Some estimates for $\theta$-type Calder\'on-Zygmund operators and linear commutators on certain weighted amalgam spaces

0.3 MB·

by  Hua Wang

θ
Some estimates for -type Calder´on–Zygmund
7
operators and linear commutators on certain
1
0
weighted amalgam spaces
2

n
a Hua Wang ∗
J

4 CollegeofMathematics andEconometrics,HunanUniversity,Changsha410082, P.R.China
2

]
A
Abstract
C
. Inthispaper,wefirstintroducesomenewkindsofweightedamalgam
h
spaces. Then we discuss the strong type and weak type estimates for
t
a a class of Caldero´n–Zygmund type operators Tθ in these new weighted
m spaces. Furthermore, the strong type estimate and endpoint estimate of
[ linear commutators [b,Tθ] formed by b and Tθ are established. Also we
study related problems about two-weight, weak type inequalities for Tθ
1 and [b,Tθ] in the weighted amalgam spaces and give some results.
v MSC(2010): 42B20; 42B35; 46E30; 47B47
8 Keywords: θ-typeCaldero´n–Zygmund operators; commutators; weighted
0
amalgam spaces; Muckenhouptweights; Orlicz spaces.
5
7
0 1 Introduction
.
1
0
Caldero´n–Zygmundsingular integraloperatorsand their generalizationson the
7 Euclidean space Rn have been extensively studied (see [4, 11, 21, 24] for in-
1
: stance). Inparticular,Yabuta[24]introducedcertainθ-typeCaldero´n–Zygmund
v
operators to facilitate his study of certain classes of pseudo-differential opera-
i
X tors. FollowingtheterminologyofYabuta[24],weintroducetheso-calledθ-type
r Caldero´n–Zygmundoperators.
a
Definition 1.1. Let θ be a non-negative, non-decreasing function on R+ =
(0,+∞) with
1 θ(t)
dt<∞. (1.1)
t
Z0
A measurable function K(·,·) on Rn ×Rn\{(x,x) : x ∈ Rn} is said to be a
∗E-mailaddress: [email protected].
1
θ-type kernel if it satisfies
C
(i) K(x,y) ≤ , for any x6=y; (1.2)
|x−y|n
(cid:12) (cid:12) C |x−z|
(cid:12) (cid:12)
(ii) K(x,y)−K(z,y) + K(y,x)−K(y,z) ≤ ·θ , (1.3)
|x−y|n |x−y|
(cid:12) (cid:12) (cid:12) (cid:12) (cid:16) (cid:17)
fo(cid:12)r |x−z|<|x−y(cid:12)|/2.(cid:12) (cid:12)
Definition 1.2. Let T be a linear operator from S(Rn) into its dual S′(Rn).
θ
We say that T is a θ-type Caldero´n–Zygmund operator if
θ
(1) T can be extended to be a bounded linear operator on L2(Rn);
θ
(2) There is a θ-type kernel K(x,y) such that
T f(x):= K(x,y)f(y)dy (1.4)
θ
ZRn
forallf ∈C∞(Rn)andforallx∈/ suppf,whereC∞(Rn)isthespaceconsisting
0 0
of all infinitely differentiable functions on Rn with compact supports.
Note that the classical Caldero´n–Zygmund operator with standard kernel
(see[4,11])isaspecialcaseofθ-typeoperatorT whenθ(t)=tδ with0<δ ≤1.
θ
Definition1.3. Givenalocally integrablefunctionbdefinedonRn, andgiven a
θ-type Caldero´n–Zygmund operator T , the linear commutator [b,T ] generated
θ θ
by b and T is defined for smooth, compactly supported functions f as
θ
[b,T ]f(x):=b(x)·T f(x)−T (bf)(x)
θ θ θ
(1.5)
= b(x)−b(y) K(x,y)f(y)dy.
ZRn
(cid:2) (cid:3)
We firstgivethefollowingweightedresultofT obtainedbyQuekandYang
θ
in [19].
Theorem 1.1 ([19]). Suppose that θ is a non-negative, non-decreasing function
on R+ =(0,+∞) satisfying condition (1.1). Let 1≤p<∞ and w ∈A . Then
p
the θ-type Caldero´n–Zygmund operator T is bounded on Lp(Rn) for p>1, and
θ w
bounded from L1(Rn) into WL1(Rn) for p=1.
w w
Since linear commutator has a greater degree of singularity than the corre-
sponding θ-type Caldero´n–Zygmund operator, we need a slightly stronger con-
dition (1.6) givenbelow. The followingweightedendpoint estimate for commu-
tator [b,T ] of the θ-type Caldero´n–Zygmund operator was established in [26]
θ
under a stronger versionof condition (1.6) assumed on θ, if b∈BMO(Rn) (for
the unweighted case, see [15]). Let us now recall the definition of the space
of BMO(Rn) (see [4, 13]). BMO(Rn) is the Banach function space modulo
constants with the norm k·k defined by
∗
1
kbk :=sup |b(x)−b |dx<∞,
∗ B
|B|
B ZB
2
wherethe supremumistakenoverallballsB inRn andb standsforthe mean
B
value of b over B, that is,
1
b := b(y)dy.
B
|B|
ZB
Theorem 1.2 ([26]). Suppose that θ is a non-negative, non-decreasing function
on R+ =(0,+∞) satisfying (1.1) and
1 θ(t)·|logt|
dt<∞, (1.6)
t
Z0
let w ∈ A and b ∈ BMO(Rn). Then for all λ > 0, there is a constant C > 0
1
independent of f and λ>0 such that
|f(x)|
w x∈Rn : [b,T ](f)(x) >λ ≤C Φ ·w(x)dx,
θ
ZRn (cid:18) λ (cid:19)
(cid:0)(cid:8) (cid:12) (cid:12) (cid:9)(cid:1)
where Φ(t)=t·(1+(cid:12)log+t) and l(cid:12)og+t=max logt,0 .
We equip the n-dimensional Euclidean spa(cid:8)ce Rn w(cid:9)ith the Euclidean norm
|·| and the Lebesgue measure dx. For any r > 0 and y ∈ Rn, let B(y,r) =
x∈Rn :|x−y|<r denotetheopenballcenteredaty withradiusr, B(y,r)c
denoteitscomplementand|B(y,r)|betheLebesguemeasureoftheballB(y,r).
(cid:8) (cid:9)
We also use the notation χ for the characteristic function of B(y,r). Let
B(y,r)
1 ≤ p,q,α ≤ ∞. We define the amalgam space (Lp,Lq)α(Rn) of Lp(Rn) and
Lq(Rn) as the set of all measurable functions f satisfying f ∈ Lp (Rn) and
loc
f <∞, where
(Lp,Lq)α(Rn)
(cid:13) (cid:13)
(cid:13) (cid:13) 1/α−1/p−1/q q 1/q
f :=sup B(y,r) f ·χ dy
(Lp,Lq)α(Rn) B(y,r) Lp(Rn)
(cid:13) (cid:13) r>0(cid:26)ZRnh(cid:12) 1/α−1(cid:12)/p−1/q (cid:13) (cid:13) i (cid:27)
(cid:13) (cid:13) =sup B(y,(cid:12)r) (cid:12) f ·χ(cid:13) (cid:13) ,
r>0 B(y,r) Lp(Rn) Lq(Rn)
with the usual modifica(cid:13)(cid:13)(cid:13)ti(cid:12)(cid:12)on whe(cid:12)(cid:12)n p = ∞ or(cid:13)(cid:13)q = ∞. Th(cid:13)(cid:13)is ama(cid:13)(cid:13)(cid:13)lgam space was
originallyintroduced by Fofanain [9]. As provedin [9] the space(Lp,Lq)α(Rn)
is nontrivialif andonly if p≤α≤q; thus in the remainingofthe paper we will
always assume that this condition p≤α≤q is fulfilled. Note that
• For 1 ≤ p ≤ α ≤ q ≤ ∞, one can easily see that (Lp,Lq)α(Rn) ⊆
(Lp,Lq)(Rn), where (Lp,Lq)(Rn) is the Wiener amalgam space defined
by (see [10, 12] for more information)
q 1/q
(Lp,Lq)(Rn):= f : f = f ·χ dy <∞ ;
(Lp,Lq)(Rn) B(y,1) Lp(Rn)
( (cid:13) (cid:13) (cid:18)ZRnh(cid:13) (cid:13) i (cid:19) )
(cid:13) (cid:13) (cid:13) (cid:13)
3
• If 1 ≤ p < α and q = ∞, then (Lp,Lq)α(Rn) is just the classical Morrey
space Lp,κ(Rn) defined by (with κ=1−p/α, see [16])
1/p
1
Lp,κ(Rn):= f : f = sup |f(x)|pdx <∞ ;
 (cid:13) (cid:13)Lp,κ(Rn) y∈Rn,r>0 |B(y,r)|κ ZB(y,r) ! 
(cid:13) (cid:13)
• If p = α andq = ∞, then (Lp,Lq)α(Rn) reduces to the usual Lebesgue 
space Lα(Rn).
In [7] (see also [6, 8]), Feuto considered a weighted version of the amalgam
space (Lp,Lq)α(w). A weight is any positive measurable function w which is
locally integrable on Rn. Let 1 ≤ p ≤ α ≤ q ≤ ∞ and w be a weight on Rn.
Wedenoteby(Lp,Lq)α(w)theweightedamalgamspace,thespaceofalllocally
integrable functions f satisfying f <∞, where
(Lp,Lq)α(w)
f :=sup w((cid:13)(cid:13)B((cid:13)(cid:13)y,r))1/α−1/p−1/q f ·χ qdy 1/q
(cid:13) (cid:13)(Lp,Lq)α(w) r>0(cid:26)ZRnh (cid:13) B(y,r)(cid:13)Lpwi (cid:27)
(cid:13) (cid:13) =sup w(B(y,r))1/α−1/p−1/q f ·χ(cid:13) (cid:13) ,
r>0 B(y,r) Lpw Lq(Rn)
(cid:13) (cid:13)
(cid:13)(cid:13) (cid:13)(cid:13) (cid:13)(cid:13) (cid:13)(cid:13) (1.7)
with the usual modification when q = ∞ and w(B(y,r)) = w(x)dx is
B(y,r)
the weighted measure of B(y,r). Then for 1 ≤ p ≤ α ≤ q ≤ ∞, we know
R
that (Lp,Lq)α(w) becomes a Banach function space with respect to the norm
k·k . Furthermore, we denote by (WLp,Lq)α(w) the weighted weak
(Lp,Lq)α(w)
amalgam space of all measurable functions f for which (see [7])
q 1/q
f :=sup w(B(y,r))1/α−1/p−1/q f ·χ dy
(cid:13) (cid:13)(WLp,Lq)α(w) r>0(cid:26)ZRnh (cid:13) B(y,r)(cid:13)WLpwi (cid:27)
(cid:13) (cid:13) =sup w(B(y,r))1/α−1/p−1/q f ·χ(cid:13) (cid:13) <∞.
r>0 B(y,r) WLpw Lq(Rn)
(cid:13) (cid:13)
(cid:13) (cid:13) (cid:13) (cid:13) (1.8)
(cid:13) (cid:13) (cid:13) (cid:13)
Note that
• If 1 ≤ p < α and q = ∞, then (Lp,Lq)α(w) is just the weighted Morrey
space Lp,κ(w) defined by (with κ=1−p/α, see [14])
Lp,κ(w)
1/p
1
:= f : f = sup |f(x)|pw(x)dx <∞ ,
 (cid:13) (cid:13)Lp,κ(w) y∈Rn,r>0 w(B(y,r))κ ZB(y,r) ! 
(cid:13) (cid:13)
and (WLp,Lq)α(w) is just the weighted weak Morrey space WLp,κ(w)
 
defined by (with κ=1−p/α)
WLp,κ(w)
1 1/p
:= f : f = sup sup λ· w x∈B(y,r):|f(x)|>λ <∞ ;
(cid:26) (cid:13) (cid:13)WLp,κ(w) y∈Rn,r>0λ>0w(B(y,r))κ/p h (cid:0)(cid:8) (cid:9)(cid:1)i (cid:27)
(cid:13) (cid:13)
4
• If p=α and q =∞, then (Lp,Lq)α(w) reduces to the weighted Lebesgue
spaceLα(Rn),and(WLp,Lq)α(w)reducestotheweightedweakLebesgue
w
space WLα(Rn).
w
Recently,manyworksinclassicalharmonicanalysishavebeendevotedtonorm
inequalitiesinvolvingseveralintegraloperatorsinthe setting ofweightedamal-
gam spaces, see [5, 6, 7, 8, 23]. These results obtained are extensions of well-
knownanaloguesintheweightedLebesguespaces. Themainpurposeofthispa-
peristwofold. Wefirstdefinesomenewkindsofweightedamalgamspaces,and
thenwearegoingtoprovethatθ-typeCaldero´n–Zygmundoperatorandassoci-
ated linear commutator which are known to be bounded in weighted Lebesgue
spaces, are also bounded in these new weighted spaces under appropriate con-
ditions. In addition, we will study two-weight, weak type norm inequalities for
θ-type Caldero´n–Zygmundoperator and associated commutator in the context
of weighted amalgam spaces.
Throughout this paper C will denote a positive constant whose value may
change at each appearance. We also use A≈B to denote the equivalence of A
and B; that is, there exist two positive constants C , C independent of A and
1 2
B such that C A≤B ≤C A.
1 2
2 Statements of the main results
2.1 Notations and preliminaries
A weight w is said to belong to the Muckenhoupt’s class A for 1 < p < ∞, if
p
there exists a constant C >0 such that
1/p 1/p′
1 w(x)dx 1 w(x)−p′/pdx ≤C
|B| |B|
(cid:18) ZB (cid:19) (cid:18) ZB (cid:19)
for every ball B ⊂Rn, where p′ is the dual of p such that 1/p+1/p′ =1. The
class A is defined replacing the above inequality by
1
1
w(x)dx≤C·essinfw(x)
|B|ZB x∈B
for every ball B ⊂ Rn. We also define A = A . For any given ball
∞ 1≤p<∞ p
B ⊂ Rn and λ > 0, we write λB for the ball with the same center as B and
S
radius is λ times that of B. It is well known that if w ∈A with 1≤p<∞(or
p
w∈A ), then w satisfies the doubling condition; that is, for any ball B ⊂Rn,
∞
there exists an absolute constant C >0 such that (see [11])
w(2B)≤Cw(B). (2.1)
When w satisfies this doubling condition (2.1), we denote w ∈ ∆ for brevity.
2
Moreover, if w ∈ A , then for any ball B and any measurable subset E of a
∞
ballB,thereexistsanumberδ >0independentofE andB suchthat(see[11])
δ
w(E) |E|
≤C . (2.2)
w(B) |B|
(cid:18) (cid:19)
5
Given a weight w on Rn, as usual, the weighted Lebesgue space Lp(Rn) for
w
1≤p<∞ is defined as the set of all functions f such that
1/p
p
f := f(x) w(x)dx <∞.
Lpw (cid:18)ZRn (cid:19)
(cid:13) (cid:13) (cid:12) (cid:12)
We also denote by(cid:13)W(cid:13)Lp(Rn)(1 ≤(cid:12)p < ∞(cid:12) ) the weighted weak Lebesgue space
w
consisting of all measurable functions f such that
1/p
f :=supλ· w x∈Rn :|f(x)|>λ <∞.
WLpw λ>0
(cid:13) (cid:13) h (cid:0)(cid:8) (cid:9)(cid:1)i
(cid:13) (cid:13)
We next recall some basic definitions and facts about Orlicz spaces needed
fortheproofofthemainresults. Forfurtherinformationonthesubject,onecan
see[20]. AfunctionAiscalledaYoungfunctionifitiscontinuous,nonnegative,
convex and strictly increasing on [0,+∞) with A(0) = 0 and A(t) → +∞ as
t → +∞. An important example of Young function is A(t) = tp(1+log+t)p
with some 1 ≤ p < ∞. Given a Young function A, we define the A-average of
a function f over a ball B by means of the following Luxemburg norm:
1 |f(x)|
f :=inf λ>0: A dx≤1 .
A,B |B| λ
(cid:26) ZB (cid:18) (cid:19) (cid:27)
(cid:13) (cid:13)
When A(t)=(cid:13)tp(cid:13), 1≤p<∞, it is easy to see that
1/p
1 p
f = f(x) dx ;
A,B |B|
(cid:18) ZB (cid:19)
(cid:13) (cid:13) (cid:12) (cid:12)
that is, the Luxembur(cid:13)g (cid:13)norm coincides w(cid:12)ith t(cid:12)he normalized Lp norm. Given
a Young function A, we use A¯ to denote the complementary Young function
associated to A. Then the following generalized Ho¨lder’s inequality holds for
any given ball B:
1
f(x)·g(x) dx≤2 f g .
|B| A,B A¯,B
ZB
(cid:12) (cid:12) (cid:13) (cid:13) (cid:13) (cid:13)
In particular, when A(t) (cid:12)= t·(1+l(cid:12)og+t), (cid:13)we(cid:13)kno(cid:13)w (cid:13)that its complementary
Young function is A¯(t)≈exp(t)−1. In this situation, we denote
f = f , g = g .
LlogL,B A,B expL,B A¯,B
(cid:13) (cid:13) (cid:13) (cid:13) (cid:13) (cid:13) (cid:13) (cid:13)
So we have (cid:13) (cid:13) (cid:13) (cid:13) (cid:13) (cid:13) (cid:13) (cid:13)
1
f(x)·g(x) dx≤2 f g . (2.3)
|B| LlogL,B expL,B
ZB
(cid:12) (cid:12) (cid:13) (cid:13) (cid:13) (cid:13)
(cid:12) (cid:12) (cid:13) (cid:13) (cid:13) (cid:13)
6
2.2 Weighted amalgam spaces
LetusbeginwiththedefinitionsoftheweightedamalgamspaceswithLebesgue
measure in (1.7) and (1.8) replaced by weighted measure.
Definition 2.1. Let 1 ≤ p ≤ α ≤ q ≤ ∞, and let w,µ be two weights on
Rn. We denote by (Lp,Lq)α(w;µ) the weighted amalgam space, the space of all
locally integrable functions f with finite norm
q 1/q
f :=sup w(B(y,r))1/α−1/p−1/q f ·χ µ(y)dy
(cid:13) (cid:13)(Lp,Lq)α(w;µ) r>0(cid:26)ZRnh (cid:13) B(y,r)(cid:13)Lpwi (cid:27)
(cid:13) (cid:13) =sup w(B(y,r))1/α−1/p−1/q f ·χ(cid:13) (cid:13) <∞,
r>0 B(y,r) Lpw Lqµ
(cid:13) (cid:13)
(cid:13) (cid:13) (cid:13) (cid:13)
with the usual modifica(cid:13)tion when q = ∞. T(cid:13)hen we ca(cid:13)n se(cid:13)e that the space
(Lp,Lq)α(w;µ) equipped with the norm · is a Banach function
(Lp,Lq)α(w;µ)
space.
(cid:13) (cid:13)
(cid:13) (cid:13)
Definition 2.2. Let 1 ≤ p ≤ α ≤ q ≤ ∞, and let w,µ be two weights on
Rn. We denote by (WLp,Lq)α(w;µ) the weighted weak amalgam space of all
measurable functions f for which
1/q
q
f :=sup w(B(y,r))1/α−1/p−1/q f ·χ µ(y)dy
(cid:13) (cid:13)(WLp,Lq)α(w;µ) r>0(cid:26)ZRnh (cid:13) B(y,r)(cid:13)WLpwi (cid:27)
(cid:13) (cid:13) =sup w(B(y,r))1/α−1/p−1/q f ·χ(cid:13) (cid:13) <∞.
r>0 B(y,r) WLpw Lqµ
(cid:13) (cid:13)
(cid:13) (cid:13) (cid:13) (cid:13)
We are going to prov(cid:13)e that θ-type Caldero´n(cid:13)–Zygmund(cid:13)opera(cid:13)tor which is
known to be bounded on weighted Lebesgue spaces, is also bounded on these
new weighted spaces for Muckenhoupt’s weights. Our first two results in this
paper can be formulated as follows.
Theorem 2.1. Let 1<p≤α<q ≤∞ and w ∈A , µ∈∆ . Then the θ-type
p 2
Caldero´n–Zygmund operator T is bounded on (Lp,Lq)α(w;µ).
θ
Theorem 2.2. Let p = 1, 1 ≤ α < q ≤ ∞ and w ∈ A , µ ∈ ∆ . Then
1 2
the θ-type Caldero´n–Zygmund operator T is bounded from (L1,Lq)α(w;µ) into
θ
(WL1,Lq)α(w;µ).
Letθbeanon-negative,non-decreasingfunctiononR+ =(0,+∞)satisfying
conditions (1.1) and (1.6), and let [b,T ] be the commutator formed by T and
θ θ
BMO function b. For the strong type estimate of linear commutator [b,T ] on
θ
the weighted spaces (Lp,Lq)α(w;µ) with 1<p≤α<q, we will prove
Theorem 2.3. Let 1 < p ≤ α < q ≤ ∞ and w ∈ A , µ ∈ ∆ . Assume that θ
p 2
satisfies (1.6) and b∈BMO(Rn), then the linear commutator [b,T ] is bounded
θ
on (Lp,Lq)α(w;µ).
7
To obtain endpoint estimate for the linear commutator [b,T ], we first need
θ
to define the weighted A-average of a function f over a ball B by means of the
weightedLuxemburg norm; that is, given a Young function A and w ∈A , we
∞
define (see [20, 25])
1 |f(x)|
f :=inf σ >0: A ·w(x)dx ≤1 .
A(w),B w(B) σ
(cid:26) ZB (cid:18) (cid:19) (cid:27)
(cid:13) (cid:13)
WhenA(cid:13)(t)(cid:13)=t,this normis denotedby k·k , whenA(t)=t·(1+log+t),
L(w),B
thisnormisalsodenotedbyk·k . ThecomplementaryYoungfunction
LlogL(w),B
oft·(1+log+t)isexp(t)−1withmeanLuxemburgnormdenotedbyk·k .
expL(w),B
For w∈A and for every ball B in Rn, we can also show the weighted version
∞
of (2.3). Namely, the following generalized Ho¨lder’s inequality in the weighted
setting
1
|f(x)·g(x)|w(x)dx ≤C f g (2.4)
w(B) LlogL(w),B expL(w),B
ZB
(cid:13) (cid:13) (cid:13) (cid:13)
is valid (see [25] for instance). Furtherm(cid:13)or(cid:13)e, we now(cid:13)int(cid:13)roduce new weighted
spaces of LlogL type as follows.
Definition 2.3. Let p=1, 1≤α≤q ≤∞, and let w,µ be two weights on Rn.
We denote by (LlogL,Lq)α(w;µ) the weighted amalgam space of LlogL type,
the space of all locally integrable functions f defined on Rn with finite norm
f .
(LlogL,Lq)α(w;µ)
(cid:13) (cid:13)
(cid:13) (cid:13) (LlogL,Lq)α(w;µ):= f : f <∞ ,
(LlogL,Lq)α(w;µ)
n (cid:13) (cid:13) o
where (cid:13) (cid:13)
q 1/q
f :=sup w(B(y,r))1/α−1/q f µ(y)dy
(LlogL,Lq)α(w;µ) LlogL(w),B(y,r)
(cid:13) (cid:13) r>0(cid:26)ZRnh (cid:13) (cid:13) i (cid:27)
(cid:13) (cid:13) =sup w(B(y,r))1/α−1/q f (cid:13) (cid:13) .
r>0 LlogL(w),B(y,r) Lqµ
(cid:13) (cid:13)
(cid:13) (cid:13) (cid:13) (cid:13)
Observethatt≤t·(1+(cid:13)log+t)for allt>0(cid:13). T(cid:13)henforanyba(cid:13)ll B(y,r)⊂Rn
and w ∈ A , we have f ≤ f by definition, i.e.,
∞ L(w),B(y,r) LlogL(w),B(y,r)
the inequality
(cid:13) (cid:13) (cid:13) (cid:13)
(cid:13) (cid:13) (cid:13) (cid:13)
1
f = |f(x)|·w(x)dx≤ f (2.5)
L(w),B(y,r) w(B(y,r)) LlogL(w),B(y,r)
ZB(y,r)
(cid:13) (cid:13) (cid:13) (cid:13)
h(cid:13)old(cid:13)s for any ball B(y,r)⊂Rn. Hence, for 1≤α≤q ≤(cid:13) ∞(cid:13), we can further see
the following inclusion:
(LlogL,Lq)α(w;µ)⊂(L1,Lq)α(w;µ).
Fortheendpointcase,wewillprovethefollowingweaktypeLlogLestimate
of the linear commutator [b,T ] in our weighted amalgam spaces.
θ
8
Theorem 2.4. Let p = 1, 1 ≤ α < q ≤ ∞ and w ∈ A , µ ∈ ∆ . Assume
1 2
that θ satisfies (1.6) and b∈BMO(Rn), then for any given λ>0 and any ball
B(y,r)⊂ Rn with y ∈Rn, r >0, there exists a constant C >0 independent of
f, B(y,r) and λ>0 such that
w(B(y,r))1/α−1−1/q ·w x∈B(y,r): [b,T ](f)(x) >λ
θ
Lqµ
(cid:13)(cid:13)(cid:13)≤C· Φ |f| (cid:0)(cid:8) , (cid:12)(cid:12) (cid:12)(cid:12) (cid:9)(cid:1)(cid:13)(cid:13)(cid:13)
λ
(cid:13) (cid:18) (cid:19)(cid:13)(LlogL,Lq)α(w;µ)
(cid:13) (cid:13)
(cid:13) (cid:13)
where Φ(t) =(cid:13)t·(1+log(cid:13)+t) and the norm k·kLqµ is taken with respect to the
variable y, i.e.,
w(B(y,r))1/α−1−1/q ·w x∈B(y,r): [b,T ](f)(x) >λ
θ
Lqµ
=(cid:13)(cid:13)(cid:13) w(B(y,r))1/α−1−1(cid:0)/(cid:8)q ·w x∈B(y(cid:12)(cid:12),r): [b,T ]((cid:12)(cid:12)f)(x(cid:9))(cid:1)>(cid:13)(cid:13)(cid:13) λ qµ(y)dy 1/q.
θ
(cid:26)ZRn(cid:20) (cid:21) (cid:27)
(cid:0)(cid:8) (cid:12) (cid:12) (cid:9)(cid:1)
Remark 2.1. From the above definitions and(cid:12)Theorem 2.(cid:12)4, we can roughly
say that the linear commutator [b,T ] is bounded from (LlogL,Lq)α(w;µ) into
θ
(WL1,Lq)α(w;µ) whenever 1≤α<q ≤∞, w ∈A and µ∈∆ .
1 2
3 Proofs of Theorems 2.1 and 2.2
Proof of Theorem 2.1. Let 1 < p ≤ α < q ≤ ∞ and f ∈ (Lp,Lq)α(w;µ) with
w∈A and µ∈∆ . We fix y ∈Rn and r>0, and set B =B(y,r) for the ball
p 2
centered at y and of radius r, 2B =B(y,2r). We represent f as
f =f ·χ +f ·χ :=f +f ;
2B (2B)c 1 2
by the linearity of the θ-type Caldero´n–Zygmundoperator T , we write
θ
w(B(y,r))1/α−1/p−1/q T (f)·χ
θ B(y,r) Lpw
(cid:13) (cid:13) 1/p
=w(B(y,r))1/α−1/p−(cid:13)1/q T (cid:13)(f)(x) pw(x)dx
θ
(cid:18)ZB(y,r) (cid:19)
(cid:12) (cid:12)
(cid:12) (cid:12) 1/p
≤w(B(y,r))1/α−1/p−1/q T (f )(x) pw(x)dx
θ 1
(cid:18)ZB(y,r) (cid:19)
(cid:12) (cid:12)
(cid:12) (cid:12) 1/p
+w(B(y,r))1/α−1/p−1/q T (f )(x) pw(x)dx
θ 2
(cid:18)ZB(y,r) (cid:19)
(cid:12) (cid:12)
:=I1(y,r)+I2(y,r). (cid:12) (cid:12) (3.1)
9
Below we will give the estimates of I (y,r) and I (y,r), respectively. By the
1 2
weighted Lp boundedness of T (see Theorem 1.1), we have
θ
I (y,r)≤w(B(y,r))1/α−1/p−1/q T (f )
1 θ 1 Lpw
(cid:13) (cid:13) 1/p
≤C·w(B(y,r))1/α−1/p−(cid:13)1/q (cid:13) |f(x)|pw(x)dx
(cid:18)ZB(y,2r) (cid:19)
=C·w(B(y,2r))1/α−1/p−1/q f ·χ
B(y,2r) Lpw
w(B(y,r))1/α−1/p−1/q (cid:13) (cid:13)
× . (cid:13) (cid:13) (3.2)
w(B(y,2r))1/α−1/p−1/q
Moreover,since 1/α−1/p−1/q <0 and w ∈A with 1<p<∞, then by the
p
inequality (2.1), we obtain
w(B(y,r))1/α−1/p−1/q
≤C. (3.3)
w(B(y,2r))1/α−1/p−1/q
Substituting the above inequality (3.3) into (3.2), we thus obtain
I (y,r)≤C·w(B(y,2r))1/α−1/p−1/q f ·χ . (3.4)
1 B(y,2r) Lpw
As for the term I (y,r), it is clear that when x ∈(cid:13)B(y,r) and(cid:13)z ∈ (B(y,2r))c,
2 (cid:13) (cid:13)
we get |x−z|≈|y−z|. We then decompose Rn into a geometrically increasing
sequence of concentric balls, and deduce the following pointwise estimate:
|f (z)| |f(z)|
2
T (f )(x) ≤C dz ≤C dz
θ 2 ZRn |x−z|n ZB(y,2r)c |y−z|n
(cid:12) (cid:12) ∞
(cid:12) (cid:12) |f(z)|
=C dz
|y−z|n
j=1ZB(y,2j+1r)\B(y,2jr)
X
∞
1
≤C |f(z)|dz. (3.5)
|B(y,2j+1r)|
j=1 ZB(y,2j+1r)
X
From this estimate (3.5), it follows that
∞
1
I (y,r)≤C·w(B(y,r))1/α−1/q |f(z)|dz.
2 |B(y,2j+1r)|
j=1 ZB(y,2j+1r)
X
By using Ho¨lder’s inequality and A condition on w, we get
p
1
|f(z)|dz
|B(y,2j+1r)|
ZB(y,2j+1r)
1/p 1/p′
≤ 1 |f(z)|pw(z)dz w(z)−p′/pdz
|B(y,2j+1r)|
(cid:18)ZB(y,2j+1r) (cid:19) ZB(y,2j+1r) !
1/p
≤C |f(z)|pw(z)dz ·w B(y,2j+1r) −1/p.
(cid:18)ZB(y,2j+1r) (cid:19)
(cid:0) (cid:1)
10