximal operator Mω is defined by
1
f y dy.
Mω fx sup
2.11
x∈B ωB B
The fractional weighted maximal operator Mβ,r,ω is defined by
Mβ,r,ω fx sup
x∈B
1
ωB1−βr/n
r f y ω y dy
1/r
,
2.12
B
where 0 ≤ β < n and r ≥ 1. For any f ∈ Lp Rn , p ≥ 1, the sharp maximal function ML# f
associated the generalized approximations to the identity {e−tL , t > 0} is given by Martell 7
as follows:
1
ML# fx sup
f y − e−tB L f y dy,
2.13
x∈B |B| B
Abstract and Applied Analysis
5
where tB rB2 and rB is the radius of the ball B. For 0 < δ < 1, we introduce the δ-sharp
#
as
maximal operator ML,δ
1/δ
δ
#
ML,δ
f ML# f ,
2.14
which is a modification of the sharp maximal operator M# of Stein and Murphy 9. Set
Mδ f M|f|δ 1/δ . Using the same methods as those of 9, 12, we can get the following.
Lemma 2.4. Assume that the semigroup e−tL has a kernel pt x, y which satisfies the upper bound
1.1. Let λ > 0 and f ∈ Lp Rn for some 1 < p < ∞. Suppose that ω ∈ A∞ , then for every 0 < η < 1,
there exists a real number γ > 0 independent of γ, f such that one has the following weighted version
of the local good λ inequality, for η > 0, A > 1,
#
fx ≤ γλ ≤ ηω x ∈ Rn : Mδ fx > λ ,
2.15
ω x ∈ Rn : Mδ f > Aλ, ML,δ
where A > 1 is a fixed constant which depends only on n.
If μ, ν ∈ A∞ , 1 < p < ∞, 0 ≤ k < 1, then
f Lp,k μ,ν
# ≤ Mδ f Lp,k μ,ν ≤ CML,δ
f
Lp,k μ,ν
.
2.16
In particular, when μ ν ω and ω ∈ A∞ , we have
#
f p,k ≤ Mδ f p,k ≤ C
f
M
L,δ L ω
L ω
Lp,k ω
.
2.17
3. Proof of Theorem 1.1
To prove Theorem 1.1, we need the following lemmas.
Lemma 3.1 see 1. Assume that the semigroup e−tL has a kernel pt x, y which satisfies the upper
bound 1.1. Then for 0 < α < 1, the difference operator L−α/2 − e−tL L−α/2 has an associated kernel
Kα,t x, y which satisfies
t
C
Kα,t x, y ≤ n−α .
x − y
x − y2
3.1
Lemma 3.2 see 4. Let 0 < α β < n, 1 < p < n/α β, 1/q 1/p − α β/n, and ω ∈ A1 .
Then for every 0 < k < p/q and 1 < r < p, one has
Mαβ,r f q,kq/p q ≤ Cf p,q p q .
L ω ,ω L
ω 3.2
Lemma 3.3 see 5. Let 0 < β < n, 1 < p < n/β, 1/s 1/p − β/n, and ω ∈ Ap,s . Then for every
0 < k < p/s, one has
Mβ,1 f s,ks/p s ≤ Cf p,k p s .
L ω ,ω L
ω 3.3
6
Abstract and Applied Analysis
Lemma 3.4 see 4. Let 0 < α β < n, 1 < p < n/α β, 1/q 1/p − α/n, 1/s 1/q − β/n,
and ωq ∈ A1 . Then for every 0 < k < p/s, one has
Mβ,1 f Ls,ks/p ωs ≤ Cf Lq,kq/p ωq ,ωs .
3.4
Lemma 3.5. Let 0 < α β < n, 1 < p < n/α β, 1/q 1/p − α/n, 1/s 1/q − β/n, and
ωq ∈ A1 . Then for every 0 < k < pβ/n, one has
−α/2 f
L
Lq,kq/p ωq ,ωs ≤ Cf Lp,k ωp ,ωs .
3.5
Proof. Since the semigroup e−tL has a kernel pt x, y which satisfies the upper bound 1.1, it
is easy to check that L−α/2 fx ≤ CIα |f|x for all x ∈ Rn . Together with the result cf. 4,
that is,
Iα fLq,kq/p ωq ,ωs ≤ CfLp,k ωp ,ωs ,
3.6
we can get the desired result.
Remark 3.6. Using the boundedness property of Iα , we also know L−α/2 is bounded from L1
to weak Ln/n−α . It is easy to check that Lemmas 3.2–3.5 also hold when k 0.
The following lemma plays an important role in the proof of Theorem 1.1.
Lemma 3.7. Let 0 < δ < 1, 0 < α < n, 0 < β < 1, and b ∈ Lipβ Rn . Then for all r > 1 and for all
x ∈ Rn , one has
#
ML,δ
b, L−α/2 f x
≤ CbLipβ Rn Mβ,1 L−α/2 f x Mαβ,r fx Mαβ,1 fx .
3.7
The same method of proof as that of Lemma 4.7 see below, one omits the details.
Proof of Theorem 1.1. We first prove a. We only prove Theorem 1.1 in the case 0 < α < 1. For
the general case 0 < α < n, the method is the same as that of 1. We omit the details.
For 0 < α β < n and 1 < p < n/α β, we can find a number r such that 1 < r < p. By
2.17 and Lemma 3.7, we obtain the following:
b, L−α/2 f Lq,kq/p ωq #
b, L−α/2 f ≤ CML,δ
Lq,kq/p ωq ≤ CbLipβ Rn Mβ,1 L−α/2 f 3.8
Lq,kq/p ωq Mαβ,r f Lq,kq/p ωq Mαβ,1 f Lq,kq/p ωq .
Abstract and Applied Analysis
7
Let 1/q1 1/p − α/n and 1/q 1/q1 − β/n. Since ωq ∈ A1 , then by 2.5, we have ω ∈ Ap,q .
Since 0 < k < min{p/q, pβ/n}, by Lemmas 3.2–3.5, we yield that
b, L−α/2 f q,kq/p q ≤ CbLipβ Rn L−α/2 f q ,kq /p q q
L
ω L
1
1
ω 1 ,ω f Lp,k ωp ,ωq 3.9
≤ CbLipβ Rn f Lp,k ωp ,ωq .
Now we prove b. Let L −Δ be the Laplacian on Rn , then L−α/2 is the classical
fractional integral Iα . Let k 0 and weight ω ≡ 1, then Lp,k ωp , ωq Lp and Lq,kq/p ωq , ω Lq . From 2, the Lp , Lq boundedness of b, Iα implies that b ∈ Lipβ Rn .
Thus Theorem 1.1 is proved.
4. Proof of Theorem 1.2
We also need some Lemmas to prove Theorem 1.2.
Lemma 4.1 see 4. Let 0 < α β < n, 1 < p < n/α β, 1/q 1/p − α/n, 1/s 1/q − β/n,
and ωs/p ∈ A1 . Then if 0 < k < p/s and rω > 1/p/q − k, one has
Mβ,1 f s,ks/p s/p ≤ Cf q,kq/p q/p .
4.1
ω
L
ω
L
,ω
,ω
Lemma 4.2 see 4. Let 0 < α < n, 1 < p < n/α, 1/q 1/p − α/n, and ωq/p ∈ A1 . Then if
0 < k < p/q and rω > 1 − k/p/q − k, one has
Mα,1 f q,kq/p q/p ≤ Cf p,k .
4.2
L
ω ,ω
L ω
Lemma 4.3 see 4. Let 0 < α < n, 1 < p < n/α, 1/q 1/p − α/n, 0 < k < p/q, and ω ∈ A∞ .
For any 1 < r < p, one has
Mα,r,ω f q,kq/p q/p ≤ Cf p,k .
4.3
L
ω ,ω
L ω
Lemma 4.4. Let 0 < α < n, 1 < p < n/α, 1/q 1/p − α/n, and ωq/p ∈ A1 . Then if 0 < k < p/q
and rω > 1 − k/p/q − k, one has
−α/2 f q,kq/p q/p ≤ Cf Lp,k ω .
L
4.4
L
ω
,ω
Proof. As before, we know that L−α/2 fx ≤ CIα |f|x for all x ∈ Rn . Using the boundedness
property of Iα on weighted Morrey space cf. 4, we have
−α/2 f
≤ Iα f q,kq/p q/p ≤ Cf p,k ,
L
4.5
Lq,kq/p ωq/p ,ω
L
ω
,ω
L
ω
where 1 < p < n/α and 1/q 1/p − α/n.
Remark 4.5. It is easy to check that the above lemmas also hold for k 0.
8
Abstract and Applied Analysis
Lemma 4.6. Assume that the semigroup e−tL has a kernel pt x, y which satisfies the upper bound
1.1, and let b ∈ Lipβ ω, ω ∈ A1 . Then, for every function f ∈ Lp Rn , p > 1, x ∈ Rn , and
1 < r < ∞, one has
1
sup
|B|
x∈B
−tB L b y − b2B f y dy ≤ CbLipβ ω ωxMβ,r,ω fx.
e
B
4.6
Proof. Fix f ∈ Lp Rn , 1 < p < ∞ and x ∈ B. Then,
1
|B|
−tB L b − b2B f y dy
e
B
1
≤
|B|
≤
1
|B|
Rn
B
B
1
|B|
2B
pt y, z bz − b2B fzdzdy
B
pt y, z bz − b2B fzdzdy
B
∞ B k1
2k1 B\2k B
4.7
pt y, z bz − b2B fzdzdy
B
.
M N.
It follows from y ∈ B and z ∈ 2B that
pt y, z ≤ Ct−n/2 ≤ C 1 .
B
B
|2B|
4.8
Thus, Hölder’s inequality and Definition 2.2 lead to the following:
M≤C
≤C
1
|2B|
1
|2B|
bz − b2B fzdz
2B
1/r |bz − b2B |r ωz1−r dz
2B
≤ CbLipβ ω
2B
1/r
1
1
fzr ωzdz
ω2Bβ/n1/r ω2B1/r
ω2B 2B
|2B|
≤ CbLipβ ω
1/r
fzr ωzdz
1
1
ω2Bβ/n1
ω2B
|2B|
≤ CbLipβ ω ωx
1
ω2B1−βr/n
≤ CbLipβ ω ωxMβ,r,ω fx.
1/r
fzr ωzdz
2B
1/r
r
fz ωzdz
2B
4.9
Abstract and Applied Analysis
9
Moreover, for any y ∈ B and z ∈ 2k1 B \ 2k B, we have |y − z| ≥ 2k−1 rB and |ptB y, z| ≤
2k−1
Ce−c2
2k1n /|2k1 B|
1
N
|B|
∞ B k1
2k1 B\2k B
pt y, z bz − b2B fzdzdy
B
∞ −c22k−1 k1n e
2
bz − b2B fzdz
≤C
2k1 B
2k1 B
k1
≤C
∞ −c22k−1 k1n e
2
bz − b2k1 B fzdz
k1
2 B
2k1 B
4.10
k1
∞ −c22k−1 k1n e
2
b2k1 B − b2B fzdz
C
2k1 B
k1
2 B
k1
.
N1 N2 .
We will estimate the values of terms N1 and N2 , respectively.
Using Hölder’s inequality and Remark 2.3, we have the following:
N1 ≤ C
∞ −c22k−1 k1n
e
2
2k1 B
k1
r
×
2k1 B
≤C
∞
|bz − b2k1 B | ωz
1−r 1/r dz
1/r
fzr ωzdz
2k1 B
2k1n e−c2
2k−1
4.11
k1
1/r
r
ω 2k1 B
1
× bLipβ ω k1 1−βr/n k1 fz ωzdz
2 B
2 B
ω 2k1 B
≤ CbLipβ ω ωxMβ,r,ω fx.
By a simple calculation, we have
β/n
.
|b2k1 B − b2B | ≤ CkωxbLipβ ω ω 2k1 B
Since ω ∈ A1 , by the Hölder inequality, we get
N2 ≤ C
∞
k1n −c22k−1
2
e
k1
≤C
∞
k1
β/n
kω 2k1 B
fzdz
ωxbLipβ ω
2k1 B
2k1 B
k1n −c22k−1
k2
e
β/n 1/r
r
ω 2k1 B
fz dz
ωxbLipβ ω 2k1 B1/r
2k1 B
4.12
10
Abstract and Applied Analysis
C
∞
k2k1n e−c2
2k−1
k1
1/r
r
ω2k1 B
1
fz dz
× ωxbLipβ ω
|2k1 B| ω 2k1 B 1−βr/n 2k1 B
1/r
∞
r
1
k1n −c22k−1
fz ωzdz
≤ C k2
e
ωxbLipβ ω
ω2k1 B1−βr/n 2k1 B
k1
≤ CbLipβ ω ωxMβ,r,ω fx.
4.13
Thus, Lemma 4.6 is proved.
Lemma 4.7. Let 0 < δ < 1, 0 < α < 1, ω ∈ A1 , and b ∈ Lip β ω. Then for all r > 1 and for all
x ∈ Rn , one has
#
ML,δ
b, L−α/2 f x
≤ CbLipβ ω × ωx1β/n Mβ,1 L−α/2 f x
ωx1−α/n Mαβ,r,ω fx ωx1β/n Mαβ,1 fx .
4.14
Proof. For any given x ∈ Rn , fix a ball B Bx0 , rB which contains x. We decompose f f1 f2 , where f1 fχ2B . Observe that
b, L−α/2 f b − b2B L−α/2 f − L−α/2 b − b2B f1 − L−α/2 b − b2B f2
4.15
e−tB L b, L−α/2 f e−tB L b − b2B L−α/2 f − L−α/2 b − b2B f1 − L−α/2 b − b2B f2 .
Then
1
|B|
δ 1/δ
−α/2
−tB L
−α/2
f y −e
b, L
f y dy
b, L
B
1/δ
δ
1
≤C
b y − b2B L−α/2 f y dy
|B| B
1/δ
δ
1
−α/2 C
b − b2B f1 y dy
L
|B| B
δ 1/δ
1
−tB L
−α/2
C
f y dy
b − b2B L
e
|B| B
1/δ
δ
1
−tB L −α/2 L
C
y dy
b − b2B f1
e
|B| B
1/δ
δ
1
−α/2
−tB L −α/2
L
C
−
e
L
dy
y
−
b
f
b
2B 2
|B| B
.
I II III IV V.
4.16
Abstract and Applied Analysis
11
We are going to estimate each term, respectively. Fix 0 < δ < 1 and choose a real
number τ such that 1 < τ < 2 and τ δ < 1. Since ω ∈ A1 , then it follows from Hölder’s
inequality that
1/τδ 1/τ δ
τδ
1
−α/2 τ δ
dy
f y dy
b y − b2B
I≤C
L
|B| B
B
1
−α/2 ≤C
f y dy
b y − b2B dy
L
|B| 2B
B
1
−α/2 1β/n
≤ CbLipβ ω
f y dy
ω2B
L
|2B|
B
≤ CbLipβ ω ωx1β/n Mβ,1 L−α/2 f x.
4.17
For II, using Hölder’s inequality, Kolmogorov’s inequality see page 485 8, and
Remark 3.6, then we deduce that
II ≤ C
1
|B|
−α/2 b − b2B f1 y dy
L
B
1
|B|α/n L−α/2 b − b2B f1 n/n−α,∞
L
|B|
1
≤ C 1−α/n b y − b2B f1 y dy
|B|
B
1/r 1/r r r −r /r
1
f y w y dy
b y − b2B w y
≤ C 1−α/n
dy
|B|
2B
2B
ω2B 1−α/n
≤ CbLipβ ω Mαβ,r,ω fx
|2B|
≤C
4.18
≤ CbLipβ ω ωx1−α/n Mαβ,r,ω fx,
where we have used the condition that ω ∈ A1 .
Using Hölder’s inequality and Lemma 4.6, we obtain that
III ≤ CbLipβ ω ωxMβ,r,ω L−α/2 f x.
4.19
For IV , using the estimate in II, we get
IV ≤
C
|B|
B
2B
pt y, z L−α/2 b − b2B f zdzdy
B
C
−α/2 ≤
b − b2B f zdz
L
|2B| 2B
≤ CbLipβ ω ωx1−α/n Mαβ,r,ω fx.
4.20
12
Abstract and Applied Analysis
By virtue of Lemma 3.1, we have the following:
V ≤
C
|B|
2Bc
B
Kα,t y, z bz − b2B fzdzdy
B
∞ rB2 1
bz − b2B fzdz
n−α
2
k
k1
|x0 − z|
k1 2 rB ≤|x0 −z|<2 rB |x0 − z|
∞
1
bz − b2B fzdz
≤ C 2−2k 1−α/n
2k1 B
2k1 B
k1
∞
1
−2k
bz − b2k1 B fzdz
≤C 2 1−α/n
k1
k1
2 B
2 B
k1
∞
1
−2k
fzdz
C 2 |b2k1 B − b2B | 1−α/n
2k1 B
2k1 B
k1
≤C
4.21
.
V I V II.
Making use of the same argument as that of II, we have
V I ≤ CbLipβ ω ωx1−α/n Mαβ,r,ω fx.
4.22
β/n
.
|b2k1 B − b2B | ≤ CkωxbLipβ ω ω 2k1 B
4.23
Note that ω ∈ A1 ,
So, the value of V II can be controlled by
CbLipβ ω ωx1β/n Mαβ,1 fx.
4.24
Combining the above estimates for I–V, we finish the proof of Lemma 4.7.
Proof of Theorem 1.2. We first prove a. As before, we only prove Theorem 1.2 in the case 0 <
α < 1. For 0 < α β < n and 1 < p < n/α β, we can find a number r such that 1 < r < p. By
Lemma 4.7, we obtain the following:
b, L−α/2 f Lq,kq/p ω1−1−α/nq ,ω
#
b, L−α/2 f ≤ CML,δ
Lq,kq/p ω1−1−α/nq ,ω
Abstract and Applied Analysis
≤ CbLipβ ω ω·1β/n Mβ,1 L−α/2 f ω·1−α/n Mαβ,r,ω f 13
Lq,kq/p ω1−1−α/nq ,ω
Lq,kq/p ω1−1−α/nq ,ω
ω·1β/n Mαβ,1 f Lq,kq/p ω1−1−α/nq ,ω
≤ CbLipβ ω Mβ,1 L−α/2 f Lq,kq/p ωq/p ,ω
Mαβ,r,ω f Lq,kq/p ω Mαβ,1 f Lq,kq/p ωq/p ,ω .
4.25
Let 1/q1 1/p − α/n and 1/q 1/q1 − β/n. Lemmas 4.1–4.4 yield that
b, L−α/2 f Lq,kq/p ω1−1−α/nq ,ω
≤ CbLipβ ω L−α/2 f Lq1 ,kq1 /p ωq1 /p ,ω
f Lp,k ω
4.26
≤ CbLipβ ω f Lp,k ω .
Now we prove b. Let L −Δ be the Laplacian on Rn , then L−α/2 is the classical
fractional integral Iα . We use the same argument as Janson 13. Choose Z0 ∈ Rn so that
|Z0 | 3. For x ∈ BZ0 , 2, |x|−αn can be written as the absolutely convergent Fourier series,
|x|−αn m∈Zn am ei
νm ,x with m |am | < ∞ since |x|−αn ∈ C∞ BZ0 , 2. For any x0 ∈ Rn and
ρ > 0, let B Bx0 , ρ and BZ0 Bx0 Z0 ρ, ρ,
bx − bBZ0 dx B
where sx sgn
ρ−αn
ρn
BZ0
ρ
−α
m∈Zn
−αn n−α
x−y
dy dx,
bx − b y x − y
4.27
BZ0
−αn
bx − b y x − y
BZ0
0
bx − bydy. Fix x ∈ B and y ∈ BZ0 , then y − x/ρ ∈ BZ0 ,2 , hence,
sx
B
bx − b y dydx
B BZ
sx
B
1
n
ρ
1
|BZ0 |
am
ρ
dy dx
n−α i
ν ,y/ρ
m
e
dy e−i
νm ,x/ρ dx
bx − b y x − y
sx
B
x − y n−α
BZ0
−α −α/2
i
νm ,·/ρ
−i
νm ,x/ρ
χBZ0 e
≤ρ dx
xχB xe
|am | sx b, L
m∈Zn
B
14
Abstract and Applied Analysis
≤ρ
−α
m∈Zn
≤ Cρ
−α
|am | b, L−α/2 χBZ0 ei
νm ,·/ρ m∈Zn
Lq,0 ω1−1−α/nq ,ω
|am |χBZ0 Lp,0 ω
q 1/q −α/n
ωx
ωx
q 1/q −α/n
1/q
dx
B
1/q
dx
B
≤ CωB1/p1/q −α/n CωB1β/n .
4.28
This implies that b ∈ Lipβ ω. Thus, b is proved.
Acknowledgment
The authors thank the referee for the useful suggestions. Z. Si was supported by Doctoral
Foundation of Henan Polytechnic University. F. Zhao was supported by Shanghai Leading
Academic Discipline Project Grant no. J50101.
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