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Hindawi Publishing Corporation
Journal of Inequalities and Applications
Volume 2010, Article ID 643948, 41 pages
doi:10.1155/2010/643948
Research Article
Boundedness of Littlewood-Paley Operators
Associated with Gauss Measures
Liguang Liu and Dachun Yang
School of Mathematical Sciences, Beijing Normal University, Laboratory of Mathematics and
Complex Systems, Ministry of Education, Beijing 100875, China
Correspondence should be addressed to Dachun Yang,
Received 16 December 2009; Accepted 17 March 2010
Academic Editor: Shusen Ding
Copyright q 2010 L. Liu and D. Yang. 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.
Modeled on the Gauss measure, the authors introduce the locally doubling measure metric space
X,d,μ
ρ
, which means that the set X is endowed with a metric d and a locally doubling regular
Borel measure μ satisfying doubling and reverse doubling conditions on admissible balls defined
via the metric d and certain admissible function ρ. The authors then construct an approximation
of the identity on X,d,μ
ρ
, which further induces a Calder
´
on reproducing formula in L
p
X for
p ∈ 1, ∞. Using this Calder
´
on reproducing formula and a locally variant of the vector-valued
singular integral theory, the authors characterize the space L
p
X for p ∈ 1, ∞ in terms of the
Littlewood-Paley g-function which is defined via the constructed approximation of the identity.
Moreover, the authors also establish the Fefferman-Stein vector-valued maximal inequality for
the local Hardy-Littlewood maximal function on X,d,μ
ρ
. All results in this paper can apply
to various settings including the Gauss measure metric spaces with certain admissible functions
related to the Ornstein-Uhlenbeck operator, and Euclidean spaces and nilpotent Lie groups of
polynomial growth with certain admissible functions related to Schr
¨
odinger operators.
1. Introduction
The Littlewood-Paley theory on R
n
nowadays becomes a very important tool in harmonic
analysis, partial differential equations, and other related fields. Especially, t he extent to
which the Littlewood-Paley theory characterizes function spaces is very remarkable; see, for
example, Stein 1, Frazier, et al. 2, and Grafakos 3, 4. Moreover, Han and Sawyer 5
established a Littlewood-Paley theory essentially on the Ahlfors 1-regular metric measure
space with a quasimetric, which means that the measure of any ball is comparable with
its radius. This theory was further generalized to the RD-space in 6, namely, a space of
homogeneous type in the sense of Coifman and Weiss 7, 8 with an additional property that
2 Journal of Inequalities and Applications
the measure satisfies the reverse doubling condition. Tolsa 9 established a Littlewood-Paley
theory with the nondoubling measure μ on R
n
, which means that μ is a Radon measure on R
n
and satisfies that μBx, r ≤ Cr
d
for all x ∈ R
n
, r>0, and some fixed d ∈ 0,n. Furthermore,
these Littlewood-Paley theories were used to establish the corresponding Besov and Triebel-
Lizorkin spaces on these different underlying spaces; see 5, 6, 10.
Let R
n
, |·|,dγ be the Gauss measure metric space, namely, the n-dimensional
Euclidean space R
n
endowed with the Euclidean norm |·|and the Gauss measure dγx ≡
π
−n/2
e
−|x|
2
dx for all x ∈ R
n
. Such an underlying space naturally appears in the study of
the Ornstein-Uhlenbeck operator; see, for example, 11–18. In particular, via introducing
some local BMO γ space and Hardy space H
1
γ associated to admissible balls defined
via the Euclidean metric and the admissible function ρx ≡ min{1, 1/|x|} for x ∈ R
n
,
Mauceri and Meda 12 developed a theory of singular integrals on R
n
, |·|,dγ
ρ
, which plays
for the Ornstein-Uhlenbeck operator the same role as that the theory of classical Calder
´
on-
Zygmund operators plays for the Laplacian on classical Euclidean spaces. The results of 12
are further generalized to some kind of nondoubling measure metric spaces by Carbonaro et
al. in 18, 19.
It is well known that the Gauss measure metric space is beyond the space of
homogeneous type in the sense of Coifman and Weiss, a fortiori, the RD-space. To be precise,
the Gauss measure is known to be only locally doubling see 12. In this paper, modeled on
the Gauss measure, we introduce the locally doubling measure metric space X,d,μ
ρ
, which
means that the set X is endowed with a metric d and a locally doubling regular Borel measure
μ satisfying the doubling and reverse doubling conditions on admissible balls defined via
the metric d and certain admissible function ρ. An interesting phenomenon is that even in
such a weak setting, we are able to construct an approximation of the identity on X,d,μ
ρ
,
which further induces a Calder
´
on reproducing formula in L
p
X for p ∈ 1, ∞.Usingthis
Calder
´
on reproducing formula and a locally variant of the vector-valued singular integral
theory, we then characterize the space L
p
X for p ∈ 1, ∞ in terms of the Littlewood-Paley
g-function which is defined by the aforementioned constructed approximation of the identity.
As a byproduct, we establish the Fefferman-Stein vector-valued maximal inequality for the
local Hardy-Littlewood maximal function on X,d,μ
ρ
, which together with the Calder
´
on
reproducing formula paves the way for further developing a theory of local Besov and
Triebel-Lizorkin spaces on X,d,μ
ρ
.
To be precise, motivated by 12,inSection 2, we introduce locally doubling measure
metric space X,d,μ
ρ
;seeDefinition 2.1 below. The reasonabilities of Definition 2.1 are given
by Propositions 2.3 and 2.5. Some geometric properties of these spaces are also presented in
Section 2.
To develop a Littlewood-Paley theory on the space X,d,μ
ρ
, one of the main
difficulties is the construction of appropriate approximations of the identity. In Section 3,by
subtly modifying Coifman’s idea in 20see 3.2 through 3.4 below, for any given
0
∈ Z,
we construct an approximation of the identity, {S
k
}
∞
k
0
, associated to ρ;seeProposition 3.2
below. Indeed, we not only modify the operators appearing in the construction of Coifman to
the setting associated with the given admissible function ρ, but also use an adjoint operator in
our construction as in Tolsa 9. Some basic estimates on such approximations of the identity
are given in Lemma 3.4 and Proposition 3.5 below. We remark that, although the Gauss
measure is a nondoubling measure considered by Tolsa 9, due to its advantage-locally
doubling property, the construction of the corresponding approximation of the identity
here does not appeal to the complicated constructions of some special doubling cubes and
associated “dyadic” cubes as in 9.
Journal of Inequalities and Applications 3
In Section 4, invoking some ideas of 3, 7, 11, we establish the L
p
X-boundedness
for p ∈ 1, ∞ and weak-1, 1 estimate of local vector-valued singular integral operators on
X,d,μ
ρ
;seeTheorem 4.1 below. As a consequence, in Theorem 4.4 below, we also obtain the
Fefferman-Stein vector-valued maximal function inequality with respect to the noncentered
local Hardy-Littlewood maximal operator see 2.20.
The existence of the approximation of the identity guarantees that we obtain some
Calder
´
on reproducing formulae in L
p
X for p ∈ 1, ∞ in 5.2 and Corollary 5.4,byusing
the methods developed in 20. Applying such formula, we then establish the Littlewood-
Paley characterization for L
p
X with p ∈ 1, ∞ on X,d,μ
ρ
in terms of Littlewood-Paley
g-function; see Theorem 5.6 below.
Some typical examples of locally doubling measure metric spaces in Definition 2.1 are
presented in Section 6. These typical examples include the aforementioned Gauss measure
metric spaces with certain admissible functions related to the Ornstein-Uhlenbeck operator,
and Euclidean spaces and nilpotent Lie groups of polynomial growth with certain admissible
functions related to Schr
¨
odinger operators; see 21–25. All results, especially, Theorems 4.4
and 5.6, are new even for these typical examples.
It should be pointed out that all results in Section 2 through Section 4 are exempt from
using the reverse locally doubling condition 2.3;seeRemark 2.2iii below.
We make the following conventions on notation. Let N ≡{1, 2, }. For any p ∈ 1, ∞,
denote by p
the conjugate index, namely, 1/p 1/p
1. In general, we use B to denote
a Banach space, and B
a
with a>0 to denote a collection of admissible balls. For any set
E ⊂X, denote by χ
E
the characteristic function of E,andby#E the cardinality of E,and
set E
≡X\E. For any operator T, denote by T
∗
its dual operator. For any a, b ∈ R,set
a ∧ b ≡ min{a, b} and a ∨ b ≡ max{a, b}. Denote by C a positive constant independent of main
parameters involved, which may vary at different occurrences. Constants with subscripts do
not change through the whole paper. We use f g and f g to denote f ≤ Cg and f ≥ Cg,
respectively. If f g f, we then write f ∼ g.
2. Locally Doubling Measure Metric Spaces
Let X,d,μ be a set X endowed with a regular Borel measure μ such that all balls defined by
the metric d have finite and positive measures. Here, the regular Borel measure μ means that
open sets are measurable and every set is contained in a Borel set with the same measure; see,
for example, 26. For any x ∈Xand r>0, set Bx, r ≡{y ∈X: dx, y <r}. For a ball
B ⊂X,weusec
B
and r
B
to denote its center and radius, respectively, and for κ>0, we set
κB ≡ Bc
B
,κr
B
. Now we introduce the precise definition of locally doubling measure metric
spaces.
Definition 2.1. A function ρ : X→0, ∞ is called admissible if for any given τ ∈ 0, ∞, there
exists a constant Θ
τ
≥ 1 such that for all x, y ∈Xsatisfying dx, y ≤ τρx,
Θ
τ
−1
ρ
y
≤ ρ
x
≤ Θ
τ
ρ
y
. 2.1
For each a>0, denote by B
a
the set of all balls B ⊂Xsuch that r
B
≤ aρc
B
. Balls in B
a
are
referred to as admissible balls with scale a. The triple X,d,μ
ρ
is called a locally doubling
4 Journal of Inequalities and Applications
metric space associated with admissible function ρ if for every a>0, there exist constants
D
a
,K
a
,R
a
∈ 1, ∞ such that for all B ∈B
a
,
μ
2B
≤ D
a
μ
B
locally doubling condition
, 2.2
and
μ
K
a
B
≥ R
a
μ
B
locally reverse doubling condition
. 2.3
Remark 2.2. i Another notion of admissible functions was introduced in 25 in the following
way: a function ρ : X→0, ∞ is called admissible if there exist positive constants C and ν
such that for all x, y ∈X,
ρ
y
≤ C
ρ
x
1/1ν
ρ
x
d
x, y
ν/1ν
. 2.4
By 25, Lemma 2.1,anyρ satisfying 2.4 also satisfies 2.1, while the converse may be not
true; see Example 6.5 below.
ii Obviously, any constant function is admissible. When ρ ≡ 1, if {D
a
}
a>0
has upper
bound, then X,d,μ
ρ
is the space of homogeneous type in the sense of Coifman and Weiss
7, 8; furthermore, if {K
a
}
a>0
has upper bound and {R
a
}
a>0
has lower bound away from 1,
then X,d,μ
ρ
is just the RD-space in 6. Conversely, any RD-space is obviously a locally
doubling measure metric space with ρ ≡ 1.
iii We remark that the locally reverse doubling condition 2.3 is a mild requirement
of the underlying space. Indeed, if a>0andX is path connected on all balls contained in B
2a
and 2.2 holds for certain a>0, then 2.3 holds; see Proposition 2.3vi below. Moreover,
2.3 is required only in Section 5, that is, all results in Section 2 through Section 4 are true
by only assuming that ρ is an admissible function satisfying 2.1 and that X,d,μ
ρ
satisfies
2.2.
iv Let d be a quasimetric, which means that there exists A
0
≥ 1 such that for all
x, y, z ∈X,dx, y ≤ A
0
dx, zdz, y. Recall that Mac
´
ıas and Segovia 27, Theorem 2
proved that there exists an equivalent quasimetric
d such that all balls corresponding to
d are
open in the topology induced by
d, and there exist constants
A
0
> 0andθ ∈ 0, 1 such that
for all x, y, z ∈X,
d
x, z
−
d
y, z
≤
A
0
d
x, y
θ
d
x, z
d
y, z
1−θ
. 2.5
If the metric d in Definition 2.1 is replaced by
d, then all results in this paper have
corresponding generalization on the space X,
d, μ
ρ
. To simplify the presentation, we always
assume d to be a metric in this paper.
Proposition 2.3. Fix a ∈ 0, ∞. Then the following hold:
i the condition 2.2 is equivalent to the following: there exist K>1 and
D
a
> 1 such that
for all B ∈B
2/Ka
, μKB ≤
D
a
μB;
Journal of Inequalities and Applications 5
ii the condition 2.2 is equivalent to the following: there exist C
a
> 1 and n
a
> 0,which
depend on a, such that for all λ ∈ 1, ∞ and λB ∈B
2a
, μλB ≤ C
a
λ
n
a
μB;
iii the following two statements are equivalent:
a there exists R
a
> 1 such that for all B ∈B
a
, μ2B ≥ R
a
μB;
b there exist K
1
∈ 1, 2 and
R
a
> 1 such that μK
1
B ≥
R
a
μB for all B ∈B
2/K
1
a
;
iv if 2.3 holds, then there exist
C
a
∈ 0, 1 and κ
a
> 0 such that for all λ>1 and λB ∈B
aK
a
,
μλB ≥
C
a
λ
κ
a
μB;
v if 2.3 holds, then K
a
B \ B
/
∅ for all B ∈B
a
;
vi if there exists a
0
> 1 such that a
0
B \ B
/
∅ for all B ∈B
2a
, and 2.2 holds for all B ∈B
a
with a ≡ a/21 4a
0
Θ
2a
0
a
, then for any given a
1
>a
0
, there exists a positive constant
C depending on a
0
and a such that for all B ∈B
a
, μa
1
B ≥
CμB.
Proof. The sufficiency of i follows from letting K 2. To see its necessity, we consider K ∈
1, 2 and K ∈ 2, ∞, respectively. When K ∈ 1, 2, there exists a unique N ∈ N such that
K
N
< 2 ≤ K
N1
, which implies t hat for all B ∈B
a
,
μ
2B
μ
K
N1
2
K
N1
B
≤
D
a
N1
μ
2
K
N1
B
≤
D
a
1log
2
K
μ
B
. 2.6
When K ∈ 2, ∞, for any B ∈B
a
, we have 2/KB ∈B
2/Ka
and μ2B ≤
D
a
μ2/KB ≤
D
a
μB,thus,2.2 holds. Therefore, we obtain i.
Nowweassume2.2 and prove the sufficiency of ii. For any λ>1, choose N ∈ N
such that 2
N−1
<λ≤ 2
N
. Then, for all λB ∈B
2a
, we have λ/2
j
B ∈B
a
for all 1 ≤ j ≤ N;
we therefore apply 2.2 N times and obtain μλB ≤ D
a
N
μλ/2
N
B ≤ D
a
λ
n
a
μB, where
n
a
≡ log
2
D
a
. The necessity of ii is obvious.
Next we prove iii.Ifa holds, then b follows from setting K
1
2. Conversely, if
b holds, then for any B ∈B
a
, we have 2/K
1
B ∈B
2/K
1
a
and
μ
2B
μ
K
1
2
K
1
B
≥
R
a
μ
2
K
1
B
≥
R
a
μ
B
, 2.7
which implies a.
To prove iv, for any λ>1, there exists a unique N ∈ N such that K
a
N−1
<λ≤
K
a
N
. This combined with the fact that λ/K
a
B ∈B
a
implies that
μ
λB
μ
K
a
N−1
λ
K
a
N−1
B
≥
R
a
N−1
μ
λ
K
a
N−1
B
≥
R
a
log
K
a
λ−1
μ
B
≡
C
a
λ
κ
a
μ
B
,
2.8
where
C
a
≡ R
a
−1
and κ
a
≡ log
K
a
R
a
.Thus,iv holds.
Notice that v is obvious. To show vi, without loss of generality, we may assume
that a
1
∈ a
0
, 2a
0
.Setσ ≡ a
1
− a
0
/1 a
0
. Observe that 0 <σ<1. Thus, for any B ∈B
a
,
we have 1 σB ∈B
2a
and a
0
1 σB \ 1 σB
/
∅. Choose y ∈ a
0
1 σB \ 1 σB.Itis
6 Journal of Inequalities and Applications
easy to check that By, σr
B
∩ B ∅ and By, σr
B
⊂ a
1
B ⊂ By, σ 2a
0
1 σr
B
.Notice
that r
B
≤ aρc
B
≤ aΘ
2a
0
a
ρy and By, σ 2a
0
1 σr
B
∈B
2a
. This combined with 2.2
and i of Proposition 2.3 yields that
μ
a
1
B
≥ μ
B
μ
B
y, σr
B
≥ μ
B
C
a
−1
σ
σ 2a
0
1 σ
n
a
μ
B
y,
σ 2a
0
1 σ
r
B
≥ μ
B
C
a
−1
σ
σ 2a
0
1 σ
n
a
μ
a
1
B
,
2.9
which further implies that μa
1
B ≥
CμB with
C ≡{1 − C
a
−1
σ/σ 2a
0
1 σ
n
a
}
−1
> 1.
This finishes the proof of vi, and hence the proof of Proposition 2.3.
Remark 2.4. i By Proposition 2.3i, there is no essential difference whether we define the
locally doubling condition 2.2 by using 2B or KB for some constant K>0.
ii The assumption K
1
∈ 1, 2 in b of Proposition 2.3iii cannot be replaced by
K
1
∈ 1, ∞;seeProposition 2.5 below. Therefore, in Definition 2.1, it is more reasonable to
require 2.3 rather than a of Proposition 2.3iii.
In the following Proposition 2.5, we temporarily consider the Gauss measure space
R
n
, |·|,γ
ρ
, where ρ is given by ρx ≡ min{1, 1/|x|} and dγx ≡ π
−n/2
e
−|x|
2
dx for all x ∈ R
n
.
In this case, for any ball B centered at c
B
andisofradiusr
B
, we have B ≡{x ∈ R
n
: |x − c
B
| <
r
B
}, and moreover, B ∈B
a
if and only if r
B
≤ aρc
B
;see12.
Proposition 2.5. Let a ∈ 0, ∞ and R
n
, |·|,γ
ρ
be the Gauss measure space. Then,
a there exist positive constants K
a
> 1 and C
a
> 1, which depend on a, such that for all
B ∈B
a
, γK
a
B ≥ C
a
γB;
b there exists a sequence of balls, {B
j
}
j∈N
⊂B
a
, such that lim
j →∞
γ2B
j
/γB
j
1.
Proof. Recall that for all B ∈B
a
and x ∈ B, it was proved in 12, Proposition 2.1,thate
−2a−a
2
≤
e
|c
B
|
2
−|x|
2
≤ e
2a
. From this, it follows that for any K
a
> 0,
γ
B
B
π
−n/2
e
−|x|
2
dx ≤ π
−n/2
e
−|c
B
|
2
2a
|
B
|
,
γ
K
a
B
K
a
B
π
−n/2
e
−|x|
2
dx ≥ π
−n/2
e
−|c
B
|
2
−2a−a
2
K
a
n
|
B
|
,
2.10
where and in what follows, we denote by |B| the Lebesgue measure of the ball B.Thus,
γK
a
B ≥ K
a
n
e
−4a−a
2
γB. Hence, a holds by choosing K
a
>e
4aa
2
/n
.
Journal of Inequalities and Applications 7
To show b, for simplicity, we may assume n 1. Consider the ball B
y
≡ By, e
−y
,
where y ≥ 1 such that e
−y
≤ a/y.Thus,B
y
∈B
a
for any such chosen y. A simple calculation
yields that lim
y →∞
γB
y
0. Therefore, using the L
-Hospital rule, we obtain
lim
y →∞
γ
2B
y
γ
B
y
lim
y →∞
y2e
−y
y−2e
−y
e
−|x|
2
dx
ye
−y
y−e
−y
e
−|x|
2
dx
lim
y →∞
1 − 2e
−y
e
−y2e
−y
2
−
1 2e
−y
e
−y−2e
−y
2
1 − e
−y
e
−ye
−y
2
−
1 e
−y
e
−y−e
−y
2
lim
y →∞
e
−3e
−2y
−2ye
−y
1 − 2e
−y
−
1 2e
−y
e
8ye
−y
1 − e
−y
−
1 e
−y
e
4ye
−y
1,
2.11
which implies the desired result of b. This finishes the proof of Proposition 2.5.
Next we present some properties concerning the underlying space X,d,μ
ρ
. In what
follows, for any x,y ∈Xand δ>0, set V
δ
x ≡ μBx, δ and V x, y ≡ μBx, dx, y.
Proposition 2.6. Let τ>0, η>0, a>0, and B ∈B
a
. Then the following hold:
a for any given τ
∈ 0,τ,ifx, y ∈Xsatisfy dx, y ≤ τ
ρx,thendx, y ≤ τ
Θ
τ
ρy,
V
τ
ρx
x
∼ V
τ
ρy
y
∼ V
τ
ρy
x
∼ V
τ
ρx
y
, 2.12
and V x, y ∼ V y, x with equivalent constants depending only on τ;
b for all x, y ∈Xsatisfying dx, y ≤ ηρx,
V
τρx
x
V
x, y
∼ V
τρy
y
V
x, y
∼ μ
B
x, τρ
x
d
x, y
, 2.13
with equivalent constants depending on η and τ;
c
dz,x<r
dz, x
a
1/V z, x dμz ≤ Cr
a
uniformly in x ∈Xand r ∈ 0,τρx;
d for any ball B
satisfying B
∩ B
/
∅ and r
B
≤ τr
B
, B
∈B
τaΘ
1τa
;
e there exists a positive constant D
a,τ
depending only on a and τ such that if B
∩ B
/
∅ and
r
B
≤ τr
B
,thenμB
≤ D
a,τ
μB.
Proof. We first show a. For all τ
∈ 0,τ,ifdx, y ≤ τ
ρx, then dx, y ≤ τΘ
τ
ρy by 2.1.
Since
B
x, τ
ρ
x
⊂ B
y, 2τ
ρ
x
⊂ B
y, 2τ
Θ
τ
ρ
y
, 2.14
by 2.2,weobtainV
τ
ρx
x ≤ D
Θ
τ
V
τ
ρy
y. A similar argument together with 2.1 and
2.2 shows the rest estimates of a as well b. The details are omitted.
8 Journal of Inequalities and Applications
To prove c,bya and 2.2,weobtain
dz,x<r
d
z, x
a
V
z, x
dμ
z
∼
dz,x<r
d
z, x
a
μ
B
x, d
z, x
dμ
z
≤
∞
j0
2
−j−1
r≤dz,x<2
−j
r
2
−j
r
a
μ
B
x, 2
−j−1
r
dμ
z
≤
∞
j0
2
−ja
D
τ
r
a
r
a
,
2.15
which implies c.
To see d,byB ∩ B
/
∅ and r
B
≤ τr
B
, we have dc
B
,c
B
<r
B
r
B
< 1 τr
B
, which
combined with 2.1 and the fact B ∈B
a
implies that
r
B
≤ τr
B
≤ τaρ
c
B
≤ τaΘ
1τa
ρ
c
B
. 2.16
Thus, d holds.
To show e,noticethatB
⊂ Bc
B
, 2τ 1r
B
∈B
2τ1a
. Choose N ∈ N such that
2
N−1
< 2τ 1 ≤ 2
N
. Then, by 2.2,weobtainμB
≤ μ2
N
B ≤ D
2τ1a
N
μB, which implies
e by setting D
a,τ
≡ D
2τ1a
1log
2
2τ1
. This finishes the proof of Proposition 2.6.
A geometry covering lemma on X,d,μ
ρ
is as follows.
Lemma 2.7. Let ρ be an admissible function. For any λ>0, there exists a sequence of balls,
{Bx
j
,λρx
j
}
j
, such that
i X
j
B
j
,whereB
j
≡ Bx
j
,λρx
j
;
ii the balls {
B
j
}
j
are pairwise disjoint, where
B
j
≡ Bx
j
, Θ
λ
2
1
−1
λρx
j
;
iii for any τ>0, there exists a positive constant M depending on τ and λ such that any point
x ∈Xbelongs to no more than M balls of {τB
j
}
j
.
Proof. Let I be the maximal set of balls,
B
j
≡ Bx
j
, Θ
λ
2
1
−1
λρx
j
⊂X, such that for all
k
/
j,
B
j
∩
B
k
∅. The existence of such a set is guaranteed by the Zorn lemma. We claim that
I is at most countable.
Indeed, we choose x
0
∈X,andsetX
N
≡ Bx
0
,Nρx
0
and J
N
≡{j :
B
j
∩X
N
/
∅}.
For any j ∈ J
N
, denote by w
j
an arbitrary point in
B
j
∩X
N
.From2.1, it follows that ρx
j
∼
ρw
j
∼ ρx
0
with constants depending only on N and λ;thus,forallz ∈
B
j
,
d
z, x
0
≤ d
z, x
j
d
x
j
,w
j
d
w
j
,x
0
≤ C
λ,N
ρ
x
0
, 2.17
Journal of Inequalities and Applications 9
for some positive constant C
λ,N
. This implies that
j∈J
N
B
j
⊂ Bx
0
,C
λ,N
ρx
0
. Likewise, there
exists a positive constant
C
λ,N
such that for all j ∈ J
N
, Bx
0
,C
λ,N
ρx
0
⊂
C
λ,N
B
j
. By this and
2.2,weobtain
#
J
N
μ
B
x
0
,C
λ,N
ρ
x
0
j∈J
N
μ
B
j
∼ μ
⎛
⎝
j∈J
N
B
j
⎞
⎠
μ
B
x
0
,C
λ,N
ρ
x
0
, 2.18
and hence #J
N
1. This combined with the fact that X
∞
N1
X
N
implies the claim.
For any z ∈X, by the maximal property of I, there exists some j such that
B
z,
Θ
λ
2
1
−1
λρ
z
∩ B
x
j
,
Θ
λ
2
1
−1
λρ
x
j
/
∅, 2.19
which combined with 2.1 implies that ρz ≤ Θ
λ
2
ρx
j
and dz, x
j
<λρx
j
. This proves
i.
For any z ∈X,setJz ≡{j : z ∈ τB
j
}.By2.1, ρx
j
∼ ρz for all j ∈ Jz. Then
by an argument similar to the proof for the above claim, we obtain iii, which completes the
proof of Lemma 2.7.
For any a>0, we consider the noncentered local Hardy-Littlewood maximal operator M
a
on X,d,μ
ρ
, which is defined by setting, for all locally integrable functions f and x ∈X,
M
a
f
x
≡ sup
B∈B
a
x
1
μ
B
B
f
y
dμ
y
, 2.20
where B
a
x is the collection of balls B ∈B
a
containing x. Observe that if X,d,μ
ρ
is the
Gauss measure metric space and ρx ≡ min{1, 1/|x|}, then 2.20 is exactly the noncentered
local Hardy-Littlewood maximal function introduced in 12, 3.1;seealso18, 7.1.
Theorem 2.8. i For any a>0, the operator M
a
in 2.20 is of weak type 1, 1 and bounded on
L
p
X for p ∈ 1, ∞.
ii For any locally integrable function f and almost all x ∈X,
lim
r → 0
1
μ
B
x, r
Bx,r
f
y
− f
x
dμ
y
0. 2.21
Proof. A similar argument as in 26, Theorem 2.2 together with 2.2 shows i. Following
the procedure in 26, Theorem 1.8, we obtain that for almost all x ∈X,
lim
r → 0
1
μ
B
x, r
Bx,r
f
y
dμ
y
f
x
, 2.22
which together with an argument similar to that of the Euclidean case see 28 yields ii.
This finishes the proof of Theorem 2.8.
10 Journal of Inequalities and Applications
3. Approximations of the Identity
Motivated by 6, 20, we introduce the f ollowing inhomogeneous approximation of the
identity on the locally doubling measure metric space X,d,μ
ρ
.
Definition 3.1. Let
0
∈ Z. A sequence of bounded linear operators, {S
k
}
∞
k
0
,onL
2
X is called
an
0
-approximation of the identity on X,d,μ
ρ
for short,
0
-AOTI if there exist positive
constants C
1
and C
2
may depend on
0
such that for all k ≥
0
and all x, x
, y and y
∈X,
S
k
x, y, the integral kernel of S
k
, is a measurable function from X×Xto C satisfying that
i S
k
x, y0ifdx, y ≥ C
1
2
−k
ρx ∧ ρy and |S
k
x, y|≤C
2
1/V
2
−k
ρx
x
V
2
−k
ρy
y;
ii |S
k
x, y − S
k
x
,y|≤C
2
dx, x
/2
−k
ρx1/V
2
−k
ρx
xV
2
−k
ρy
y if
dx, x
≤ C
1
∨ 12
−k1
ρx;
iii |S
k
x, y − S
k
x, y
|≤C
2
dy, y
/2
−k
ρy1/V
2
−k
ρx
xV
2
−k
ρy
y if
dy, y
≤ C
1
∨ 12
−k1
ρy;
iv |S
k
x, y − S
k
x, y
− S
k
x
,y − S
k
x
,y
|≤ C
2
dx, x
/2
−k
ρxdy, y
/
2
−k
ρy1/V
2
−k
ρx
xV
2
−k
ρy
y if dx, x
≤ C
1
∨ 12
−k1
ρx and dy, y
≤
C
1
∨ 12
−k1
ρy;
v
X
S
k
x, wdμw1
X
S
k
w, ydμw for all k ≥
0
.
The existence of the approximation of the identity on X,d,μ
ρ
follows from a subtle
modification on the construction of Coifman in 20, Lemma 2.2see also 6 .Different from
20, here we define S
k
M
k
T
k
W
k
T
∗
k
M
k
, where T
k
is an integral operator whose kernel is
defined via the admissible function ρ,andM
k
and W
k
are the operators of multiplication by
1/T
k
1 and T
∗
k
1/T
k
1
−1
, respectively; see 3.2, 3.3,and3.4 below. We remark that the
idea of using the dual operator T
∗
k
here was used before by Tolsa 9.
Proposition 3.2. For any given
0
∈ Z, there exists a nonnegative symmetric
0
-AOTI {S
k
}
∞
k
0
,
where the symmetric means that S
k
x, yS
k
y, x for all k ≥
0
and x,y ∈X. Moreover, there
exists a positive constant C
3
(may depend on
0
) such that for all k ≥
0
and x, y ∈Xsatisfying
dx, y ≤ 2
−k
ρx,
C
3
V
2
−k
ρx
x
S
k
x, y
≥ 1. 3.1
Proof. Let h be a differentiable radial function on R satisfying χ
0,a
0
≤ h ≤ χ
0,2a
0
with a
0
≡
2Θ
2
−
0
. For any k ≥
0
, f ∈ L
1
loc
X,andu ∈X, define
T
k
f
u
≡
X
h
d
u, w
2
−k
ρ
w
f
w
dμ
w
, 3.2
and its dual operator
T
∗
k
f
u
≡
X
h
d
u, w
2
−k
ρ
u
f
w
dμ
w
. 3.3
Journal of Inequalities and Applications 11
Then, for all x,y ∈X,set
S
k
x, y
≡
1
T
k
1
x
X
h
d
x, z
2
−k
ρ
z
1
T
∗
k
1/T
k
1
z
h
d
z, y
2
−k
ρ
z
dμ
z
1
T
k
1
y
. 3.4
It is easy to see that S
k
is nonnegative, S
k
x, yS
k
y, x,and
X
S
k
x, ydμy1.
The support condition of h together with 2.1 and 2.2 implies that for any u ∈X,
T
k
1
u
∼ V
2
−k
ρu
u
∼ T
∗
k
1
u
, 3.5
with constants depending on
0
.
If S
k
x, y
/
0, then by 3.4, there exists z ∈Xsuch that dx, z ≤ a
0
2
−k1
ρz and
dz, y ≤ a
0
2
−k1
ρz, which together with 2.1 implies that
d
x, y
≤ a
0
Θ
a
0
2
−
0
1
2
−k2
ρ
x
∧ ρ
y
, 3.6
and that the integral domain in 3.4 is Bx, a
0
Θ
a
0
2
−
0
1
2
−k1
ρx.
For any z ∈ Bx, a
0
Θ
a
0
2
−
0
1
2
−k1
ρx,by3.5, 2.2, the support condition of h,and
Proposition 2.6a,weobtain
T
∗
k
1
T
k
1
z
X
h
d
z, w
2
−k
ρ
z
1
T
k
1
w
dμ
w
Bz,2
−k
ρz
1
V
2
−k
ρw
w
dμ
w
1,
T
∗
k
1
T
k
1
z
Bz,a
0
2
−k1
ρz
1
T
k
1
w
dμ
w
1,
3.7
which further implies t hat for all z ∈ Bx, a
0
Θ
a
0
2
−
0
1
2
−k1
ρx,
T
∗
k
1
T
k
1
z
∼ 1. 3.8
By 3.5, 3.8, Proposition 2.6a, and the fact that the integral domain in 3.4 is
Bx, a
0
Θ
a
0
2
−
0
1
2
−k1
ρx,weobtain
0 ≤ S
k
x, y
1
V
2
−k
ρx
x
1
V
2
−k
ρx
x
V
2
−k
ρy
y
. 3.9
Thus, i of Definition 3.1 holds with positive constants C
1
and C
2
depending only on
0
.
To show 3.1, by the fact h ≥ χ
0,a
0
and 3.8, we obtain that when dx, y ≤ 2
−k
ρx,
S
k
x, y
1
T
k
1
x
X
χ
{dx,z≤a
−1
0
2
−k
ρz}
z
h
d
z, y
2
−k
ρ
z
dμ
z
1
T
k
1
y
. 3.10
12 Journal of Inequalities and Applications
When dx, y ≤ 2
−k
ρx and dx, z ≤ a
−1
0
2
−k
ρz,by2.1 and the fact that a
0
> 1, we have
Θ
2
−
0
−1
ρz ≤ ρx ≤ Θ
2
−
0
ρz and
d
y, z
≤ d
y, x
d
x, z
≤ 2
−k
ρ
x
a
−1
0
2
−k
Θ
2
−
0
ρ
x
≤ 2
−k1
ρ
x
≤ a
0
2
−k
ρ
z
, 3.11
which implies that hdz, y/2
−k
ρz 1. Inserting this into 3.10 and then using 3.5,we
obtain 3.1.
NowweshowthatS
k
satisfies the desired regularity in the first variable when
dx, x
≤ C
1
∨ 12
−k1
ρx. Notice that in this case, S
k
x, y − S
k
x
,y
/
0 implies that
dx, y 2
−k
ρx, and hence ρy ∼ ρx ∼ ρx
by 2.1.Write
S
k
x, y
− S
k
x
,y
1
T
k
1
x
−
1
T
k
1
x
X
h
d
x, z
2
−k
ρ
z
1
T
∗
k
1/T
k
1
z
h
d
z, y
2
−k
ρ
z
dμ
z
1
T
k
1
y
1
T
k
1
x
X
h
d
x, z
2
−k
ρ
z
− h
d
x
,z
2
−k
ρ
z
1
T
∗
k
1/T
k
1
z
h
d
z, y
2
−k
ρ
z
dμ
z
×
1
T
k
1
y
≡ Z
1
Z
2
.
3.12
If dx, x
≤ C
1
∨ 12
−k1
ρx, then by the mean value theorem, 2.1, 2.2, 3.5,and
Proposition 2.6a,
1
T
k
1
x
−
1
T
k
1
x
≤
1
T
k
1
x
T
k
1
x
X
h
d
x, z
2
−k
ρ
z
− h
d
x
,z
2
−k
ρ
z
dμ
z
1
V
2
−k
ρx
x
V
2
−k
ρx
x
dx,z≤a
0
2
−k1
ρz
or dx
,z≤a
0
2
−k1
ρz
d
x, x
2
−k
ρ
z
dμ
z
d
x, x
2
−k
ρ
x
1
V
2
−k
ρx
x
.
3.13
By this, 3.5, 3.8, ρx
∼ ρx, dx, y 2
−k
ρx,andProposition 2.6a,weobtain
Z
1
d
x, x
2
−k
ρ
x
1
V
2
−k
ρx
x
∼
d
x, x
2
−k
ρ
x
1
V
2
−k
ρx
x
V
2
−k
ρy
y
. 3.14
NowweestimateZ
2
.IfZ
2
/
0, from the support condition of h and Proposition 2.6a,we
deduce that dx, z ≤ C2
−k
ρx for some positive constant C that depends on
0
. Therefore,
by the mean value theorem and 3.8,
Z
2
1
T
k
1
x
dx,z≤C2
−k
ρx
d
x, x
2
−k
ρ
z
h
d
z, y
2
−k
ρ
z
dμ
z
1
T
k
1
y
, 3.15
Journal of Inequalities and Applications 13
which combined with 2.1, 3.5, dx, y 2
−k
ρx, dx, x
≤ C2
−k
ρx,and
Proposition 2.6a further implies that
Z
2
d
x, x
2
−k
ρ
x
1
V
2
−k
ρx
x
V
2
−k
ρy
y
.
3.16
Combining the estimates of Z
1
and Z
2
yields that S
k
satisfies ii of Definition 3.1.
We finally prove that S
k
satisfies iv of Definition 3.1 if dx, x
≤ C
1
∨ 12
−k1
ρx
and dy, y
≤ C
1
∨ 12
−k1
ρy. In this case, S
k
x, y − S
k
x
,y − S
k
x, y
− S
k
x
,y
/
0
implies that dx, y 2
−k
ρx and hence ρx
∼ ρx ∼ ρy ∼ ρy
by 2.1.Write
S
k
x, y
− S
k
x
,y
−
S
k
x, y
− S
k
x
,y
1
T
k
1
x
−
1
T
k
1
x
X
h
d
x, z
2
−k
ρ
z
1
T
∗
k
1/T
k
1
z
h
d
z, y
2
−k
ρ
z
dμ
z
×
1
T
k
1
y
−
1
T
k
1
y
1
T
k
1
x
−
1
T
k
1
x
1
T
k
1
y
×
X
h
d
x, z
2
−k
ρ
z
1
T
∗
k
1/T
k
1
z
h
d
z, y
2
−k
ρ
z
− h
d
z, y
2
−k
ρ
z
dμ
z
1
T
k
1
x
X
h
d
x, z
2
−k
ρ
z
− h
d
x
,z
2
−k
ρ
z
1
T
∗
k
1/T
k
1
z
h
d
z, y
2
−k
ρ
z
dμ
z
×
1
T
k
1
y
−
1
T
k
1
y
1
T
k
1
x
1
T
k
1
y
×
X
h
d
x, z
2
−k
ρ
z
− h
d
x
,z
2
−k
ρ
z
1
T
∗
k
1/T
k
1
z
×
h
d
z, y
2
−k
ρ
z
− h
d
z, y
2
−k
ρ
z
dμ
z
≡ Z
3
Z
4
Z
5
Z
6
.
3.17
By 3.13, 3.5, 3.8, 3.6, the fact ρx
∼ ρx ∼ ρy ∼ ρy
,andProposition 2.6a,we
obtain
Z
3
d
x, x
2
−k
ρ
x
1
V
2
−k
ρx
x
d
y, y
2
−k
ρ
y
1
V
2
−k
ρy
y
μ
B
x, a
0
Θ
a
0
2
−
0
1
2
−k
ρ
x
d
x, x
2
−k
ρ
x
d
y, y
2
−k
ρ
y
1
V
2
−k
ρx
x
V
2
−k
ρy
y
.
3.18
14 Journal of Inequalities and Applications
The estimates for Z
4
through Z
5
are similar to those of Z
3
or Z
2
and hence omitted. Therefore,
S
k
satisfies iv of Definition 3.1. This finishes the proof of Proposition 3.2.
Remark 3.3. a It should be mentioned that 3.1 is crucial in establishing the vector-valued
Fefferman-Stein maximal function inequality; see Theorem 4.4 below.
b Let
0
∈ Z.Givenanyτ>0, if {S
k
}
∞
k
0
satisfy i and ii of Definition 3.1, then by
2.1 and 2.2, we have that there exists a positive constant C depending on τ such that for
all k ≥
0
and all dx, x
≤ τ2
−k
ρx,
S
k
x, y
− S
k
x
,y
≤ C
d
x, x
2
−k
ρ
x
1
V
2
−k
ρx
x
V
2
−k
ρy
y
. 3.19
If {S
k
}
∞
k
0
satisfy i and iii of Definition 3.1, then a symmetric estimate as in 3.19 holds
for the second variable. Analogously, if {S
k
}
∞
k
0
satisfy i through iv of Definition 3.1, then
for all dx, x
≤ τ2
−k
ρx and dy, y
≤ τ2
−k
ρy,
S
k
x, y
− S
k
x, y
−
S
k
x
,y
− S
k
x
,y
≤ C
d
x, x
2
−k
ρ
x
d
y, y
2
−k
ρ
y
1
V
2
−k
ρx
x
V
2
−k
ρy
y
.
3.20
The following technical lemma in some sense illustrates that the composition of two
0
-AOTI’s is still an
0
-AOTI except Definition 3.1v.
Lemma 3.4. Let
0
∈ Z and let {S
k
}
∞
k
0
and {E
k
}
∞
k
0
be two
0
-AOTI’s. Set D
0
≡ S
0
, Q
0
≡ E
0
,
D
k
≡ S
k
− S
k−1
, and Q
k
≡ E
k
− E
k−1
for k>
0
. Then for any η, σ, δ ∈ 0, 1 and σ δ ∈ 0, 1,there
exists a positive constant C, depending on η, σ, δ, C
1
, and C
2
, such that the kernel of D
k
Q
j
,whichis
still denoted by D
k
Q
j
, satisfies that for all k, j ≥
0
,
i if D
k
Q
j
x, y
/
0,thendx, y ≤ C
4
2
−k∧j
ρx ∧ ρy with C
4
≡ 4C
1
Θ
C
1
2
−
0
1
;
ii for all x, y ∈X,
D
k
Q
j
x, y
≤ C2
−|k−j|
1
V
2
−k∧j
ρx
x
V
2
−k∧j
ρy
y
; 3.21
iii for all x, y, y
∈Xsatisfying dy, y
≤ C
4
∨ 12
−k∧j1
ρy,
D
k
Q
j
x, y
− D
k
Q
j
x, y
≤ C2
−|k−j|1−η
d
y, y
2
−k∧j
ρ
y
η
1
V
2
−k∧j
ρx
x
V
2
−k∧j
ρy
y
;
3.22
Journal of Inequalities and Applications 15
iv for all x,y,x
∈Xsatisfying dx, x
≤ C
4
∨ 12
−k∧j1
ρx,
D
k
Q
j
x, y
− D
k
Q
j
x
,y
≤ C2
−|k−j|1−η
d
x, x
2
−k∧j
ρx
η
1
V
2
−k∧j
ρx
x
V
2
−k∧j
ρy
y
;
3.23
v for all x, y, x
, y
∈Xsatisfying dx, x
≤ C
4
∨ 12
−k∧j1
ρx and dy, y
≤ C
4
∨
12
−k∧j1
ρy,
D
k
Q
j
x, y
− D
k
Q
j
x
,y
−
D
k
Q
j
x, y
− D
k
Q
j
x
,y
≤ C2
−|k−j|1−ησδ
d
x, x
2
−k∧j
ρ
x
η1−σ
d
y, y
2
−k∧j
ρy
η1−δ
×
1
V
2
−k∧j
ρx
x
V
2
−k∧j
ρy
y
;
3.24
vi for all x,y ∈X,
X
D
k
Q
j
x, ydμx
X
D
k
Q
j
x, ydμy0 when k ∨ j >
0
; 1
when k j
0
.
Proof. Without loss of generality, we may assume that j ≥ k ≥
0
.ByDefinition 3.1i, for all
j ≥
0
, Q
j
x, y
/
0 implies that
d
x, y
≤ C
1
2
−j−1
ρ
x
∧ ρ
y
, 3.25
likewise for D
k
. Therefore, if D
k
Q
j
x, y
X
D
k
x, zQ
j
z, ydμz
/
0, then there exists z ∈
X such that dx, z ≤ C
1
2
−k−1
ρx∧ρz and dz, y ≤ C
1
2
−j−1
ρz∧ρy, which together
with 2.1 yields i.
The support and size conditions of S
0
and E
0
together with 2.1, 2.2,and
Proposition 2.6a imply that ii holds when j k
0
. To show that ii holds when j>
0
,
by the fact
X
Q
j
z, ydμz0, 3.25, the size condition of Q
j
, and the regularity of D
k
,we
obtain that for all x, y ∈X,
D
k
Q
j
x, y
X
D
k
x, z
− D
k
x, y
Q
j
z, y
dμ
z
≤
dz,y≤C
1
2
−j1
ρy
D
k
x, z
− D
k
x, y
Q
j
z, y
dμ
z
1
2
−k
ρ
y
1
V
2
−k
ρx
x
dz,y≤C
1
2
−j1
ρy
d
y, z
V
2
−j
ρy
y
V
y, z
dμ
z
,
3.26
16 Journal of Inequalities and Applications
which combined with Proposition 2.6c and j ≥ k further implies that
D
k
Q
j
x, y
1
2
−k
ρ
y
1
V
2
−k
ρx
x
dz,y≤C
1
2
−j1
ρy
d
z, y
V
z, y
dμ
z
2
k−j
1
V
2
−k
ρx
x
.
3.27
This together with i of this lemma and Proposition 2.6a yields ii.
The proofs for iii and iv are similar and we only show iii. To this end, it suffices
to prove that when dy, y
≤ C
4
∨ 12
−k1
ρy,
D
k
Q
j
x, y
− D
k
Q
j
x, y
d
y, y
2
−k
ρ
y
1
V
2
−k
ρx
x
V
2
−k
ρy
y
. 3.28
To see this, notice that if D
k
Q
j
x, y − D
k
Q
j
x, y
/
0, then the assumption of iii combined
with i and 2.1 yields that
ρ
x
∼ ρ
y
∼ ρ
y
,d
x, y
2
−k
ρ
x
∧ ρ
y
. 3.29
This together with ii and Proposition 2.6a further implies that
D
k
Q
j
x, y
− D
k
Q
j
x, y
2
−|k−j|
χ
{dx,y≤C
4
2
−k
ρx}
x, y
1
V
2
−k
ρx
x
1
V
2
−k
ρx
x
2
−|k−j|
1
V
2
−k
ρx
x
V
2
−k
ρy
y
.
3.30
Taking the geometric mean between 3.28 and 3.30 gives the desired estimate of iii.
Now we verify 3.28. Indeed, by the observation 3.19 on the regularity of the second
variable, it sufficestoshow3.28 for dy, y
≤ C
1
2
−k
ρy/4. In fact, we show that 3.28
holds for dy, y
≤ C
1
2
−k
ρydx, y/4. To this end, by Definition 3.1v, we write
D
k
Q
j
x, y
− D
k
Q
j
x, y
X
D
k
x, z
− D
k
x, y
Q
j
z, y
− Q
j
z, y
dμ
z
≤
2
i1
W
i
D
k
x, z
− D
k
x, y
Q
j
z, y
− Q
j
z, y
dμ
z
≡
2
i1
Z
i
,
3.31
where W
1
≡{z ∈X: dy, y
≤ C
1
2
−j
ρydz, y/2} and W
2
≡{z ∈X: dy, y
>
C
1
2
−j
ρydz, y/2}.
We first estimate Z
1
.Ifz ∈ W
1
and Q
j
z, y − Q
j
z, y
/
0, then either dz, y ≤
C
1
2
−j
ρz ∧ ρy or dz, y
≤ C
1
2
−j
ρz ∧ ρy
, which together with 2.1 yields that
Journal of Inequalities and Applications 17
dy, y
2
−j
ρy and dz, y 2
−j
ρz ∧ ρy 2
−k
ρy. These f acts and 3.29 together
with Proposition 2.6a and the regularities of {D
k
}
∞
k
0
and {Q
j
}
∞
j
0
yield that
Z
1
d
y, y
2
−j
ρ
y
W
2
d
z, y
2
−k
ρ
y
1
V
2
−k
ρy
y
χ
j
z, y
V
2
−k
ρz
z
V
2
−k
ρy
y
dμ
z
χ
k
x, y
d
y, y
2
−k
ρ
x
1
V
2
−k
ρx
x
V
2
−k
ρy
y
,
3.32
where and in what follows, χ
j
z, y ≡ χ
{dz,y2
−j
ρz∧ρy}
z, y for all j ≥
0
and z, y ∈X.
To estimate Z
2
, notice that f or any z ∈ W
2
,by3.29 and 2.1, we have
d
z, y
≤ 2d
y, y
≤
C
1
2
−k
ρ
y
d
x, y
2
2
−k
ρ
y
∼ 2
−k
ρ
x
. 3.33
This combined with 3.19 and Proposition 2.6b gives that
Z
2
χ
k
x, y
W
2
d
z, y
2
−k
ρ
x
1
V
2
−k
ρy
y
Q
j
z, y
Q
j
z, y
dμ
z
χ
k
x, y
W
2
d
z, y
2
−k
ρ
x
d
y, y
C
1
2
−j
ρ
y
d
z, y
1
V
2
−k
ρy
y
Q
j
z, y
Q
j
z, y
dμ
z
χ
k
x, y
d
y, y
2
−k
ρ
y
1
V
2
−k
ρy
y
X
Q
j
z, y
Q
j
z, y
dμ
z
d
y, y
2
−k
ρ
y
1
V
2
−k
ρx
x
V
2
−k
ρy
y
.
3.34
Combining the estimates of Z
1
and Z
2
yields 3.28 and hence iii holds.
When j ≥ k, to prove v,itsuffices to verify that for any η ∈ 0, 1, dx, x
≤ C
4
∨
12
−k1
ρx and dy, y
≤ C
4
∨ 12
−k1
ρy,
D
k
Q
j
x, y
− D
k
Q
j
x
,y
−
D
k
Q
j
x, y
− D
k
Q
j
x
,y
d
x, x
2
−k
ρx
η
d
y, y
2
−k
ρy
η
1
V
2
−k
ρx
x
V
2
−k
ρy
y
.
3.35
To see this, notice that if D
k
Q
j
x, y
− D
k
Q
j
x
,y
/
0, then by i and the assumption
dx, x
≤ C
4
∨ 12
−k1
ρx together with 2.1, we have dx, y
2
−k
ρx, which combined
18 Journal of Inequalities and Applications
with dy, y
≤ C
4
∨ 12
−k
ρx further implies that dx, y 2
−k
ρx.Bythis,iv of this
lemma, 3.19,andProposition 2.6a,weobtain
D
k
Q
j
x, y
− D
k
Q
j
x
,y
−
D
k
Q
j
x, y
− D
k
Q
j
x
,y
2
−|k−j|1−η
d
x, x
2
−k
ρx
η
1
V
2
−k
ρx
x
V
2
−k
ρy
y
χ
k
x, y
V
2
−k
ρx
x
V
2
−k
ρy
y
2
−|k−j|1−η
dx, x
2
−k
ρx
η
1
V
2
−k
ρx
x
V
2
−k
ρy
y
.
3.36
Using iii of this lemma and a symmetric argument, we obtain that
D
k
Q
j
x, y
− D
k
Q
j
x
,y
−
D
k
Q
j
x, y
− D
k
Q
j
x
,y
2
−|k−j|1−η
d
y, y
2
−k
ρy
η
1
V
2
−k
ρx
x
V
2
−k
ρy
y
.
3.37
Then the geometric mean among 3.35, 3.36,and3.37 gives the desired estimate of v.
By the observation 3.20, we only need to show 3.35 for dy, y
≤ C
1
2
−k
ρy/8
and dx, x
≤ C
1
2
−k
ρy/8. Actually, we now establish 3.35 for dy, y
≤ C
1
2
−k
ρy
dx, y/8anddx, x
≤ C
1
2
−k
ρydx, y/8. To this end, notice that if |D
k
Q
j
x, y −
D
k
Q
j
x
,y − D
k
Q
j
x, y
− D
k
Q
j
x
,y
/
0, then i of this lemma implies that at least
one of the following four inequalities holds: dx, y ≤ C
4
2
−k
ρx ∧ ρy, dx
,y ≤
C
4
2
−k
ρx
∧ ρy, dx, y
≤ C
4
2
−k
ρx ∧ ρy
,anddx
,y
≤ C
4
2
−k
ρx
∧ ρy
.This
and 2.1 together with the assumptions dy, y
≤ C
1
2
−k
ρydx, y/8anddx, x
≤
C
1
2
−k
ρydx, y/8 imply that
d
x, y
2
−k
ρ
x
,ρ
x
∼ ρ
y
∼ ρ
x
∼ ρ
y
, 3.38
and hence
d
x, x
2
−k
ρ
x
,d
y, y
2
−k
ρ
y
. 3.39
Journal of Inequalities and Applications 19
Then we write
D
k
Q
j
x, y
− D
k
Q
j
x
,y
−
D
k
Q
j
x, y
− D
k
Q
j
x
,y
X
D
k
x, z
− D
k
x
,z
−
D
k
x, y
− D
k
x
,y
Q
j
z, y
− Q
j
z, y
dμ
z
≤
2
i1
U
i
D
k
x, z
− D
k
x
,z
−
D
k
x, y
− D
k
x
,y
Q
j
z, y
− Q
j
z, y
dμ
z
≡
2
i1
J
i
,
3.40
where U
1
≡{z ∈X: dy, y
≤ C
1
2
−j
ρydz, y/2} and U
2
≡{z ∈X: dy, y
>
C
1
2
−j
ρydz, y/2}.
If z ∈ U
1
and Q
j
z, y − Q
j
z, y
/
0, then by the support condition of Q
j
and the fact
dy, y
≤ C
1
2
−j
ρydz, y/2 together with 3.38, we h ave
d
z, y
2
−j
ρ
y
2
−k
ρ
y
, 3.41
and hence dy, y
2
−j
ρy.Bythis,3.41, 3.39, the second-order difference condition of
D
k
,andRemark 3.3b, we then obtain
J
1
U
1
d
x, x
2
−k
ρ
x
d
z, y
2
−k
ρ
y
1
V
2
−k
ρx
x
V
2
−k
ρy
y
d
y, y
2
−j
ρ
y
χ
{dz,y2
−j
ρy}
z
V
2
−j
ρz
z
V
2
−j
ρy
y
dμ
z
d
x, x
2
−k
ρ
x
d
y, y
2
−k
ρ
y
1
V
2
−k
ρx
x
V
2
−k
ρy
y
.
3.42
If z ∈ U
2
, then by 3.39, we have dz, y ≤ 2dy, y
2
−k
ρy. This and 3.39
together with the second-order difference condition of D
k
and 3.19 yield that
J
2
U
2
d
y, y
C
1
2
−j
ρ
y
d
z, y
d
x, x
2
−k
ρ
x
d
z, y
2
−k
ρ
y
1
V
2
−k
ρx
x
V
2
−k
ρy
y
×
Q
j
z, y
− Q
j
z, y
dμ
z
d
x, x
2
−k
ρ
x
d
y, y
2
−k
ρ
y
1
V
2
−k
ρx
x
V
2
−k
ρy
y
X
Q
j
z, y
− Q
j
z, y
dμ
z
d
x, x
2
−k
ρ
x
d
y, y
2
−k
ρ
y
1
V
2
−k
ρx
x
V
2
−k
ρy
y
.
3.43
Combining the estimates of J
1
and J
2
yields 3.35. Hence, v holds.
20 Journal of Inequalities and Applications
Property vi can be obtained simply by using Definition 3.1v. This finishes the proof
of Lemma 3.4.
We conclude this section with some basic properties of
0
-AOTI, which are used in
Section 5. For all f ∈ L
p
X with p ∈ 1, ∞ and x ∈X,setS
k
fx ≡
X
S
k
x, yfydμy.
Denote by L
∞
b
X the collection of all f ∈ L
∞
X with bounded support.
Proposition 3.5. Let
0
∈ Z and {S
k
}
∞
k
0
be an
0
-AOTI as in Definition 3.1.
i There exists a positive constant C depending only
0
such that for all x, y ∈Xand k ≥
0
,
X
|S
k
x, y|dμy ≤ C and
X
|S
k
x, y|dμx ≤ C.
ii There exists a positive constant C depending only on
0
such that for all k ≥
0
, locally
integrable functions f, and x ∈X, |S
k
fx|≤CM
C
1
2
−
0
fx,whereC
1
is the constant
appearing in Definition 3.1(i).
iii For p ∈ 1, ∞, there exists a positive constant C
p
, depending on p and
0
, s uch that for all
k ≥
0
and f ∈ L
p
X, S
k
f
L
p
X
≤ C
p
f
L
p
X
.
iv Set D
0
≡ S
0
and D
k
≡ S
k
− S
k−1
for k>
0
.ThenI
∞
k
0
D
k
in L
p
X,where
p ∈ 1, ∞ and I is the identity operator on L
p
X.
Proof. i can be easily deduced from the support and size conditions of S
k
together with
Proposition 2.6c. We can easily show ii by using 2.20 and Definition 3.1i. Property iii
is a simple corollary of i and H
¨
older’s inequality.
To prove iv,itsufficestoshowthatlim
N →∞
f −
N
k
0
D
k
f
L
p
X
0 for all f ∈
L
p
X with p ∈ 1, ∞. Since f −
N
k
0
D
k
f
L
p
X
f − S
N
f
L
p
X
, it is enough to show
lim
N →∞
X
f
x
− S
N
f
x
p
dμ
x
0. 3.44
Now we prove 3.44 for p ∈ 1, ∞.Letx ∈Xbe a point such that Theorem 2.8ii
holds for f. Then using v and i of Definition 3.1,weobtain
f
x
− S
N
f
x
≤
X
S
N
x, y
f
x
− f
y
dμ
y
1
μ
B
x, C
1
2
−N
ρ
x
Bx,C
1
2
−N
ρx
f
x
− f
y
dμ
y
,
3.45
which tends to 0 as N →∞,byTheorem 2.8ii. This and |S
N
fx| M
C
1
2
−
0
fx together
with the dominated convergence theorem and Theorem 2.8i imply that 3.44 holds for p ∈
1, ∞.
To prove 3.44 for the case p 1, we first consider f ∈ L
∞
b
X. Assume that supp f ⊂
Bx
0
,r
0
ρx
0
for some x
0
∈Xand r
0
> 0. Combining this with 2.1 gives supp S
k
f ⊂
Bx
0
, C
1
Θ
r
0
r
0
ρx
0
.ByH
¨
older’s inequality and L
∞
b
X ⊂ L
2
X together with the fact
that 3.44 holds f or p 2, we obtain that for all f ∈ L
∞
b
X,
lim
N →∞
S
N
f
− f
L
1
X
≤ lim
N →∞
μ
B
x
0
,
C
1
Θ
r
0
r
0
ρ
x
0
1/2
S
N
f
− f
L
2
X
0,
3.46
Journal of Inequalities and Applications 21
which combined with the density of L
∞
b
X in L
1
X and Proposition 3.5iii yields that 3.44
holds for p 1. Thus, we obtain iv, which completes the proof of Proposition 3.5.
4. Local Vector-Valued Singular Integral Operators
In this section, let X,d be a metric space and μ a regular Borel measure satisfying 2.2.
Denote by B a complex Banach space with norm ·
B
,andbyB
∗
its dual space with norm
·
B
∗
.AfunctionF defined on a σ-finite measure space X,μ and taking values in B is
called B-measurable if there exists a measurable subset X
0
of X such that μX\X
0
0and
FX
0
is contained in some separable subspace B
0
of B, and for every u
∗
∈ B
∗
, the complex-
valued map x →u
∗
,Fx is measurable. From this definition and the theorem in 29, page
131, it follows that the function x →Fx
B
on X is measurable.
For any p ∈ 0, ∞, we define L
p
X, B to be the space of all B-measurable functions
F on X satisfying F
L
p
X,B
< ∞, where F
L
p
X,B
{
X
Fx
p
B
dμx}
1/p
with a usual
modification made when p ∞. Define L
p,∞
X, B to be the space of all B-measurable
functions F on X satisfying F
L
p,∞
X,B
< ∞, where
F
L
p,∞
X,B
sup
α>0
α
μ
{
x ∈X:
F
x
B
>α
}
1/p
.
4.1
Denote by L
∞
b
X, B the set of all functions in L
∞
X, B with bounded support. For p ∈
0, ∞,letL
p
X ⊗ B be the set of all finite linear combinations of elements of B with
coefficients in L
p
X, that is, elements of the form,
F f
1
u
1
··· f
m
u
m
, 4.2
where m ∈ N, f
j
∈ L
p
X,andu
j
∈ B for j ∈{1, ,m}.BothL
∞
b
X, B and L
p
X ⊗ B are
dense in L
p
X, B; see, for example, 3 or 30, Lemma 2.1.GivenF ∈ L
1
X ⊗ B as in 4.2,
we define its integral to be the following element of B:
X
F
x
dμ
x
≡
m
j1
X
f
j
x
dμ
x
u
j
. 4.3
Therefore, for any F ∈ L
1
X, B, the integral
X
Fxdμx, as a unique extension of the
integral of functions in L
1
X ⊗ B, is well defined; it is not difficult to show that
X
F
x
dμ
x
B
≤
X
F
x
B
dμ
x
; 4.4
see, for instance, 3 or 29. Here we refer the reader to 3, 31, 32 for more detailed
knowledge on Banach space-valued f unctions.
In what follows, we consider a kernel
−→
K defined on X×X \ Δ with Δ{x, x :
x ∈X}that takes values in the space LB
1
, B
2
of all bounded linear operators from Banach
space B
1
to Banach space B
2
. Then
−→
Kx, y is a bounded linear operator from B
1
to B
2
22 Journal of Inequalities and Applications
whose norm is denoted by
−→
Kx, y
B
1
→ B
2
. Assume that
−→
Kx, y is LB
1
, B
2
-measurable
and locally integrable on X×X \ Δ such that the integral
−→
T
F
x
X
−→
K
x, y
F
y
dμ
y
4.5
is well defined as an element of B
2
for all F ∈ L
∞
b
X, B
1
and x
/
∈ supp F.SetA ≡
lim inf
τ → 0
Θ
τ
. Suppose that there exist constants C
5
> 2A
2
Θ
A
2
Θ
A
A and C
6
> 0 such that
for all x, y ∈Xsatisfying dx, y ≤ C
5
ρx,
−→
K
x, y
B
1
→ B
2
≤ C
6
1
V
x, y
,
4.6
dx,z≥2dx,y
−→
Kz, x −
−→
Kz, y
B
1
→ B
2
dμ
z
≤ C
6
, 4.7
dx,z≥2dx,y
−→
Kx, z −
−→
Ky, z
B
1
→ B
2
dμ
z
≤ C
6
.
4.8
Let N≡{x, y ∈X×X: dx, y ≤ ρx ∧ ρy} and N
x
≡{y ∈X: x, y ∈N}.
Then for all x ∈X,set
−→
T
local
F
x
≡
−→
T
χ
N
x
F
x
, 4.9
where χ
N
x
represents the characteristic function of the set N
x
. A conclusion concerned such
locally vector-valued singular integrals is as follows.
Theorem 4.1. Let B
1
and B
2
be Banach spaces. Suppose that
−→
T given by 4.5 is a bounded linear
operator from L
r
X, B
1
to L
r
X, B
2
for some r ∈ 1, ∞ with norm A
r
> 0. Assume that
−→
K
satisfies 4.6 through 4.8.Then
−→
T
local
as in 4.9 has well-defined extensions on L
p
X, B
1
for all
p ∈ 1, ∞. Moreover, there exists a positive constant C depending on X, p, and C
5
such that
i whenever p ∈ 1,r, for all F ∈ L
p
X, B
1
,
−→
T
local
F
L
p,∞
X,B
2
≤ C
C
6
A
r
F
L
p
X,B
1
; 4.10
ii whenever p ∈ 1, ∞, for all F ∈ L
p
X, B
1
,
−→
T
local
F
L
p
X,B
2
≤ C
C
6
A
r
F
L
p
X,B
1
. 4.11
Proof. It suffices to show the theorem for F ∈ L
∞
b
X, B
1
,sinceL
∞
b
X, B
1
is dense in
L
p
X, B
1
for p ∈ 1, ∞. We further assume that μX < ∞, since the proof for the case
μX∞ is similar and simpler.
Journal of Inequalities and Applications 23
Suppose that r<∞ and p ∈ 1,r.If0<A
−1
r
λ ≤F
L
p
X,B
1
/μX
1/p
this happens
only when μX < ∞, then
μ
x ∈X:
−→
T
local
F
x
B
2
>λ
≤ μ
X
≤
A
r
λ
p
F
p
L
p
X,B
1
. 4.12
Assume now that A
−1
r
λ>F
L
p
X,B
1
/μX
1/p
.ByLemma 2.7, for any given sufficiently
small positive number t, which will be determined later, there exists a sequence of balls,
B
j
≡ Bx
j
,tρx
j
, such that X
j
B
j
and {ηB
j
}
j
has finite overlapping property whenever
η>0. Set B
∗
j
≡ Bx
j
, Θ
t
tρx
j
. It follows easily from 2.1 that for any given j,
x∈B
j
N
x
⊂ B
∗
j
. Therefore, by 4.9,
μ
x ∈X:
−→
T
local
F
x
B
2
>λ
≤
j
μ
x ∈ B
j
:
−→
T
χ
N
x
F
x
B
2
>λ
≤
j
μ
x ∈ B
j
:
−→
T
χ
B
∗
j
F
x
B
2
>
λ
2
j
μ
x ∈ B
j
:
−→
T
χ
B
∗
j
\N
x
F
x
B
2
>
λ
2
≡
j
Y
j
Y.
4.13
Observe that if y ∈ B
∗
j
and x ∈ B
j
, then dx, y ≤ Θ
t
2tρx
j
≤ Θ
t
Θ
t
2tρx,sodx, y <
C
5
ρx if we choose t>0sufficiently small. Thus, for any x ∈ B
j
,by4.4 and 4.6, we have
−→
T
χ
B
∗
j
\N
x
F
x
B
2
B
∗
j
\N
x
−→
K
x, y
F
y
dμ
y
B
2
≤ C
6
B
∗
j
\N
x
F
y
B
1
V
x, y
dμ
y
.
4.14
If x ∈ B
j
and y ∈ B
∗
j
\N
x
, then 2.1 implies that B
∗
j
⊂ Bx, Θ
t
Θ
t
2tρx and dx, y >
Θ
t
−1
Θ
Θ
t
t
−1
ρx. This combined with 4.14 and 2.2 yields that there exists a positive
constant
C, depending only on X and t, such that
−→
T
χ
B
∗
j
\
N
x
F
x
B
2
≤
C
C
6
M
a
F
·
p
B
1
χ
B
∗
j
x
1/p
,
4.15
24 Journal of Inequalities and Applications
where a ≡ Θ
t
Θ
t
2t.By4.15, Theorem 2.8i, Lemma 2.7ii, the finite overlapping
property of {B
∗
j
}
j
, and the fact that for all κ>1and{a
j
}
j∈N
⊂ C,
j∈N
a
j
κ
≤
⎛
⎝
j∈N
a
j
⎞
⎠
κ
,
4.16
we then obtain
Y ≤
j
μ
x ∈ B
∗
j
: M
a
F
·
p
B
1
χ
B
∗
j
x
>
λ
p
CC
6
C
6
j
Fχ
B
∗
j
p
L
p
X,B
1
λ
p
C
6
F
p
L
p
X,B
1
λ
p
.
4.17
Now we estimate
j
Y
j
.Setf
j
≡F·
B
1
χ
B
∗
j
. Then supp f
j
⊂ B
∗
j
. We claim that there
exists a positive number t sufficiently small such that
supp
M
a
f
p
j
1/p
⊂ B
x
j
,C
5
ρ
x
j
. 4.18
In fact, for any x ∈X,from2.20, we deduce that if M
a
f
p
j
x
1/p
/
0, then there exists a
ball B x satisfying that B ∩ B
∗
j
/
∅ and r
B
≤ aρc
B
. From this and 2.1 together with the
triangular inequality of d, it follows that dx, x
j
≤ 2aΘ
a
Θ
Θ
t
t
Θ
t
tρx
j
. Combined this
with the facts C
5
> 2A
2
Θ
A
2
Θ
A
A, A ≡ lim inf
τ → 0
Θ
τ
,anda Θ
t
Θ
t
2t,weobtainthat
x ∈ Bx
j
,C
5
ρx
j
if t is sufficiently small. Thus, 4.18 holds.
For any λ>0, the set Ω
λ
≡{x ∈X: M
a
f
p
j
x
1/p
>λ} is open and, by 4.18, Ω
λ
is contained in the ball Bx
j
,C
5
ρx
j
. Following the procedure of the proof for the Whitney
covering lemma see 33, page 277 and 7, pages 70–71, we obtain that for any fixed j,
there exists a sequence {B
i
j
}
i∈I
j
of balls, where I
j
is an index set depending on j and a positive
number M depending on D
C
5
in 2.2 with a C
5
,butnotonj such that
iΩ
A
−1
r
λ
i∈I
j
B
i
j
⊂ Bx
j
,C
5
ρx
j
;
ii r
B
i
j
≤ C
5
ρx
j
/2Θ
C
5
2;
iii every point of X belongs to no more than M balls of {3B
i
j
}
i∈I
j
;
iv the balls {1/4B
i
j
}
i∈I
j
are mutually disjoint and 3Θ
C
5
1B
i
j
∩ Ω
A
−1
r
λ
/
∅.
Journal of Inequalities and Applications 25
For any given j, i ∈ I
j
,andx ∈X,wesetζ
i
j
x ≡ χ
B
i
j
x/
i∈I
j
χ
B
i
j
x,anddefine
g
j
x
≡ F
x
χ
B
∗
j
\Ω
A
−1
r
λ
x
i∈I
j
⎧
⎨
⎩
1
μ
B
i
j
B
i
j
F
y
χ
B
∗
j
y
ζ
i
j
y
dμ
y
⎫
⎬
⎭
χ
B
i
j
x
, 4.19
h
i
j
x
≡ F
x
χ
B
∗
j
x
ζ
i
j
x
−
⎧
⎨
⎩
1
μ
B
i
j
B
i
j
F
y
χ
B
∗
j
y
ζ
i
j
y
dμ
y
⎫
⎬
⎭
χ
B
i
j
x
. 4.20
By Properties i, iii,andiv above together with Lemma 2.7,itisnotdifficult to show that
there exists a positive constant C, independent of j, such that
v for all x ∈X, Fxχ
B
∗
j
xg
j
xh
j
x, where h
j
x
i∈I
j
h
i
j
x;
vi for almost every x ∈X, g
j
x
B
1
≤ CA
−1
r
λ;
vii g
j
L
p
X,B
1
≤ CFχ
B
∗
j
L
p
X,B
1
;
viii for any i ∈ I
j
,supph
i
j
⊂ B
i
j
and
i∈I
j
μB
i
j
≤ CFχ
B
∗
j
p
L
p
X,B
1
/A
−1
r
λ
p
;
ix for any i ∈ I
j
,
X
h
i
j
xdμxθ
B
1
, where θ
B
1
denotes the zero element of B
1
;
x
i∈I
j
h
i
j
L
p
X,B
1
≤ CFχ
B
∗
j
L
p
X,B
1
;
xi
i∈I
j
h
i
j
L
1
X,B
1
≤ CA
−1
r
λ
1−p
Fχ
B
∗
j
p
L
p
X,B
1
by viii.
Then, by Property v,weobtain
Y
j
≤ μ
x ∈ B
j
:
−→
T
g
j
x
B
2
>
λ
2
μ
x ∈ B
j
:
−→
T
h
j
x
B
2
>
λ
2
≤
2
r
λ
r
−→
T
g
j
·
B
2
r
L
r
X
μ
⎛
⎝
i∈I
j
3B
i
j
⎞
⎠
μ
⎛
⎝
⎧
⎨
⎩
x ∈ B
j
\
i∈I
j
3B
i
j
:
T
h
x
B
2
>
λ
2
⎫
⎬
⎭
⎞
⎠
≡ L
j
H
j
N
j
.
4.21
Recall that p ∈ 1,r. The boundedness of
−→
T from L
r
X, B
1
to L
r
X, B
2
together with
Properties vi and vii implies that
L
j
A
r
r
λ
r
g
j
r
L
r
X,B
1
A
r
λ
p
g
j
p
L
p
X,B
1
A
r
λ
p
Fχ
B
∗
j
p
L
p
X,B
1
. 4.22
By 2.2 and viii together with Theorem 2.8i, we have
H
j
i∈I
j
μ
3B
i
j
i∈I
j
μ
B
i
j
A
r
λ
p
Fχ
B
∗
j
p
L
p
X,B
1
. 4.23
