A Class of Rough Generalized Marcinkiewicz Integrals on Product Domains

: 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.


Introduction
Throughout this article, we let d ≥ 2 (d = n or m) and R d be a Euclidean space of dimensions d. Furthermore, we let S d−1 be the unit sphere in R d equipped with the normalized Lebesgue surface measure dµ d (·) ≡ dµ.
When α = 2, h ≡ 1, and λ 1 = 1 = λ 2 , we denote the operator M (α) Ω,h by M Ω . In this case, M Ω is essentially the classical Marcinkiewicz integral on product spaces. The study of the L p boundedness of the operator M Ω was started by Ding in [1], in which he established the L 2 boundedness of M Ω whenever Ω lies in the space L(log L) 2 (S n−1 × S m−1 ). Thereafter, the boundedness of M Ω has been studied by many researchers. For example, Choi in [2] proved the L 2 boundeness of M Ω if Ω satisfies the weaker condition Ω ∈ L(log L)(S n−1 × S m−1 ). In [3], the authors proved that M Ω is bounded on L p (R n × R m ) for all p ∈ (1, ∞) if Ω ∈ L(log L) 2 (S n−1 × S m−1 ). Later on, the authors of [4] improved and extended the above results. In fact, they showed that the operator M Ω is of type (p, p) for all 1 < p < ∞, provided that Ω ∈ L(log L)(S n−1 × S m−1 ). Furthermore, they found that by adapting the technique employed in [5] to the product space setting, the condition Ω ∈ L(log L)(S n−1 × S m−1 ) is optimal in the sense that it cannot be replaced by a weaker condition Ω ∈ L(log L) 1−ε (S n−1 × S m−1 ) for some ε ∈ (0, 1). On the other hand, Al-Qassem in [6] showed that M Ω is bounded on is optimal in the sense that we cannot replace it by By using an extrapolation argument, the authors of [8] proved that the L p boundedness of M Ω,h for all |1/2 − 1/p| < min{1/γ , 1/2} whenever h ∈ ∆ γ (R + × R + ) for some γ > 1 and Ω lies in either the space indicates the class of measurable functions h which are defined on R + × R + and satisfy Recently, the authors of [9] established that if h ≡ 1 and Ω ∈ L(log L) 2/α (S n−1 × S m−1 ) or for all p ∈ (1, ∞).
It is well known that the Marcinkiewicz integral, M Ω , on product spaces naturally generalizes the Marcinkiewicz integral in one parameter setting which was introduced by E. Stein in [10]. The singularity of M Ω is along the diagonals {x = ω} and {y = υ}. The study of singular integrals on product spaces and the study of M Ω 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, where φ k (x) = 2 −kn E(2 −k x) for k ∈ Z, ψ j (y) = 2 −jm J(2 −j y) for j ∈ Z, and the functions E ∈ C ∞ 0 (R n ) and J ∈ C ∞ 0 (R m ) are radial functions satisfying the following proprieties: It was shown in [13] that the space where p denotes the exponent conjugate to p, that is, 1/p + 1/p = 1 whenever 1 < p < ∞ and p := 1 or p := +∞ for p := +∞ or p := 1, respectively. In light of the results in [8] concerning the boundedness of the operator M Ω,h and of the results in [9] concerning the boundedness of the generalized operator M Ω,h bounded under the same assumptions in [8] with replacing α = 2 by α > 1?
The main purpose of this work is to answer the above question affirmatively. Precisely, we have the following: and Ω ∈ L q S n−1 × S m−1 for some q ∈ (1, 2]. Then, there is a constant C p,Ω,h such that By employing the estimates in Theorems 1 and 2 and employing an extrapolation argument as in [14] (see also [15,16]), we obtain the following: Theorem 3. Let h be given as in Theorem 1.

Remark 1.
(1) The conditions assumed for Ω in Theorems 3 and 4 are the weakest conditions in their respective classes for the case α = 2 and h ≡ 1 (see [4,6]).
(3) The result in Theorem 3 in the case α = 2 and 1 < γ ≤ 2 essentially improves Theorem 2 in [8], in which the authors proved the L p boundedness of M (2) Ω,h for p ∈ ( 2γ γ −2 , 2γ 2−γ ). Hence, the range of p in Theorem 3 is better than the range of that obtained in [8].
Henceforward, the constant C signifies a positive real number that could be different at each occurrence but is independent of all essential variables.

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 θ ≥ 2, consider the family of measures {µ K Ω,h ,r,s := µ r,s : r, s ∈ R + } and its corresponding maximal operators µ * h and S h,θ on R n × R m by R n ×R m f dµ r,s = 1 r λ 1 s λ 2 where |µ r,s | is defined in the same way as µ r,s but with Ωh replaced by |Ωh| . and where C p,h, Proof. Thanks to Hölder's inequality, we obtain that
The next lemma is found in [8] with very minor modifications. We omit the proof.
In order to prove our main results, we need to prove the following lemmas.
Proof. By duality, there is a set of functions {M j,k (ω, υ, r, s)} defined on R n × R m × R + × where Since γ ≥ 2 ≥ γ ≥ α, then by Hölder's inequality we obtain Since p > α , there exists a function Therefore, a simple change in variable together with Lemmas 1 and (19) give Therefore, by (18) and (20), Inequality (17) is proved. Consequently, the proof of Lemma 4 is complete.
Therefore, to prove Theorem 1, it is sufficient to prove that there exists a positive constant ε such that for all p ∈ ( αγ α+γ −1 , α γ α −γ ) with γ ≥ α, and also for all γ < p < ∞ with γ ≤ α. Let us first estimate the norm of A t,i ( f ) for the case p = α = 2. Indeed, by Plancherel's theorem, Fubini's theorem, and Lemma 2, we obtain