1. Introduction and Main Results
Suppose
is a nilpotent Lie group, which has the multiplication, inverse, expansion and norm configurations
for
, respectively, then we call
being a homogeneous group (see [
1] or [
2]). The multiplication and inverse operations are polynomials and
t-action is an automorphism of the group structure, where
t is of the form
for some constants
. Besides,
is a norm linked to the expansion configuration. We call the value
N the dimensionality of
. In addition to the Euclidean structure,
is equipped with a homogeneous nilpotent Lie group structure, where Lebesgue measure is a bi-invariant Haar measure, the identity is the origin
and multiplication
, satisfies
(1) ;
(2) ;
(3) if , then , where and for with a polynomial depending only on
Finally, the Heisenberg group on
is an example of a homogeneous group. If we define the multiplication
, the
with this group law is the Heisenberg group
; a dilation is defined by
, that is the parameters
.
Definition 1. Let is a function on , which is non-negative locally integrable. For , we call that w is an weight, denoted by , if The supremum here is taken over of all balls . We call that the quantity is the constant of w. For , if for , then we say that w is an weight, denoted by , where M represents the Hardy-Littlewood maximal function. In addition, let , then we have Definition 2. Let , and be a non-negative locally integrable function. For , ifwhere is the conjugate exponent of p, that is Definition 3. Suppose . Let , then ifwhere and the supremum is taken over of all balls . We now review the definition of Riesz potential on homogeneous group. For
N,
and the corresponding associated maximal function
by
The reason why we study the weighted estimates for these operators is because they have a wide range of applications in partial differential equations, Sobolev embeddings or quantum mechanics (see [
3] or [
4]).
Muckenhoupt and Wheeden [
5] are the first scholars to study the Riesz potential. When
is an isotropic Euclidean space, Muckenhoupt and Wheeden [
5] show that
is bounded from
to
for
. Moreover, the sharp constant in this inequality was given in [
6]:
Definition 4. Suppose . Let be the commutator defined by The iterative commutators , are defined naturally by In 2016, Holmes, Rahm and Spencer [
7] prove that
where
. Later, the quantitative estimates for iterated commutators of fractional integrals was obtained by N. Accomazzo, J. C. Martínez-Perales and I. P. Rivera-Ríos [
8].
In 2013, Sato [
9] gave the estimates for singular integrals on homogeneous groups. In [
10], X. T. Duong, H. Q. Li and J. Li established the Bloom-type two weight estimates for the commutator of Riesz transform on stratified Lie groups. Moreover, Z. Fan and J. Li [
11] obtained the quantitative weighted estimates for rough singular integrals on homogeneous groups.
Motivated by the above estimates, we investigate the quantitative weighted estimation for the higher order commutators of fractional integral operators on homogeneous groups.
In this paper, our main result is the follow theorem.
Theorem 1. Let and defined by , and m is a positive integer. Assume that and that .
If , then For every , if is bounded from to , then with
2. Domination of the Iterated Commutators by Sparse Operators
2.1. A System of Dyadic Cubes
We define a left-unchanged analogous-distance
d on
by
, which signifies that there has a constant
such that for any
,
Next, let be the open ball which is centered on and is the radius.
Let be k-th denumerable index set. A denumerable class , of Borel sets is known as a set of dyadic cubes with arguments and if it has the characteristics below:
(1) (disjoint union) for all ;
(2) If , then either or ;
(3) For arbitrary and for any , there is a exclusive such that ;
(4) For arbitrary there exists no more that M (a settled geometric constant) such that , and ;
(5) ;
(6) If and , then . The set is called a dyadic cube of generation k with centre and side length .
From the natures of the dyadic system above, for any
and
, we get that there is a constant
such that:
2.2. Adjacent Systems of Dyadic Cubes
Let be a limited set of the dyadic families, then we call that it is a collection of neighbor systems of dyadic cubes with arguments , and if it has the following two characteristics:
(1) For any is a system of dyadic cubes with arguments and ;
(2) For any ball
with
, there have
and
of generation
k which is centered on
such that
and
2.3. Sparse Operators
We review the concept of sparse family given in [
12] on ordinary spaces of homogeneous description in the sense of Coifman and Weiss [
13], which is also suitable in the case of homogeneous groups.
Definition 5. Let , for every , we call that the collection of dyadic cubes be a η-sparse, if there exists a measurable subset such that and the sets have only limited overlap.
Definition 6. Given a sparse family, the sparse operator is defined bywhere . In this subfraction, the primary target is to reveal the following quantitative edition of Lacey’s pointwise domination inequality.
Proposition 1. Let . Let m be a nonnegative integer. For every and , there exits dyadic systems and η-sparse families such that for ,where for a sparse family is the sparse operator given by To show the Proposition 1, we need some auxiliary maximal operators. To begin with, let
be the smallest integer such that
and let
.
Next we define the grand maximal truncated operator
as follows:
where the first supremum is taken over of all balls
satisfying
. We can know that this operator is of vital importance in the following proof, Given a ball
, for
we also define a local edition of
by
Now, we claim that the following lemma is true.
Lemma 1. Let . The following pointwise estimates holds:
There exists a constant such that for ,
Using the results of Lemma 1, we then prove the Proposition 1.
Proof of Proposition 1.
In order to proof the Proposition 1, we refer to the thinking in [
8] for this domination, which is adapted to our situation of homogeneous groups.
Firstly, we suppose that
f is supported in a ball
, next we disintegrate
which respect to this ball
. We can do it as follows. We start define the annuli
and select the minimum integer
such that
Next, for any
, we select the balls
centred in
and with radius
to cover
. From the doubling property [
13], we obtain
where
is an positive constant that only relates on
and
.
We now go over the characters of these
. Denote
, where
is defines as in (
4). Then we have
, which was shown in the proof of Theorem 3.7 in [
12] that
and
Now, because of the Equation (
8) and (
9), we see that each
, at most overlap with
annuli
’s. Moreover, for every
j and
covers
.
Next by observing the (
2), there is an integer
and
such that
. Additionally, for this
, as in
Section 2.1 the ball that includes
and has comparable measure to
is represented by
. Consequently,
is overwritten by
and
, where the implicit constant relates only to
and
.
Now we claim that there exists a
-sparse family
, the set of all dyadic cubes in
-th dyadic system that are contained in
, such that for
,
where
Here,
is the dyadic cube in
for some
such that
, where
is defined as in
Section 2.1,
defined as in (
5) and
defined as in (
4).
Assume that we have already proven the assertion (
10). Let us take a partition of
as follows:
We next consider the annuli
for
and the covering
of
as in (
6). We note that for each
, there exist
and
such that
. Therefore, we acquire that for each such
, the enlargement
covers
since
covers
.
Next, we utilize (
10) to each
, then we acquire a
-sparse family
such that (
10) can be established for
.
Now, set
. Then we observe that the balls
are overlapping not more than
times, where
is the constant in (
7). Then, we can obtain that
is a
-sparse family and for
,
Since , and it is clear that ( depends only on ), we obtain that . Now, we set , then since the fact that is -sparse, we can acquire that each family is -sparse.
Now, we let
where
is a constant relating only on
. Then it follows that (
3) holds, which finishes the proof. □
Proof of the Assertion (10).
To demonstrate the assertion it suffice to attest the following recursive computation: there exist the cubes
that does not intersect each other such that
and for
,
Iterating this estimate, we acquire (
10) with
being the union of all the families
, where
as mentioned above, and
are the cubes acquired at the
k-th stage of the iterative approach. Clearly
is a
-sparse family, since let
Now we prove the recursive estimate. For any countable family
of disjoint cubes
, we have that
So we just have to reveal that we can opt for a family of pairwise disjoint cubes
such that
and that for
,
Using that
for any
, and also that
it follows that
Now we define the set
, where
with
being a positive number to be chosen.
From [
8], we can choose
big enough (depending on
) such that
where
is defined in
Section 2.1. We now utilize the Calderón-Zygmund decomposition to the function
on
at the height
, to acquire pairwise disjoint cubes
such that
and
. This implies that
Fix some
j. Since we have
, we observe that
which allows us to control the summation in
by considering the cube
.
Now by (i) in Lemma 1, we know that
Since
, we have that
These estimates allow us to control the remaining terms in , so we are done. □
Proof of Lemma 1.
Now we give the proof process of Lemma 1.
The result in the Euclidean space case can be referred to as [
8]. Now, we can adapt the proof in [
8] to our setting of homogeneous groups.
(i) Let
r is close enough to 0 such that
. Then,
the estimate for the first term follows by standard computations involving a dyadic annuli-type decomposition of the
.
Then,
the estimate in (i) is settled letting
in (
11).
(ii) Let
. Let
be the closed ball with radius
, which centered at
x. Then
, and we acquire
For the first term, since
is homogeneous of degree
, and by using the Proposition 1.7 in [
1], we get
Next, for
, we have
. Then we have
Finally, we observe that
which finishes the proof of (ii). □
Next, we review that the dyadic weighted
space associated with the system
is defined as
where
. Then according to the dyadic structure theorem studies in [
14], one has
Now, to verify a function
b is in
, it suffices to verify it belongs to each weighted dyadic
space
. Given a dyadic cube
with
, and a measurable function
f on
, we define the local mean oscillation of
f on
Q by
where
With these notation and dyadic structure theorem above, following the same proof in [
10], we also acquire that for any weight
, we have
where
C depends on
.
Proposition 2. Suppose that is a homogeneous group with dimension . Then for any cube , there exist measurable set with , such that Proof. We take ideas from N. Accomazzo, J. C. Martínez-Perales and I. P. Rivera-Ríos [
8]. In [
8], for any cube
with
, there exists a subset
with
such that for every
,
where
is a not necessarily unique number that satisfies
Let with and such that for every . Further let , then and for every .
We obtain that at least half of the set
E is contained either in
or in
since
Q is the disjoint union of
and
. Without loss of generality, we assume that half of
E is in
, then we let
, we have
and
Then if
and
, we have that
which shows that Proposition 2 holds. □
Given a dyadic grid
, define the dyadic Riesz potential operator
Proposition 3. Given , then for any dyadic grid , Proof. The result in the Euclidean setting is from the Proposition 2.1 in [
15]. Here, we can adapt the proof in [
15] to our setting of spaces of homogeneous type. □
3. Proof of Theorem 1
To proof (i), we are following the ideas in [
16] or [
8].
Let
be a dyadic system in
and let
be a sparse family from
. We know
by duality, we have that
By Lemma 3.5 in [
12], there exists a sparse family
such that
and for every cube
, for
,
where
Assume that
with
to be chosen, then we have for
,
Next, note that for each
, from [
12], for an arbitrary function
h, we have
where
and
stands for the
ℓ-th iteration of
.
Then we have
where
and
.
From (
13) and the boundedness of
, if
are as in the hypothesis of Theorem 1.1 and
, then
Observe that
is self-adjoint, then
By Hölder inequality, we have that
Applying that
(see, e.g., [
17] ),
Using (
14), we have that
and applying again
,
which, along with the previous estimate, yields
where
and
Hence, setting
, where
, it reading follows from Hölder’s inequality
Thus, we acquire that
and
Combining all the preceding estimates obtains (i).
To proof (ii), we are going to follow ideas in [
10]. Based on (
12), it suffices to show that there exists a positive constant
C such that for all dyadic cubes
,
Using Proposition 2 and Hölder inequality implies that
where we used that
.
Then from [
8], we have
so the
Now we observe that since
then by Hölder inequality,
then
Consequently, since
, we finally get
Thus, (
15) holds and hence, the proof of (ii) is complete.
Therefore, we complete the proof of Theorem 1.