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In this article, suitable estimates for a class of rough generalized Marcinkiewicz integrals on product spaces are established. By these estimates, together with employing Yano’s extrapolation technique, we obtain the boundedness of the aforementioned integral operators under weak conditions on singular kernels. A number of known previous results on Marcinkiewicz as well as generalized Marcinkiewicz operators over a symmetric space are essentially improved or extended.
Throughout this article, we let ( or m) and be a Euclidean space of dimensions d. Furthermore, we let be the unit sphere in equipped with the normalized Lebesgue surface measure .
For , we assume that
where h is a measurable function defined on and is a measurable function defined on which satisfies the following properties:
For and , we consider the generalized parametric Marcinkiewicz integral over the symmetric space
When , , and , we denote the operator by . In this case, is essentially the classical Marcinkiewicz integral on product spaces. The study of the boundedness of the operator was started by Ding in , in which he established the boundedness of whenever lies in the space . Thereafter, the boundedness of has been studied by many researchers. For example, Choi in  proved the boundeness of if satisfies the weaker condition . In , the authors proved that is bounded on for all if . Later on, the authors of  improved and extended the above results. In fact, they showed that the operator is of type for all , provided that . Furthermore, they found that by adapting the technique employed in  to the product space setting, the condition is optimal in the sense that it cannot be replaced by a weaker condition for some . On the other hand, Al-Qassem in  showed that is bounded on for all if with . Moreover, he showed that the condition is optimal in the sense that we cannot replace it by for any . Here, is a special class of block spaces introduced in .
By using an extrapolation argument, the authors of  proved that the boundedness of for all whenever for some and lies in either the space or in the space with . Here, (for ) indicates the class of measurable functions h which are defined on and satisfy
Recently, the authors of  established that if and or , then
for all .
It is well known that the Marcinkiewicz integral, , on product spaces naturally generalizes the Marcinkiewicz integral in one parameter setting which was introduced by E. Stein in . The singularity of is along the diagonals and . The study of singular integrals on product spaces and the study of as well as its generalizations, which may have singularities along subvarities, has attracted the attention of many authors in recent years. One of the principal motivations for the study of such operators is the requirements of several complex variables and large classes of “subelliptic” equations. For more background information, readers may refer to Stein’s survey articles [11,12].
Let us recall the definition of Triebel–Lizorkin spaces, . Assume that and . The homogeneous Triebel–Lizorkin space is defined to be the class of all tempered distributions f on such that
where for , for , and the functions and are radial functions satisfying the following proprieties:
if for some constant T;
It was shown in  that the space satisfies the following:
The Schwartz space is dense in ;
where denotes the exponent conjugate to p, that is, whenever and or for or , respectively.
In light of the results in  concerning the boundedness of the operator and of the results in  concerning the boundedness of the generalized operator , a natural questions arises in the following:
Question: Is the operator bounded under the same assumptions in  with replacing by ?
The main purpose of this work is to answer the above question affirmatively. Precisely, we have the following:
Let for some and for some . Then, there is a constant such that
for all if , and
for all if , where .
Assume that with and that Ω lies in with . Then, we have
for all if , and
for all if .
By employing the estimates in Theorems 1 and 2 and employing an extrapolation argument as in  (see also [15,16]), we obtain the following:
Let h be given as in Theorem 1.
If , then the inequality
holds for if , and for if .
If for some , then the inequality
holds for if , and for if .
Let for some and for some . Then, the operator is bounded on for if , and for if .
(1) The conditions assumed for Ω in Theorems 3 and 4 are the weakest conditions in their respective classes for the case and (see [4,6]).
(2) For the special case , Theorem 4 gives that is bounded on for all , provided that Ω belongs to or to , which is Theorem 2.7 in .
(3) The result in Theorem 3 in the case and essentially improves Theorem 2 in , in which the authors proved the boundedness of for . Hence, the range of p in Theorem 3 is better than the range of that obtained in .
(4) The authors of  proved the () boundedness of only for the special case and . Therefore, the results in Theorem 3 essentially improve the main results in .
(5) For the special case with , Theorem 4 leads to the boundedness of for all .
Henceforward, the constant C signifies a positive real number that could be different at each occurrence but is independent of all essential variables.
2. Auxiliary Lemmas
This section is devoted to introducing some notation and establishing some lemmas that will be needed to prove the main results of this paper. For , consider the family of measures and its corresponding maximal operators and on by
where is defined in the same way as but with replaced by .
Assume that , with and . Then, for any with , we have
Thanks to Hölder’s inequality, we obtain that
Therefore, Minkowski’s inequality for the integrals and Corollary 5 in  lead to
Inequality (7) is easily deduced from Inequality (6). □
The next lemma is found in  with very minor modifications. We omit the proof.
Let , for some and for some . Then, the following estimates hold:
where and is the total variation of .
In order to prove our main results, we need to prove the following lemmas.
Suppose that , with and with . Let and be arbitrary functions defined on . Then, there exists a positive constant such that the inequality
holds for all .
We employ a similar argument used in . First, let us consider the case . By duality, there is a non-negative function such that and
By Hölder’s inequality, it is easy to obtain that
Again, by using Hölder’s inequality and Inequalities (11) and (12), we have
where . Therefore, since , then we have
for all . For the case , we use (12) and Hölder’s inequality to obtain that
Finally, we consider the case . Define the linear operator on any function by . Then, we have
where . Define the linear operator on any function by . Hence, by interpolating between (23) and (24), we obtain
for all with . The proof of this lemma is complete. □
3. Proof of the Main Results
Proof of Theorem 1. We employ similar arguments as those in [19,20]. Assume that , with and with . By Minkowski’s inequality, we obtain
Take , then . Choose a set of functions defined on , with the following properties:
where is independent of . Define the operators and for . Therefore, we obtain that for any ,
Therefore, to prove Theorem 1, it is sufficient to prove that there exists a positive constant such that
for all with , and also for all with .
Let us first estimate the norm of for the case . Indeed, by Plancherel’s theorem, Fubini’s theorem, and Lemma 2, we obtain
where and .
However, we estimate the -norm of in the following. By Lemmas 3 and 5, together with the Littlewood–Paley theory and invoking Lemma 2.3 in , we obtain
for all with , and also for all with . Therefore, by interpolating (28) with (29), we immediately obtain (27). This ends the proof of Theorem 1.
Proof of Theorem 2. To prove this theorem, we follow the exact procedure that was used in the proof of Theorem 1, employing Lemma 4 instead of Lemma 3.
In this article, we established appropriate bounds for the generalized parametric Marcinkiewicz integral operator under the assumption that for some . Then, we used these bounds, along with Yano’s extrapolation argument, to prove the boundedness of the operator under very weak conditions on the kernel function . Such conditions on are considered to be the best possible among their respective classes. The results in this article improve and extend several known results in the field of Marcinkiewicz and generalized Marcinkiewicz operators. In fact, our results improve and extend the results in [1,2,3,4,6,8,9,17].
Formal analysis and writing-original draft preparation: M.A. and H.A.-Q. All authors have read and agreed to the published version of the manuscript.
This research received no external funding.
Data Availability Statement
No data were used to support this study.
The authors would like to express their gratitude to the referees for their valuable comments and suggestions in improving writing this paper. In addition, they are grateful to the editor for handling the full submission of the manuscript.
Conflicts of Interest
The authors declare that they have no conflict of interest.
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