On the boundedness of parabolic maximal operators on product domains.

In this article, appropriate sharp L p bounds for a certain class of rough maximal operators M Ω , γ with mixed homogeneity are established. Specifically, when the function Ω belongs to L q ( S m - 1 × S n - 1 ) with m , n ≥ 2 and q > 1 , the boundedness of the such operators is obtained. Further, the extrapolation argument employed in [1] is applied on these gotten bounds to obtain the L p boundedness of the aforementioned operators whenever the kernels are in the space L ( log ⁡ L ) 2 γ ' ( S m - 1 × S n - 1 ) or in the block space B q ( 0 , 2 γ ' - 1 ) ( S m - 1 × S n - 1 ) with 1 < γ ≤ 2 and q > 1 . Our obtained results are considered substantial extensions and improvements of what was known previously.

Introduction and statement of the results

Throughout this paper, we assume that and is the unit sphere in the Euclidean space , which is equipped with the normalized Lebesgue surface measure .

Let be a function defined by with and are fixed real numbers in the interval . For a fixed , we denote the unique solution of the equation by . The metric space is called mixed homogeneity space related to . Let (with ) be the diagonal matrix The polar coordinates transform related to the metric space is given by the following: . Hence, , where and refers to the Jacobian of the above transforms.

It was proved in [2] that and also a real constant C exists such that for all .

For a measurable function φ on , and an integrable function Ω over satisfying and we let be the kernel on defined by where and are fixed real numbers.

For , we let be the class of all measurable functions such that and we let . It is clear that for any .

For , we consider the maximal operator , given bywhere

When and , then , , , , and . Also, when , then the operator is just the classical maximal operator on the product domains (denoted by ). Ding introduced the operator in [3] in which he showed that whenever Ω belongs to the space , then is bounded on . Independently Al-Salman in [4] and Al-Qassem and Pan in [5] improved the results of Ding. Indeed, they proved the (for ) boundedness of under the condition . Furthermore, they found that the condition is optimal in the sense that the operator will lose the boundedness if we replace by for some . On the other side, Al-Qassem in [6] got another improvement to that result in [3]. In fact, he established that if Ω belongs to the block space , then is of type for all . Moreover, he satisfied the optimality of the condition in the sense that the space cannot be replaced by (with ) so that is bounded on . For recent advances in the investigation of the operators and , the readers are referred to consult [7], [8], [9], [10], [11], and the references therein.

It is worth mentioning that the operator in the one parameter case was systematically investigated in [12].

In this article, we shall give appropriate estimates for the parabolic maximal operator on the product domains under weak conditions that and , and then we shall use these obtained estimates in the extrapolation argument that employed in [1] to get some new extended results in the parabolic maximal functions. Moreover, we shall present and prove several applications of our main result. The first result of this paper is described in the following theorem.

Let for some with and satisfy the conditions (1.1) - (1.2) . Let for some . Then a positive constant exists such that for with , and where .

By the conclusions of Theorem 1.1 and applying the same extrapolation argument used in [1], [13], we obtain the following result:

Suppose that and , and suppose that . Then is bounded on for , and it is bounded on for with .

We should state here that for the case and , Theorem 1.2 extends and improves the results given in [3], [4], [5], [6]. Further, in the proof of Theorem 1.2, we employ the extrapolation method found in [1], [13], which is considered a new alternative technique. More novelty, we can utilize the boundedness of the operator to satisfy the boundedness of the singular integral operator . In fact, as a direct result of Theorem 1.2 and the fact that for any , we conclude the following:

Let Ω be given as in Theorem 1.2. Let for some . Then the singular operator given by (1.4) is bounded on for ; and it bounded on for with .

Again, when and , then the operator (denoted by ) is just the classical singular integral operator on the product spaces. Historically the operator was introduced in [14] in which the author gave the () boundedness of the such operator whenever and Ω satisfies some regularity conditions. Later on, the aforementioned operator was elaborated very much by Stein and Fefferman in [15]. Subsequently, the study of the boundedness of under some various conditions on φ and Ω has considered by many researchers. For instance, the authors of [16] proved that is bounded on for provided that and φ satisfies some certain integrability-size condition. Also, they established that the condition is optimal. For more information about the importance of such operators, we refer the readers (for example to [14], [15], [16], [17], [18], [19], among numerous reference).

We point out that the study of the parabolic singular operator in the one parametric case was introduced by Fabes and Riviére in [2] in which the authors established the () boundedness of whenever and . Later on, the authors of [20] improved the above result. Precisely, they proved that is bounded for any under the condition that .

We point out that this is the first time to investigate the boundedness of on the product domains.

Throughout the rest of this article, whenever the letter C appears, it refers to a positive bounded constant that may be different at different occurrences and independent of the essential variables.

Preliminary lemmas

In this section, we give some auxiliary lemmas which will be needed to prove Theorem 1.1. Let us begin with the following lemma, which is a special case of a theorem in [[21], p. 1248].

For , let the maximal function be given by Then, for , we have

Suppose that satisfies the conditions (1.1) - (1.2) . Define the maximal function by Then, for , we have

It is clear thatwhere . Hence, by Lemma 2.1, we get that where and . This completes the proof. □

The next lemma plays a significant role in the proof of our main result.

Let,with, and satisfy(1.1)-(1.2). For, we definebywhereThen, a positive constant C exists so thatwhere,and,are denoted to the distinct numbers of,.

It is trivial to verify that

On the other side, by using [[22], Lemma 2.2], we get that where . Combine (2.3) with the trivial estimate we deduce that for any , Thanks to Hölder's inequality, we obtain Thus, as , choose ϵ small enough so that , we have Again, combine the last inequality with the estimate (2.2), we obtain Similarly, we derive

Now, we use the cancellation condition (1.2) to get that which when combined with the trivial estimate we deduce Consequently, In the same manner, we obtain Therefore, by (2.5)-(2.8), we finish the proof of this lemma. □

Proof of Theorem 1.1

In the proof of this theorem, we employ similar arguments used in the proof of [Theorem 1.6, [1]] and [Theorem 1.1, [23]]. By the duality, where a linear operator defines by Thus, So, to prove Theorem 1.1, it suffices to show it only when and , and then apply the interpolation theorem to get for all with .

First, let us consider the case ; we estimate our inequality as follows: assume that and . Then for all , we have which gives that, for any φ with , for almost every where , which implies

Now let us consider case ; we follow arguments similar to those in [24]. Choose a collection of functions defined on that satisfy the following: Define the multiplier operators in by Hence, for any Schwartz function f on , we have where and Thus, to prove our main result for , it is enough to show for some and for all . First, we estimate the -norm of ; by Fubini's theorem, Plancherel's theorem and Lemma 2.3, where , and . Hence, when we choose ϵ small enough, the inequality (3.5) is satisfied for .

Now to prove (3.5) for , we use the duality. So, we reach that there is with such that So, by Hölder's inequality plus Lemma 2.2, we deduce that where . Thus, which when combined with the estimate (3.6) leads to that there is so that for all . Therefore, by (3.4) and (3.7), the proof of the main result is complete.

Declarations

Author contribution statement

Mohammed Ali: Conceived and designed the analysis; Analyzed and interpreted the data; Contributed analysis tools or data; Wrote the paper.

Funding statement

This research did not receive any specific grant from funding agencies in the public, commercial, or not-for-profit sectors.

Data availability statement

No data was used for the research described in the article.

Declaration of interests statement

The authors declare no conflict of interest.

Additional information

No additional information is available for this paper.