1. Introduction
Throughout this article, the conjugate index of q is denoted by , which is defined by the relation . Moreover, in the Euclidean space (), the unit sphere is equipped with the normalized Lebesgue surface measure .
For fixed
,
, let
be a function given by
with
. The mixed homogeneity metric space related to
on
is defined by the metric
, where
is the unique solution to
(see [
1]). For
, we define the diagonal
matrix
by
For the space
, we use the weighted spherical coordinates
This change of variables gives that
, where
and
is the Jacobian of the weighted spherical coordinates.
By [
1], one can show that
and
For
with
), let
be the kernel on
given by
where
g is a measurable function on
; and
is a function in
satisfying the conditions
For a nice function
, the generalized parabolic Marcinkiewicz integral operators
is defined by
for
, where
and
.
It is clear that when
, then
and
. Hence, the operator
is just the generalized Marcinkiewicz integral operator, denoted by
. Moreover, when
,
, and
, then the operator
, denoted by
, becomes the classical parametric Marcinkiewicz integral operator. Historically, Stein, in [
2], introduced the operator
. Precisely, he showed that
is bounded on
for
provided that
with
. Subsequently, a considerable amount of research has been done to obtain the
boundedness of
(see for instant [
3,
4,
5,
6]).
The study of the boundedness of
was started by Hörmander in [
7]. In fact, he proved that
is of type
for
if
and
with
. Later on, the study of the operator
under very various conditions on the kernels attractted the attention of many mathematicians. For relevant results, we advice the readers to consult [
8,
9,
10,
11,
12,
13,
14,
15,
16], among others.
Afterward, the investigation to establish the boundedness of the integral operator
began. In fact, Chen, Fan and Ying introduced the operator
in [
17], where the authors showed that if
,
,
, and
for some
, then for all
,
Recently, this result was improved by Le in [
18]. Precisely, he satisfied (3) for all
provided that
is a function in
,
and
, where
refers to the class of all measurable functions
, which satisfy
For background information and recent advances on the study of the operator
, readers may refer to [
16,
19,
20,
21,
22], and the references therein.
It is worth mentioning that the parabolic Marcinkiewicz operator
was recently introduced by Ali in [
23].
For
and a measurable functions
g on
, define
where
with
for
and
.
Now, the class of all such functions with is denoted by . However, the class of all such functions with is denoted by .
It is easy to see that for any and ; and also for a given , for any .
Here, we recall some useful spaces defined on
and related to our work. In these spaces, we only deal with measurable functions. The space
(for
) is the set of all functions
on
such that
Moreover, the block space which was introduced in [
24] is denoted by
(for
and
).
The homogeneous Triebel–Lizorkin space is defined on as follows: Let and with be a radial function satisfying:
if ;
. For
and
with
,
where
denotes the tempred distribution class on
and
for
.
The following properties of the Triebel–Lizorkin space are well known (for more details see [
25]).
is dense in ;
for ;
if ;
.
We formulate our main results as follows:
Theorem 1. Suppose that and satisfies the conditions of Equations (1) and (2) for some . Let ψ be in , and is an increasing and convex function with . Then there is such thatfor andfor . The constant is independent of Ω
, g, r, q and ψ. Theorem 2. Assume that Ω
and ψ are given as in Theorem 1. Let for some . Thenfor if and ; and for if and . Where is independent of g, Ω
, r, q and ψ. Applying the extrapolation arguments as in [
20] to the conclusions of Theorems 1 and 2, one can find the following:
Theorem 3. Let Ω satisfy the conditions of Equations (1) and (2) and ψ be given as in Theorem 1.
If for some and , then for , If for some and , then for , If and , then for , If and , then for , The constant is independent of g, Ω, and ψ.
Theorem 4. Suppose that Ω satisfies Equations (1) and (2), for some and ψ is given as in Theorem 1.
If for some , thenfor if and ; and for if and . If , thenfor if and ; and for if and . Let us presnt some results related to the optimality of our main results.
Remark 1. The authors of [3] established that is bounded on for provided that with . Furthermore, they proved that the exponent in cannot be replaced by any number so that the operator is bounded on . The author of [6] showed that is of type if . Moreover, he proved that the operator is not bounded on whenever for some . The authors of [20] established the boundedness of the parametric Marcinkiewicz operator under the same our conditions on and λ only when and . Here and henceforth, whenever the letter C appears, it refers to a positive constant whose value may be different at each appearance but independent of the basic variables.
2. Some Preliminary Lemmas
This section is devoted to establish some preliminary results and to introduce some notations. For
, a suitable function
on
, and real valued measurable functions
on
and
g on
, the family of measures
and the corresponding maximal operators
and
on
are defined by
and
where
is defined in the same way as
, but with replacing
by
and
g by
. We write
and
for the total variation of
Let us recall the following lemma due to Cheng and Ding.
Lemma 1. [26] Suppose that . Let m denote the distinct numbers of . Then for , we havewhere C is independent of v and ξ. Lemma 2. Let for some satisfy the conditions of Equations (1) and (2), and let for some . Suppose that ψ is in , increasing and convex function with . Then there exist constants C and γ with such that
- (i)
- (ii)
If , then
- (iii)
If , then
for all . The constant C is independent of j, , q, and r.
Proof. By the definition of
, it is clear that
is true. Consider the case
for some
. A simple change of variables and Hölder’s inequality implies that
where
Since
and
, then we directly get
Thanks to the Schwartz inequality, we have
where
Let
. So, by Lemma 1, we get
for any
with
. Thus, by Hölder’s inequality we reach
Choose
, so we get that the last integral is finite, which leads to
and hence, by using Equation (6), it follows
Using the cancelation property of Equation (2) and a simple change of variables, we deduce that
which, when combined with the trivial estimate
, gives
Thus, for
, we have
Therefore,
for
. To prove
, we follow the same above procedure and the fact
for
This completes the proof of Lemma 2. □
We shall need the following lemma from [
27].
Lemma 3. Let s and s be fixed real numbers, and be a function from to . Let be the maximal function related to the curve given by Then for a constant exists so thatfor all . The constant is independent of and f. By following a similar argument found in [
28], which has its roots in [
29], we establish the following lemma.
Lemma 4. Let for some satisfying Equations (1) and (2), for some and . Suppose that ψ is given as in Theorem 1. Then for any , there exists a positive constant such that Proof. Without loss of generality we may assume that
. Let
be a Schwartz function such that
for
, and
for
. Let
. Define the family of measure
on
by
Then by the proof of Lemma 2 and a simple calculation, we obtain
Moreover let
be the maximal function defined by
Therefore, by using a similar argument as in the proof of Lemma in ([
29], p. 544) together with Equations (10)–(13), Lemma 3, ([
28], Inequality (2.13)), and ([
30], Lemma 2.3) we directly establish Equations (8) and (9). □
In order to handle our main results, we shall establish the following lemmas which can be derived by following the same procedure used in [
20].
Lemma 5. Let for some , for some and . Assume that λ is a real number with and ψ is given as in Theorem 1. Then, there exists a constant such thatandhold for arbitrary functions on . The constant is independent of Ω
, g, r, and q. Proof. Let us first prove Equation (14). For a fixed
p with
, by duality, there is a non-negative function
with
such that
Hölder’s inequality and a simple change of variable lead to
Thus, by Hölder’s inequality and Equations (16) and (17), we get that
where
. Hence, by the assumption on
and Lemma 4, we obtain
for
. Now for the case
; by Hölder’s inequality and Equation (17), we have that
Therefore, Equation (14) is satisfied for the case .
Now consider the case
. By the duality, there exist functions
defined on
with
such that
where
Since
, there is a nonnegative function
such that
So, by following the same above argument, we get
where
. Therefore, Equation (15) is satisfied by using Equations (19) and (21). This completes the proof of Lemma 5. □
By combining the proofs of ([
20], Lemmas 2.4–2.5) and Lemma 5, we get the following:
Lemma 6. Let for some , for some and . Assume that ψ is given as in Theorem 1, and λ is a real number. Then for arbitrary functions on , a positive constant C exists such thatfor if ; and also, for if . 3. Proof of Theorem 1
The idea of the proof of this theorem is taken from [
21], which has its roots in [
20]. Thanks to Minkowski’s inequality, we have that
Let
. Choose a collection of smooth functions
on
satisfying the following:
Define the multiplier operators
in
by
Hence, we deduce that for
,
where
Let us now estimate the
-norm of
. First, we consider the case
. Notice that
. Thus, by Plancherel’s theorem and Lemma 2, we have that
where
,
and
. Therefore,
Now consider the case
. By Lemma 5, we obtain that
However, for the case for
, we have
Therefore, by interpolating Equation (24) with Equations (25) and (26) and using Equation (23), we finish the proof of Theorem 1.
It should be noticed that the proof of Theorem 2 can be done in the same manner as in the above argument, whereas Lemma 6 with is needed instead of Lemma 5.