Boundedness of a class of rough maximal functions.

In this work, we obtain appropriate sharp bounds for a certain class of maximal operators along surfaces of revolution with kernels in L q ( S n - 1 ) , q > 1 . By using these bounds and using an extrapolation argument, we establish the L p boundedness of the maximal operators when their kernels are in L ( log L ) α ( S n - 1 ) or in the block space B q 0 , α - 1 ( S n - 1 ) . Our main results represent significant improvements as well as natural extensions of what was known previously.

Introduction and main results

Throughout this article, let , , be the n-dimensional Euclidean space and be the unit sphere in equipped with the normalized Lebesgue surface measure . Also, let for and denote the exponent conjugate to p; that is, .

Let , where is a measurable function and Ω is a homogeneous function of degree zero on that is integrable on and satisfies the cancelation property

For , define to be the set of all measurable functions that satisfy the condition , and define .

For a suitable mapping , we define the maximal operator for by where is a real-valued polynomial.

When , we denote by . Also, when , we denote by which is the classical maximal operator that was introduced by Chen and Lim in [17]. The authors of [17] proved that when and for some , then the boundedness of is satisfied for . This result was improved by Al-Salman in [10]; he established the boundedness of for all provided that . Moreover, he pointed out that the condition is optimal in the sense that in cannot be replaced by any smaller positive number. In addition, the last result was generalized by Al-Qassem (see [4, Theorem 1.5]). Indeed, he verified that is bounded on for all and under the condition . Later on, Al-Qassem in [4] improved the above results. Precisely, he obtained that if for some , ; and ϕ is , convex and increasing function with , then is bounded on for any with ; and it is bounded on for . On the other hand, when Ω belongs to the block spaces for some , then the author of [3] showed that is bounded on for all . Furthermore, he found Ω which lies in for all such that in not bounded on . Subsequently, the study of the boundedness of under various conditions on the function has been performed by many authors. The readers can see [9, 12, 20, 21, 23–25], and [28] for the significance of considering integral operators with oscillating kernels.

We point out that the study the maximal operator was initiated by Al-Salman in his work in [11]. In fact, he investigated the () boundedness of under the condition for some . For more information about the importance and the recent advances on the study of such operators, the readers are referred to [1, 2, 5, 27], and the references therein.

In view of the results in [4] as well as the results in [11], it is natural to ask whether the parametric maximal operator is bounded on under weak conditions on Ω, ϕ, and γ. We shall obtain an answer to this question in the affirmative as described in the next theorem. Precisely, we will establish the following result.

Theorem 1.1

Suppose that , , and satisfy condition (1.1) with . Suppose also that is in , convex and increasing function with . Let be a polynomial of degree m and be given by (1.2). Then there exists a constant such that for and ; and where , , and is a positive constant that may depend on the degree of the polynomial P but it is independent of Ω, ϕ, q, and the coefficients of the polynomial P.

By the conclusion from Theorem 1.1 and applying an extrapolation argument (see [8, 11] and [26]), we get the following.

Theorem 1.2

Suppose that Ω is given as in Theorem 1.1 and is given by (1.2), where ϕ is in , convex and increasing function with . If , then is bounded on for and ; and it is bounded on for .

Here and henceforth, the letter C denotes a bounded positive constant that may vary at each occurrence but is independent of the essential variables.

Preliminary lemmas

This section is devoted to present and prove some auxiliary lemmas which will be used in the proof of Theorem 1.1. We start with the following lemma which can be derived by applying the arguments (with only minor modifications) used in [11].

Lemma 2.1

Let , , and satisfy condition (1.1) with . Assume that is an arbitrary function on , and assume also that is a polynomial of degree such that is not one of its terms and . For , define by where Then a positive constant C exists such that

Proof

On the one hand, it is clear that Also, it is easy to get that with . Without loss of generality, we may assume that . Then, we follow the same steps as in [11, (2.9)–(2.12)] to prove that the inequality holds for some constant . Therefore, combining (2.4) with the trivial estimate (2.3) leads to  □

We shall need the following lemma which can be acquired by using the argument employed in the proof of [14, Lemma 4.7].

Lemma 2.2

Let be a homogeneous function of degree zero and satisfy condition (1.1). Suppose that is in , convex and increasing function with . Let the maximal function be given by Then, for , there exists a positive number so that for every .

Using a similar argument as in the proof of [4, Theorem 1.6], we obtain the following.

Lemma 2.3

Let , , and satisfy condition (1.1) with . Assume that is given as in Theorem 1.1. Then there exists a constant such that for .

Since for , it is enough to prove this lemma for . It is clear that Let be a smooth partition of unity in adapted to . More precisely, we require the following: Define the multiplier operators in by Hence, for , we have where By using [4, ineq. (3.10)] together with Lemma 2.2, we get for some constant and for all . Therefore, by (2.6) and (2.7), we immediately satisfy inequality (2.5) for all . □

Proof of the main results

Proof of Theorem 1.1

The proof of Theorem 1.1 mainly depends on the approaches employed in the proof of [11, Theorem 1.1] and [4, Theorem 1.6]. By duality, for , we get which gives where is a linear operator defined by Now if we assume that for ; and then by applying the interpolation theorem for the Lebesgue mixed normed spaces to the last two inequalities, we directly obtain for with ; and . Thus, to prove our theorem, it is enough to prove it only for the cases and .

Case 1 (if ). Assume that and . Then, for all , we have Hence, by taking the supremum on both sides over all h with , we reach for almost every where , which implies

Case 2 (if ). We use the induction on the degree of the polynomial P. If the degree of P is 0, then by Lemma 2.3 we get that, for all , Now, assume that (1.3) is satisfied for any polynomial of degree less than or equal to m with . We need to show that (1.3) is still true if . Let be a polynomial of degree . Without loss of generality, we may assume that , and also we may assume that P does not contain as one of its terms. Let be a collection of functions satisfying the following conditions: Define the multiplier operators in by and set Thanks to Minkowski’s inequality, we have where and Let us first estimate -norm of . Define Hence, by generalized Minkowski’s inequality, it is easy to show that If , then by a simple change of variables, Plancherel’s theorem, Fubini’s theorem, and Lemma 2.1, we get that However, if , then by the duality, there exists with such that So, by Hölder’s inequality and Lemma 2.2, we conclude that where . Thus, which when combined with (3.6) gives that there is so that for all . Therefore, by (3.5) and (3.7), we obtain

Now, let us estimate the -norm of . Let . Define and by Thus, by Minkowski’s inequality, we deduce On the one hand, since , then by our assumption, for all . On the other hand, since we have then by the Cauchy–Schwarz inequality, we reach that Hence, by Lemma 2.2, we get that for all . Therefore, by (3.9)–(3.11), we obtain Consequently, by (3.4), (3.8), and (3.12), we finish the proof of Theorem 1.1. □

Proof of Theorem 1.2

Assume that Ω satisfies condition (1.1). If with , then as in [14], we can decompose Ω as a sum of functions in . In fact, we have a sequence of functions in with such that Thus, we get the following: Since , then we have for , and since then by Minkowski’s inequality and (3.13)–(3.15), we deduce that However, if with and , then where each is a complex number, each is a q-block supported in an interval on and For each μ, define the blocklike function by Then it is easy to show that has the following properties: Without loss of generality, we may assume that . So, Therefore, by Minkowski’s inequality and the above procedure, we get that for all . □

Further results

In this section, we present some additional results that follow by applying Theorems 1.1 and 1.2. The first result concerns the boundedness of oscillatory singular integrals. More precisely, we deduce the following.

Theorem 4.1

Assume that , and satisfying condition (1.1). Let for some and ϕ be given as in Theorem 1.1. Then the singular integral operator given by is bounded on for .

The proof of this case is reached by using the observation that In fact, by the last inequality and Theorem 1.2, we obtain that is bounded on for with . Furthermore, by a standard duality argument, we satisfy the boundedness of for with . So, if , then we are done. However, if , then we apply the real interpolation theorem to attain the boundedness of for (). This completes the proof. □

The generalized parametric Marcinkiewicz operator related to the operator is defined by As a direct consequence of the notice that for , it is easy to derive the following result.

Theorem 4.2

Let Ω satisfy condition (1.1) and belong to the space for some and . Suppose that ϕ and P are given as in Theorem 1.2. Then the parametric Marcinkiewicz operator is bounded on for with ; and it is bounded on for .

We point out that by specializing to the case , and , then the operator (denoted by ) is just the classical Marcinkiewicz integral operator introduced by Stain in [29] in which he showed that is of type for provided that for some . Subsequently, the operator has been studied by many authors (for instance, see [11, 13, 15, 18], as well as [19] and the references therein). For the significance and recent advances on the study of the generalized parametric Marcinkiewicz operators, we refer the readers to consult [7] and [6] among others.

It is worth mentioning that Theorem 4.1 generalizes the corresponding results in [4, 14, 16], and [22]. However, Theorem 4.2 extends and improves the results found in [11, 13, 19], and [29].