1. Introduction
To understand the Bernstein theorem for absolutely convergent Fourier transforms, Beurling [
1] researched the Herz space
in 1964. The Herz space
is further explored by Herz [
2] in 1968. In the 1990’s, Lu and Yang [
3] introduced the homogeneous Herz space
and the non-homogeneous Herz space
. In recent years, Herz spaces has been extensively studied in the fields of harmonic analysis, see [
4,
5,
6,
7,
8] and so on.
Wiener amalgam is an indispensable tool in time-frequency analysis [
9,
10] and sampling theory [
11]. At first, amalgam spaces is elaborated by Wiener in [
12]. Still, the systematic investigation of amalgam spaces should look at the study of Holland [
13], which contains the research of dual spaces and multipliers on
. Wiener amalgam spaces are generalized by Feichtinger and Weisz from
or
to Banach function spaces, see, [
14,
15,
16,
17,
18] and so on. A definition of the amalgam space
is defined by:
where
with the usual modification for
or
, denote
the open ball with centered at
y with the radius 1.
is the characteristic function of the ball
.
Very recently, the slice space
was introduced by Auscher and Mourgoglou [
19], an exceptional examples of classical amalgam spaces. Auscher and Prisuelos-Arribas [
20] researched some classical operators of harmonic analysis for generalized version slice space
in 2017, and proved that amalgam spaces and slice spaces are equivalent. In recent years, many authors studied slice-type spaces, such as, in 2019, Zhang, Yang, Yuan, and Wang [
21] introduced Orlicz-slice spaces and Orlicz-slice Hardy spaces. In 2022, Lu, Zhou and Wang [
22] introduced Herz-slice spaces. In 2023, we defined mixed-norm Herz-slice space
in [
23], and the theory of the Hardy–Littlewood maximal operator is given in this space. Given
and
, the Hardy-Littlewood maximal function
is defined by
where the supremum is taken over all balls
containing
x.
In 1961, the mixed-norm Lebesgue space
was introduced by Benedek and Panzone [
24], where
, which can be traced back to [
25]. Because mixed-norm Lebesgue spaces has a more refined structure on partial differential equations [
26,
27], more scholarly authors like to study problems on it. For example, the mixed-norm amalgam space
was introduced by Zhang and Zhou [
28] in 2022, and the boundedness of the Calderón–Zygmund operator
T and the commutators with
functions on the mixed-norm amalgam space
was established [
29]. The properties of the
function
b are referred to [
30]. A function
, we define the mean of
b over
Q by
. The bounded mean oscillation space, namely
space is defined by
the supremum is taken over all cubes
Q in
. For a deeper discussion about mixed-norm spaces, we can see [
26,
31,
32,
33] and so on.
Our motivation is based on our introduction of the mixed-norm Herz-slice space
, to consider whether we can establish the boundedness of the Calderón–Zygmund operator
T and commutators with
functions on the mixed-norm Herz-slice space
. In this paper, we get the boundedness of the Calderón–Zygmund operator
T on mixed-norm Herz-slice spaces, and demonstrate the necessary and sufficient conditions for the boundedness of the commutator
on mixed-norm Herz-slice spaces. Our results are all new when we return to the classical Herz-slice spaces, slice spaces and Lebesgue spaces. The commutator
is defined by
The remainder of this paper is organized as follows. In
Section 2, we introduce some necessary space definitions and operator notation, and we also give crucial lemma. In
Section 3, we obtain the separable, weak convergence of mixed-norm Herz-slice spaces. To get the boundedness of the Calderón–Zygmund operator
T on mixed-norm Herz-slice spaces, we explain whether
has an absolutely continuous quasi-norm and the class
is dense in
.
is the space of infinitely differentiable functions with compact support in
. Furthermore, we get the boundedness of the Calderón–Zygmund operator
T of commutators with
functions on mixed-norm Herz-slice spaces by
is bounded on
. Let
. The commutators of the Hardy-Littlewood maximal function with
functions
are defined, respectively, by
where
the ball in
centered at
with the radius
. Finally, we establish the necessary and sufficient conditions for the boundedness of the commutator
on mixed-norm Herz-slice spaces.
By
we denote the class of Lebesgue measurable functions on
. Let
. Denote the characteristic function of a set G by
. Denote the Lebesgue measure of a measurable set by
. Given
, let
with
,
when
, and
,
is the characteristic function of
S. Let
denote the collection of all Schwartz functions on
, equipped with the well-known classical topology determined by a countable family of semi-norms. We denote
by the space of infinitely differentiable complex-valued functions. Denotes the unit sphere in
by
:=
. We denote
by the space of all continuos, complex-valued functions with compact support. The letters
will denote
n-tuples of the numbers in
,
.
indicates that
∞ for each
. Moreover, for
and
, let
be its conjugate index, that is, and
satisfies
. Let
The letter is used for various constants, it is independent of the main parameters and maybe change from line to line. We denote a positive constant depending on the indicated parameters by . We write , mean that for some constant , especially, indicates that and .
3. Main Result
In this part, we first establish the separability of Herz-slice spaces with a mixed-norm, get a weak convergence on mixed-norm Herz-slice spaces, then, to show the Calderón–Zygmund operator T is bounded on mixed-norm Herz-slice spaces, we need to indicate that is well-defined on . Furthermore, we get the necessary and sufficient conditions for the boundedness of the commutator on mixed-norm Herz-slice spaces.
Now, we prove that is separable space.
Theorem 1. Let , and , . Then is separable space.
Proof. Let
and
, by using Corollary 2, there exist
such that
which implies that
is uniformly continuous. Thus, there exists dyadic cube
sequence and rational number
sequence such that
Denote
is a set of simple functions
ℏ and
with
is a dyadic cube’s sequence and
is a rational numbers’s sequence. It suffices to show that
ℏ is countable and dense in
. Thus,
is separable. The proof is complete. □
Theorem 2. Let , and , . Assume that there exists a positive constant D such thatthen there exists a subsequence that is weakly convergent in the space . Proof. For
, applying ([
23], Theorem 3.1), we can see that
Therefore, we only need to explain that there exists a subset
such that for any
,
where
.
By using Theorem 1, suppose that
is dense in
. Denote
Applying the Hölder inequality,
Then there exist convergent subsequences
via the boundedness of
. Repeat this step, we get a subsequence
of
satisfying
is convergence. Thus, for
there exists a convergent subsequences
. After a diagonal process, for any
, we know that a subsequence
is convergence and
For any
and
, there exists
have
If
k and
are large enough, then
therefore, for any
,
is a Cauchy sequence. Let
and
be a linear bounded functional on
. By using ([
23] Theorem 3.1), we see that there exist
such that
the proof is completed. □
In following that, we investigate the boundedness of the Calderón–Zygmund operator T on .
Theorem 3. Let , , , and . If the Caldeón-Zygmund operator T satisfies
- (1)
Given function f, and with , - (2)
Given function f, and with ,We have T is bounded on .
Before we come to the proof of the above theorem, we should to explain that is well-defined on .
Lemma 4. Let , and , . Then has an absolutely continuous quasi-norm.
Proof. Let
be a sequence of measurable sets. For any
,
and
. We see that,
when
, we know that
, which, together with Definition 1, we have
By Definition 4, we know that has an absolutely continuous quasi-norm. This accomplishes the desired result. □
Lemma 5. Let X be a ball quasi-Banach function space having an absolutely continuous quasi-norm and M is bounded on X. If , then in X as .
Proof. Let
. Then the assertion is trivial. Combining Lemma 3, we have
where
for all
. We know that the class
is dense in
X, see [
38], where
is continuous functions with compact support. Then there is a function
such that
based on these, when
, we have
for all
. Namely,
uniformly on compact sets as
. Thus, we get
with
compact. By using Definition 2.1, we get
Finally by using (
14) and (
15),
the proof is complete. □
As a consequence of Lemma 5, we deduce the following result.
Corollary 1. Let X be a ball quasi-Banach function space having an absolutely continuous quasi-norm. Suppose that M is bounded on X. We have the class is dense in X.
We know that
is ball quasi-Banach function space (see [
23], Proposition 3.2). We immediately get the following result, which means that
is well-defined on
.
Corollary 2. Let , and , . Then the class is dense in .
Proof of Theorem 3. By Corollary 2, we know
is well defined on
, Observe that
First estimate
, using Lemma 1 and (
13), we have
For
, we use Lemma 2, we have
First estimate
, we use Lemma 1 and (
13), we know that
We can estimate
by Lemma 2, we have
We need to pay attention to
T is bounded on
as in Definition 1 (see [
29], Corollary 4.1), to estimate
,
Finally, we to estimate
, we use Lemma 1 and (5.1), we conclude that
For
, using Lemma 2, we have
Using Lemma 1 and (
12), we know that
For
, by Lemma 2 yields
we got what we want. □
The following theorem aims is to give the necessary conditions for the boundedness of the commutator on mixed-norm Herz-slice spaces.
Theorem 4. Let and , . Let . T be an Calderón–Zygmund operator satisfies the local size conditionwhere , and with , and the conditionwhere , supp and with . We have is also bounded on , provided that and . Before we come to the proof of the above Theorem, we need give the boundedness of on .
Lemma 6. Let , and . Let . For any , then is bounded on
Proof. By [
23], Lemma 5.1, we find that
is well defined on
. Set
. We write
For the part of
, by using [
29], Theorem 2.4, we get
For
, denote
is the mean value of
b on
. Observe that if
, from the properties of
functions (see Stein [
30]), by Lemma 1 and Remark 1, we have
Therefore, for
,
For
, using Hölder’s inequality can deduce that
Since
. For part of
, when
and
, we have
Thus, by Lemmas 1 and 2, we can see that
Hence, when
,
For
, using the Hölder inequality, we have
Because . We got what we want. □
In what follows, we show commutators is also bounded on .
Proof of Theorem 4. Write
. We then have
For
, since
is bounded on
in [
29], we have
For
, when
and
, using (
16), we get
For part of
, when
and
, then
. Therefore,
Now let
. By Lemma 1 and (
17), we first deduced that, when
,
By this and (1.1), we have
because
. This accomplishes the desired result. □
Finally, we prove the other side, namely, the sufficient conditions for the boundedness of the commutator on mixed-norm Herz-slice spaces.
Theorem 5. Let , and , . Let T be an Calderón–Zygmund operator with satisfying (9), (10), and . Let b be a locally integrable function. If is bounded on , then . Using a similar way of [
23], Proposition 3.2, we immediately have the following lemma.
Lemma 7. Let and . If , then the characteristic function on satisfieswhere , . Proof of Theorem 5. Suppose that boundedness of commutator
on
. We use the same method as Janson [
39]. Let us take
and
such that
. Then for
,
such that
can be written as the absolutely convergent Fourier series,
with
, where the exact form of the vectors
is unrelated, then we have the expansion
Given cubes
and
, if
and
, then
Let
and
. Then
Setting
and
, we have
Applying Hölder’s inequality and Lemma 7, we have
then
. This completes the conclusion. □