Estimates for Generalized Parabolic Marcinkiewicz Integrals with Rough Kernels on Product Domains

: We prove L p estimates of a class of generalized Marcinkiewicz integral operators with mixed homogeneity on product domains. By using these estimates along with an extrapolation argument, we obtain the boundedness of our operators under very weak conditions on the kernel functions. Our results in this paper improve and extend several known results on both generalized Marcinkiewicz integrals and parabolic Marcinkiewicz integrals on product domains.


Introduction
Throughout this article, let s ≥ 2 (s = κ or η) and S s−1 be the unit sphere in the Euclidean space R s which is equipped with the normalized Lebesgue surface measure dσ s (·) ≡ dσ.
On the other hand, the investigation of the L p boundedness of the operator G (µ) ,h was considered by many authors. For example, Al-Salman introduced G (µ) ,h in [10] in which he proved that G . Later on, the authors of [11] improved the results presented in [10]. In fact, they proved the L p boundedness of G (2) ,h for all |1/2 − 1/p| < min{1/2, 1/ } whenever in B Here, ∆ (R + × R + ) (for > 1) refers to the set of all measurable functions h such that Let us now recall the definition of Triebel-Lizorkin spaces on product domains. Let 1 < µ, p < ∞ and − → c = (c 1 , c 2 ) ∈ R × R. The homogeneous Triebel-Lizorkin space where for s ∈ {κ, η} and x ∈ R s , ψ j,s (x) = 2 −js D s (2 −j x) and D s ∈ C ∞ 0 (R s ) is radial function satisfies the following: The authors of [12] proved that the space Recently, the authors of [13] employed the extrapolation argument of Yano [14] to prove that whenever Ω lies in the space L(log L) 2/µ (S κ−1 × S η−1 ) or in the space where refers to a special class of block spaces introduced in [15]. Very recently, the result in [13] was improved in [16] in which the authors In the view of the results in [11] regarding the boundedness of the parabolic Marcinkiewicz operator G (2) ,h and the results in [16] regarding the boundedness of the generalized parametric Marcinkiewicz operator M (µ) ,h , we have the following natural question: Is the integral operator G (µ) ,h bounded under the same conditions on h and as that was assumed in [16]? In this article, we shall answer the above question in the affirmative. In fact, we prove the following: There then exists a real number A p > 0 such that ,h (g) h ∆ (R + ×R + ) and A p is independent of , h, q, Now by using the estimates in Theorems 1 and 2 and following the same method as employed in [17] along with the extrapolation argument as in [14,18,19], we obtain the following results.

Theorem 3.
Assume that h is given as in Theorem 1.

Remark 1.
(i) For any 0 < γ ≤ 1, m > 0 and q > 1, the following inclusions hold and are proper: (ii) For the special cases h ≡ 1 and µ = 2, the authors of [7] showed that M (2) ,1 is bounded on L p (R κ × R η ) for all p ∈ (1, ∞) under the condition Ω ∈ L(log L)(S κ−1 × S η−1 ). In addition, they found that this condition is the weakest possible condition so that the boundedness of M ,1 holds. On the other hand, the L p (1 < p < ∞) boundedness of M (2) ,1 was proved in [8] if Ω ∈ B . Hence, The results in Theorem 4 are improvement as well as generalization to the results in [10,13]. (iv) When µ = with 2 < < ∞, Theorem 4 gives the boundedness of G (µ) ,h for all p ∈ (1, ∞), which obviously gives the full range of p.
Throughout the rest of the paper, the letter A represents a positive constant which is independent of the essential variables and its value is not necessarily the same at each occurrence.

Auxiliary Lemmas
In this section, we need to introduce some notations and establish some lemmas. For γ ≥ 2, consider the family of measures {σ K ,h ,s,r := σ s,r : s, r ∈ R + } and its concerning maximal operators σ * h and M h,γ on R κ × R η given by where |σ s,r | is defined in the same way as σ s,r except that h is replaced by |h | . We shall need the following two lemmas from [11].

Lemma 4.
Let and {F j,k (·, ·), j, k ∈ Z} be given as in Lemma 3. Suppose that h ∈ ∆ (R + × R + ) for some ∈ (1, ∞). Then there exists a positive constant A h, such that for all p ∈ (1, µ) if µ ≤ and γ ≥ 2; and A proof of this Lemma can be constructed by following a similar argument as that employed in the proof of Lemma 3 and following similar argument as that used in the proofs of Theorems 4-5 in [16] (with minor modifications). We omit the details.
Proof of Theorem 2. A proof can be constructed by following a similar approach as that used in the proof of Theorem 1 except that we employ Lemma 4 instead of Lemma 3. We omit the details.