BOUNDEDNESS OF MARCINKIEWICZ INTEGRALS ON PRODUCT SPACES AND EXTRAPOLATION

In this article, we establish Lp estimates for the Marcinkiewicz integral operators with rough kernels on product spaces. These estimates and extrapolation arguments improve and extend some known results on Marcinkiewicz integrals. AMS Subject Classification: 40B15, 40B20, 40B25


Introduction
Let n, m ≥ 2, and let S N −1 (N = n or m) be the unit sphere in R N equipped with the normalized Lebesgue surface measure dσ = dσ(•).Also, let x ′ = x/|x| for x ∈ R n \ {0}, y ′ = y/|y| for y ∈ R m \ {0}.Let p ′ denote to the exponent conjugate to p.
The investigations of the L p (R n × R m ) boundedness of Marcinkiewicz integral operators on product spaces began by Ding in [17] in which he established the L 2 boundedness of M Ω,c if Ω ∈ L(log L) 2 (S n−1 × S m−1 ).Subsequently, it was studied by many mathematicians.For example, the author of [14] proved that M Ω,c is bounded for all 1 < p < ∞ provided that Ω ∈ L(log L) 2 (S n−1 × S m−1 ).For more information about the importance and the recent advances on the study of such operators, the readers are refereed (for instance to [1], [3], [13], [15], [16], [29], [30], as well as [31], and the references therein).
For d = 0, we let H d be the class of all functions φ : (0, ∞) → R which are smooth and satisfy the following growth conditions: for t ∈ (0, ∞), and C 1 , C 2 , C 3 plus C 4 are positive constants independent of t.Also, for γ ≥ 1, we let ∆ γ (R + × R + ) denote the collection of all measurable functions h : R The primary focus of this paper is establishing L p estimates of M ρ,τ Ω,h,φ,ψ for various functions φ ∈ H d 1 , ψ ∈ H d 2 for some d 1 , d 2 = 0, and h ∈ ∆ γ (R + × R + ); and then apply the extrapolation argument used in [4] to obtain new improved results.Our main result is formulated as follows: Then there exists a constant C p (independent of Ω, h, γ, and q) such that for |1/p − 1/2| < min{1/2, 1/γ ′ }, where The power of our theorem lies in using its conclusion and extrapolation (see [4]) to obtain improved results.In particular, Theorem 1 and extrapolation lead to the following theorem.
(1) The class H d was introduced by Fan and Pan in [21] in their studies of singular integrals.Model functions for the φ ∈ H d are φ(t) = t d with d > 0 or φ(t) = t r with r < 0. We point out that the class H d is empty when d = 0.
(2) The authors of [3] under the condition Ω ∈ L(log L)(S n−1 × S m−1 ).Furthermore, they proved that the exponent 1 is optimal for the L 2 boundedness of M 1,1 Ω,c to hold.
(3) Al-Qassem in [1] showed that M 1,1 Ω,h is of type (p, p) for p ∈ (1, ∞) if Ω ∈ B (0,0) q (S n−1 × S m−1 ) and h ∈ L ∞ (R + × R + ).Moreover, in the same paper Al-Qassem showed that the condition Ω ∈ B (0,0) q (S n−1 × S m−1 ) is optimal in the sense that there exists Ω ∈ B (0,ν) q for some q, γ > 1, then the L p boundedness of M ρ,τ Ω,φ,ψ,h was obtained in [9] for any p satisfying (5) In the one parameter case, the authors of [10] used the extrapolation arguments to show that if Ω belongs to the class L(log L) 1/2 (S n−1 ) or to the class Here and henceforth, the letter C denotes a bounded positive constant that may vary at each occurrence but independent of the essential variables.
The following lemma can be obtained by applying a similar argument used in [7].Lemma 3. Let Ω ∈ L q S n−1 × S m−1 for some q > 1 and satisfies the cancellation conditions (1).Suppose that φ ∈ H d 1 and ψ ∈ H d 2 for some Then there are constants C and α with 0 < α < 1 2q ′ such that Proof.We prove this lemma only for d 1 , d 2 > 0 because the proof for the other cases are essentially the same and requires only minor modifications.Also we prove this lemma for the case 1 < q ≤ 2, since L q S n−1 × S m−1 ⊆ L 2 S n−1 × S m−1 for q ≥ 2. By Schwarz inequality, we get that where By Van der Corput's lemma, the assumptions on φ and integration by parts, we conclude , which when combined with the trivial estimate |J(ξ, x, u)| ≤ C leads to for any 0 < α < 1.Thus, by Hölder's inequality we have By choosing 0 < 2αq ′ < 1, we get that the last integral is finite, and hence In the same manner, we obtain Using the cancelation property of Ω, we have by a simple change of variable that Since |φ (tr)| ≤ C (tr) d 1 and 1 2 < r < 1, we achieve .
Proof.By using the definition of σ Ω,φ,ψ,h,t,s , it is easy to show that (8) holds.By Hölder's inequality, we have that and if γ > 2, then by Hölder's inequality, we deduce Thus, in either case we reach , where ω = max{2, γ ′ }; hence, by Lemma 3 we obtain Therefore, The following lemma follows immediately by applying a well known argument found in [20].
Then for 1 < p ≤ ∞, there exists a constant C p such that 2 k e −iϕ(r)ξ dr r .
Following the same approaches used in the proof of Lemma 4, we achieve that Therefore, by invoking Theorem A of [20] we conclude that M ϕ (f ) is bounded on L p (R) for 1 < p ≤ ∞.
By Lemma 5 and using the same arguments as in [ [25], Proposition 1 (pp.72)] (se also [[21], Lemma 3.1]), we immediately get the following lemma.Lemma 6.Let ϕ ∈ H d for some d = 0 and u ∈ S N −1 .Define the maximal function Then, a positive constant C p exists such that for any for some γ > 1. Assume that and φ, ψ are given as in Theorem 1. Then for any f ∈ Proof.By Hölder's inequality, we have Hence, Minkowski's inequality for integrals yields that where •)(y) and • denotes the composition of operators.Thus, by using the last inequality and Lemma 6, we finish the proof of this lemma The following lemma can be obtained by applying the arguments (with only minor modifications) used in [4] and [7].

Lemma 5 .
Let ϕ ∈ H d for some d = 0. Define the maximal function