Some estimates for commutators of fractional integral operators on weighted Morrey spaces

0.18 MB·English

by  Hua Wang

On some commutator theorems for fractional
0
1 integral operators on the weighted Morrey spaces
0
2
t Hua Wang∗
c
O
School of Mathematical Sciences, Peking University,Beijing 100871, P. R.China

3
1

]
A
Abstract
C
Let 0 < α < n and I be the fractional integral operator. In this
. α
h paper,we willshowsome weightedboundedness propertiesof commu-
at tator [b,Iα] onthe weighted MorreyspacesLp,κ(w) under appropriate
m conditions on the weight w, where the symbol b belongs to weighted
[ BMO or Lipschitz space or weighted Lipschitz space.

1 MSC(2000) 42B20; 42B35
v
Keywords: WeightedMorreyspaces;fractionalintegraloperators;com-
8
3 mutator; weighted BMO; Lipschitz function; weighted Lipschitz func-
6 tion
2
. 1. Introduction
0
1 The classical Morrey spaces Lp,λ were originally introduced by Morrey
0
in [7] to study the local behavior of solutions to second order elliptic partial
1
: differentialequations. ForthepropertiesandapplicationsofclassicalMorrey
v
spaces, we refer the readers to [7,11]. In [1], Chiarenza and Frasca showed
i
X
the boundedness of the Hardy-Littlewood maximal operator, the fractional
r integral operator and the Calder´on-Zygmund singular integral operator on
a
these spaces.
Recently, Komori and Shirai [6] defined the weighted Morrey spaces
Lp,κ(w) and studied the boundedness of the above classical operators on
these spaces. Assume that I is a fractional integral operator and b is a
α
locally integrable function on Rn, the commutator of b and I is defined as
α
follows
[b,I ]f(x)= b(x)I f(x)−I (bf)(x).
α α α
∗E-mail address: [email protected].
1
In [6], the authors proved that when 0 < α < n, 1 < p < n/α, 1/q =
1/p − α/n, 0 < κ < p/q and w ∈ A (Muckenhoupt weight class), then
p,q
[b,I ] is boundedfrom Lp,κ(wp,wq) to Lq,κq/p(wq)whenever b ∈ BMO(Rn).
α
The main purposeof this paper is to study the weighted boundedness of
commutator [b,I ] on the weighted Morrey spaces when b belongs to some
α
other function spaces. Our main results are stated as follows.
Theorem 1. Let 0 < α < n, 1 < p < n/α, 1/q = 1/p−α/n, 0 < κ < p/q
and wq/p ∈ A . Suppose that b ∈ BMO(w)(weighted BMO)and r > 1−κ ,
1 w p/q−κ
then [b,I ] is bounded from Lp,κ(w) to Lq,κq/p(w1−(1−α/n)q,w), where r
α w
denotes the critical index of w for the reverse H¨older condition.
Theorem 2. Let 0 < β < 1, 0 < α+β < n, 1 < p < n/(α+β), 1/s =
1/p − (α + β)/n, 0 < κ < min{p/s,pβ/n} and ws ∈ A . Suppose that
1
b ∈ Lip (Rn)(Lipschitz space), then [b,I ] is bounded from Lp,κ(wp,ws) to
β α
Ls,κs/p(ws).
Theorem 3. Let0 < β < 1, 0< α+β < n, 1< p < n/(α+β), 1/s = 1/p−
(α+β)/n, 0 < κ < p/s and ws/p ∈ A . Suppose that b ∈ Lip (w)(weighted
1 β
Lipschitz space) and r > 1 , then [b,I ] is bounded from Lp,κ(w) to
w p/s−κ α
Ls,κs/p(w1−(1−α/n)s,w).
2. Definitions and Notations
First let us recall some standard definitions and notations of weight
classes. A weight w is a locally integrable function on Rn which takes val-
ues in (0,∞) almost everywhere, all cubes are assumed to have their sides
parallel to the coordinate axes. Given a cube Q and λ > 0, λQ denotes
the cube with the same center as Q whose side length is λ times that of
Q, Q = Q(x ,r) denotes the cube centered at x with side length r. For a
0 0
given weight function w, we denote the Lebesgue measure of Q by |Q| and
the weighted measure of Q by w(Q), where w(Q) = w(x)dx.
Q
We shall give the definitions of three weight classRes as follows.
Definition 1 ([8]). A weight function w is in the Muckenhoupt class A
p
with 1 < p < ∞ if for every cube Q in Rn, there exists a positive constant
C which is independent of Q such that
p−1
1 1 − 1
w(x)dx w(x) p−1 dx ≤ C.
(cid:18)|Q|Z (cid:19)(cid:18)|Q| Z (cid:19)
Q Q
When p = 1, w ∈A , if
1
1
w(x)dx ≤ Cessinfw(x).
|Q|Z
Q x∈Q
2
When p = ∞, w ∈ A , if there exist positive constants δ and C such that
∞
given a cube Q and E is a measurable subset of Q, then
δ
w(E) |E|
≤ C .
w(Q) (cid:18)|Q|(cid:19)
Definition 2 ([9]). A weight function w belongs to A for 1 < p < q <
p,q
∞ if for every cube Q in Rn, there exists a positive constant C which is
independent of Q such that
1/q 1/p′
1 w(x)qdx 1 w(x)−p′dx ≤ C,
(cid:18)|Q| Z (cid:19) (cid:18)|Q| Z (cid:19)
Q Q
where p′ denotes the conjugate exponent of p > 1; that is, 1/p+1/p′ = 1.
Definition 3 ([3]). A weight function w belongs to the reverse H¨older class
RH if there exist two constants r > 1 and C > 0 such that the following
r
reverse H¨older inequality
1/r
1 1
w(x)rdx ≤ C w(x)dx
(cid:18)|Q| Z (cid:19) (cid:18)|Q|Z (cid:19)
Q Q
holds for every cube Q in Rn.
Itiswellknownthatifw ∈ A with1 < p < ∞,thenw ∈ A forallr > p,
p r
andw ∈ A forsome1 < q < p. Ifw ∈ A with1≤ p < ∞,thenthereexists
q p
r > 1 such that w ∈ RH . It follows from H¨older’s inequality that w ∈ RH
r r
implies w ∈ RH for all 1 < s < r. Moreover, if w ∈ RH , r > 1, then we
s r
have w ∈ RH for some ε > 0. We thus write r ≡sup{r > 1: w ∈RH }
r+ε w r
to denote the critical index of w for the reverse H¨older condition.
We state the following results that we will use frequently in the sequel.
Lemma A ([3]). Let w ∈ A , p ≥ 1. Then, for any cube Q, there exists an
p
absolute constant C such that
w(2Q) ≤ Cw(Q).
In general, for any λ > 1, we have
w(λQ) ≤ Cλnpw(Q),
where C does not depend on Q nor on λ.
3
Lemma B ([3,4]). Let w ∈ A ∩RH , p ≥ 1 and r > 1. Then there exist
p r
constants C , C > 0 such that
1 2
p (r−1)/r
|E| w(E) |E|
C ≤ ≤ C
1 2
(cid:18)|Q|(cid:19) w(Q) (cid:18)|Q|(cid:19)
for any measurable subset E of a cube Q.
Lemma C ([5]). Let s> 1, 1 ≤ p < ∞ and As = {w : ws ∈ A }. Then
p p
As = A ∩RH .
p 1+(p−1)/s s
In particular,
As = A ∩RH .
1 1 s
Next we shall introduce the Hardy-Littlewood maximal operator and
several variants, the fractional integral operator and some function spaces.
Definition 4. The Hardy-Littlewood maximal operator M is defined by
1
M(f)(x) = sup |f(y)|dy.
|Q| Z
x∈Q Q
For 0 < β < n, r ≥ 1, we define the fractional maximal operator M by
β,r
1/r
1
M (f)(x) = sup |f(y)|rdy .
β,r x∈Q(cid:18)|Q|1−βnr ZQ (cid:19)
Let w be a weight. The weighted maximal operator M is defined by
w
1
M (f)(x) = sup |f(y)|w(y)dy.
w
w(Q) Z
x∈Q Q
For 0 < β < n and r ≥ 1, we define the fractional weighted maximal operator
M by
β,r,w
1/r
1
M (f)(x) = sup |f(y)|rw(y)dy ,
β,r,w x∈Q(cid:18)w(Q)1−βnr ZQ (cid:19)
where the supremum is taken over all cubes Q containing x.
Definition 5 ([13]). For 0 < α < n, the fractional integral operator I is
α
defined by
Γ(n−α) f(y)
I (f)(x) = 2 dy.
α 2απn2Γ(α) ZRn |x−y|n−α
2
4
Let 1≤ p < ∞ andw beaweight function. Alocally integrable function
b is said to be in BMO (w) if
p
1/p
1
kbk = sup |b(x)−b |pw(x)1−pdx ≤ C < ∞,
BMOp(w) (cid:18)w(Q) Z Q (cid:19)
Q Q
whereb = 1 b(y)dy andthesupremumis taken over allcubesQ ⊂ Rn.
Q |Q| Q
We denote simpRly by BMO(w) when p = 1.
Let 0 < β < 1 and 1 ≤ p < ∞. A locally integrable function b is said to
be in Lipp(Rn) if
β
1/p
1 1
kbkLippβ = suQp|Q|β/n (cid:18)|Q|ZQ|b(x)−bQ|pdx(cid:19) < ∞.
We denote simply by Lip (Rn) when p = 1.
β
Let 0 < β < 1, 1 ≤ p < ∞ and w be a weight function. A locally
p
integrable function b is said to belong to Lip (w) if
β
1/p
1 1
kbkLippβ(w) = suQpw(Q)β/n (cid:18)w(Q) ZQ|b(x)−bQ|pw(x)1−pdx(cid:19) < ∞.
We also denote simply by Lip (w) when p = 1.
β
Lemma D ([2,10]). (i) Let w ∈ A . Then for any 1 ≤ p < ∞, there exists
1
an absolute constant C > 0 such that kbk ≤ Ckbk .
BMOp(w) BMO(w)
(ii) Let 0 < β < 1. Then for any 1 ≤ p < ∞, there exists an absolute
constant C > 0 such that kbkLippβ ≤ CkbkLipβ.
(iii) Let 0 < β < 1 and w ∈ A . Then for any 1 ≤ p < ∞, there exists an
1
absolute constant C > 0 such that kbkLippβ(w) ≤ CkbkLipβ(w).
We are going to conclude this section by defining the weighted Morrey
space. For further details, we refer the readers to [6].
Definition 6. Let 1 ≤ p < ∞, 0< κ < 1 and w be a weight function. Then
the weighted Morrey space is defined by
Lp,κ(w) = {f ∈ Lp (w) : kfk < ∞},
loc Lp,κ(w)
where
1/p
1
kfk = sup |f(x)|pw(x)dx
Lp,κ(w) (cid:18)w(Q)κ Z (cid:19)
Q Q
and the supremum is taken over all cubes Q in Rn.
5
Remark. Equivalently, we could define the weighted Morrey space with balls
instead of cubes. Hence we shall use these two definitions of weighted Morrey
space appropriate to calculations.
In order to deal with the fractional order case, we need to consider the
weighted Morrey space with two weights.
Definition 7. Let 1 ≤ p < ∞ and 0 < κ < 1. Then for two weights u and
v, the weighted Morrey space is defined by
Lp,κ(u,v) = {f ∈ Lp (u) : kfk < ∞},
loc Lp,κ(u,v)
where
1/p
1
kfk = sup |f(x)|pu(x)dx .
Lp,κ(u,v) (cid:18)v(Q)κ Z (cid:19)
Q Q
We shall need the following estimate given in [6].
Theorem E. If 0 < β < n, 1 < p < n/β, 1/s = 1/p−β/n, 0 < κ < p/s
and w ∈ A , then M is bounded from Lp,κ(wp,ws) to Ls,κs/p(ws).
p,s β,1
Throughout this article, we will use C to denote a positive constant,
which is independent of the main parameters and not necessarily the same
at each occurrence. By A∼ B, we mean that there exists a constant C > 1
such that 1 ≤ A ≤ C. Moreover, we will denote the conjugate exponent of
C B
r > 1 by r′ = r/(r−1).
3. Proof of Theorem 1
We shall adopt the same method given in [12]. For 0 < δ < 1, we define
#
the δ-sharp maximal operator M as
δ
M#(f)= M#(|f|δ)1/δ,
δ
whichis amodificationof thesharpmaximaloperator M# ofFefferman and
Stein [14]. We also set M (f) = M(|f|δ)1/δ. Suppose that w ∈ A , then
δ ∞
for any cube Q, we have the following weighted version of the local good λ
inequality(see [14])
λ
#
w {x ∈ Q : M f(x)> λ,M f(x) ≤ λε} ≤ Cε·w {x ∈ Q : M f(x)> } ,
δ δ δ 2
(cid:0) (cid:1) (cid:0) (cid:1)
for all λ,ε > 0. As a consequence, by using the standard arguments(see
[14,15]), we can establish the following estimate, which will play an impor-
tant role in the proof of our main results.
6
Proposition 3.1. Let 0 < δ < 1, 1 < p < ∞ and 0 < κ < 1. If u,v ∈ A ,
∞
then we have
#
kM (f)k ≤ CkM (f)k
δ Lp,κ(u,v) δ Lp,κ(u,v)
for all functions f such that the left hand side is finite. In particular, when
u = v = w and w ∈A , then we have
∞
#
kM (f)k ≤ CkM (f)k
δ Lp,κ(w) δ Lp,κ(w)
for all functions f such that the left hand side is finite.
Next we are going to prove a series of lemmas which will be used in the
proof of our main theorems.
Lemma 3.2. Let 0 < α < n, 1 < p < n/α, 1/q = 1/p−α/n and w ∈ A .
∞
Then for every 0 < κ< p/q, we have
kM (f)k ≤ Ckfk .
α,1,w Lq,κq/p(w) Lp,κ(w)
Proof. Fix a cube Q ⊆ Rn and decompose f = f +f , where f = fχ ,
1 2 1 2Q
χ denotes the characteristic function of 2Q. Since M is a sublinear
2Q α,1,w
operator, then we have
1 1/q
M f(x)qw(x)dx
α,1,w
w(Q)κ/p(cid:16)ZQ (cid:17)
1 1/q
≤ M f (x)qw(x)dx
α,1,w 1
w(Q)κ/p(cid:16)ZQ (cid:17)
1 1/q
+ M f (x)qw(x)dx
α,1,w 2
w(Q)κ/p(cid:16)ZQ (cid:17)
=I +I .
1 2
As we know, the fractional weighted maximal operator M is bounded
α,1,w
from Lp(w) to Lq(w) provided that w ∈A . This together with Lemma A
∞
yield
1 1/p
I ≤ C |f(x)|pw(x)dx
1
w(Q)κ/p(cid:16)Z2Q (cid:17)
w(2Q)κ/p (1)
≤ Ckfk ·
Lp,κ(w) w(Q)κ/p
≤ Ckfk .
Lp,κ(w)
We now turntoestimate thetermI . Asimplegeometric observation shows
2
that for any x ∈ Q, we have
7
1
M (f )(x) ≤ sup |f(y)|w(y)dy.
α,1,w 2
w(R)1−α/n Z
R:Q⊆3R R
When Q ⊆ 3R, then by Lemma A, we have w(Q) ≤ Cw(R). It follows from
H¨older’s inequality that
1
|f(y)|w(y)dy
w(R)1−α/n Z
R
1 1/p 1/p′
≤ |f(y)|pw(y)dy w(y)dy
w(R)1−α/n(cid:16)ZR (cid:17) (cid:16)ZR (cid:17)
≤Ckfk ·w(R)(κ−1)/p+α/n
Lp,κ(w)
≤Ckfk ·w(Q)(κ−1)/p+α/n,
Lp,κ(w)
where in the last inequality we have used the fact that (κ−1)/p+α/n < 0.
Hence
I ≤ Ckfk ·w(Q)(κ−1)/p+α/nw(Q)1/qw(Q)−κ/p ≤ Ckfk . (2)
2 Lp,κ(w) Lp,κ(w)
Combining the above inequality (2) with (1) and taking the supremum over
all cubes Q ⊆ Rn, we obtain the desired result.
Lemma 3.3. Let 0 < α < n, 1 < p < n/α, 1/q = 1/p−α/n, 0 < κ < p/q
and w ∈ A . Then for any 1< r < p, we have
∞
kM (f)k ≤ Ckfk .
α,r,w Lq,κq/p(w) Lp,κ(w)
Proof. With the notations mentioned earlier, we know that
M (f) = M (|f|r)1/r.
α,r,w αr,1,w
From the definition, we readily see that
kM (f)k = kM (|f|r)k1/r .
α,r,w Lq,κq/p(w) αr,1,w Lq/r,κq/p(w)
Since 1/q = 1/p−α/n, then for any 1 < r < p, we have r/q = r/p−αr/n.
Hence, by Lemma 3.2, we can obtain
M (|f|r) 1/r ≤ C |f|r 1/r ≤ Ckfk .
αr,1,w Lq/r,κq/p(w) Lp/r,κ(w) Lp,κ(w)
(cid:13) (cid:13) (cid:13) (cid:13)
(cid:13) (cid:13) (cid:13) (cid:13)
We are done.
8
Lemma 3.4. Let 0< α < n, 1 < p < n/α, 1/q = 1/p−α/n and wq/p ∈ A .
1
Then if 0< κ < p/q and r > 1−κ , we have
w p/q−κ
kM (f)k ≤ Ckfk .
α,1 Lq,κq/p(wq/p,w) Lp,κ(w)
Proof. FixaballB = B(x ,r ) ⊆ Rn,whereB(x ,r )denotestheballwith
0 B 0 B
the center x and radius r . We decompose f = f +f , where f = fχ .
0 B 1 2 1 2B
Since M is a sublinear operator, then we have
α,1
1 1/q
M f(x)qw(x)q/pdx
α,1
w(B)κ/p(cid:16)ZB (cid:17)
1 1/q
≤ M f (x)qw(x)q/pdx
α,1 1
w(B)κ/p(cid:16)ZB (cid:17)
1 1/q
+ M f (x)qw(x)q/pdx
α,1 2
w(B)κ/p(cid:16)ZB (cid:17)
=I +I .
3 4
For any function f it is easy to see that
M (f)(x) ≤ CI (|f|)(x). (3)
α,1 α
From the definition, we can easily check that
w ∈Ap,q if and only if wq ∈ A1+q/p′. (4)
Since wq/p ∈ A , then by (4), we have w1/p ∈ A . It is well known that the
1 p,q
fractional integral operator I is bounded from Lp(wp) to Lq(wq) whenever
α
w ∈ A (see [9]). This together with Lemma A imply
p,q
1 1/p
I ≤ C |f(x)|pw(x)dx
3 w(B)κ/p(cid:16)Z2B (cid:17)
w(2B)κ/p (5)
≤ Ckfk ·
Lp,κ(w) w(B)κ/p
≤ Ckfk .
Lp,κ(w)
We now turn to deal with I . Note that when x ∈ B, y ∈ (2B)c, then we
4
have |y−x|∼ |y−x |. Since q/p > 1 and wq/p ∈ A , then by Lemma C, we
0 1
get w ∈ A ∩RH . It follows from the inequality (3), H¨older’s inequality
1 q/p
and the condition A that
p
|f(y)|
M (f )(x) ≤C dy
α,1 2 Z |x−y|n−α
(2B)c
∞
1
≤C |f(y)|dy
|2j+1B|1−α/n Z
Xj=1 2j+1B
9
∞
1
≤C ·|2j+1B|w(2j+1B)−1/p
|2j+1B|1−α/n
Xj=1
1/p
· |f(y)|pw(y)dy
(cid:16)Z2j+1B (cid:17)
∞
≤Ckfk |2j+1B|α/nw(2j+1B)(κ−1)/p.
Lp,κ(w)
Xj=1
Hence
wq/p(B)1/q ∞
I ≤ Ckfk · |2j+1B|α/nw(2j+1B)(κ−1)/p
4 Lp,κ(w) w(B)κ/p
Xj=1
|B|−α/nw(B)1/p ∞
≤ Ckfk · |2j+1B|α/nw(2j+1B)(κ−1)/p
Lp,κ(w) w(B)κ/p
Xj=1
∞ |2j+1B|α/n w(B)(1−κ)/p
= Ckfk · .
Lp,κ(w) |B|α/n w(2j+1B)(1−κ)/p
Xj=1
Since r > 1−κ , then we can find a suitable number r such that r > 1−κ
w p/q−κ p/q−κ
and w ∈ RH . Consequently, by Lemma B, we can get
r
(r−1)/r
w(B) |B|
≤ C .
w(2j+1B) (cid:18)|2j+1B|(cid:19)
Therefore
∞
I ≤ Ckfk (2jn)α/n−(r−1)(1−κ)/pr
4 Lp,κ(w)
Xj=1 (6)
≤ Ckfk ,
Lp,κ(w)
where the last series is convergent since α/n−(r−1)(1−κ)/pr < 0. Com-
bining the above inequality (6) with (5) and taking the supremum over all
balls B ⊆ Rn, we get the desired result.
It should be pointed out that from the above proof of Lemma 3.4, the
same conclusion also holds for the fractional integral operator I ; that is,
α
kI (f)k ≤ Ckfk .
α Lq,κq/p(wq/p,w) Lp,κ(w)
Lemma 3.5. Let 0< α < n, 1 < p < n/α, 1/q = 1/p−α/n and wq/p ∈ A .
1
Then if 0< κ < p/q and r > 1−κ , we have
w p/q−κ
kM (f)k ≤ Ckfk .
w Lq,κq/p(wq/p,w) Lq,κq/p(wq/p,w)
10