ROUGH SINGULAR INTEGRALS ON PRODUCT SPACES

where, p.v. denotes the principal value. It is known that if Φ is of finite type at 0 (see Definition 2.2) and Ω ∈ 1(Sn−1), then TΦ,Ω is bounded on Lp for 1<p <∞ [15]. Moreover, it is known that TΦ,Ω may fail to be bounded on Lp for any p if the finite-type condition is removed. In [8], Fan et al. showed that the Lp boundedness of the operator TΦ,Ω still holds if the condition Ω ∈ 1(Sn−1) is replaced by the weaker condition Ω ∈ Lq(Sn−1) for some q > 1. Subsequently, the Lp (1< p <∞) boundedness of TΦ,Ω was established under conditions much weaker than Ω ∈ Lq(Sn−1) [1, 6]. In particular, Al-Qassem et al. [1] established the Lp boundedness of TΦ,Ω under the condition that the function Ω belongs to the block space B 0,0 q (Sn−1) introduced by Jiang and Lu in (see [14]). In fact, they proved the following theorem.


Introduction and results.
For n ∈ N, n ≥ 2, let K(•) be a Calderón-Zygmund kernel defined on R n , that is, where Ω ∈ L 1 (S n−1 ) is a homogeneous function of degree zero that satisfies with dσ (•) being the normalized Lebesgue measure on the unit sphere.Let B n (0, 1) be the unit ball centered at the origin in R n .For a Ꮿ ∞ mapping Φ : B n (0, 1) → R d , d ≥ 1, consider the singular integral operator Bn(0,1) f x − Φ(y) K Ω (y)dy, (1.3) where, p.v. denotes the principal value.
It is known that if Φ is of finite type at 0 (see Definition 2.2) and Ω ∈ Ꮿ 1 (S n−1 ), then T Φ,Ω is bounded on L p for 1 < p < ∞ [15].Moreover, it is known that T Φ,Ω may fail to be bounded on L p for any p if the finite-type condition is removed.In [8], Fan et al. showed that the L p boundedness of the operator T Φ,Ω still holds if the condition Ω ∈ Ꮿ 1 (S n−1 ) is replaced by the weaker condition Ω ∈ L q (S n−1 ) for some q > 1. Subsequently, the L p (1 < p < ∞) boundedness of T Φ,Ω was established under conditions much weaker than Ω ∈ L q (S n−1 ) [1,6].In particular, Al-Qassem et al. [1] established the L p boundedness of T Φ,Ω under the condition that the function Ω belongs to the block space B 0,0 q (S n−1 ) introduced by Jiang and Lu in (see [14]).In fact, they proved the following theorem.
If Φ is of finite type at 0, then for 1 < p < ∞ there exists a constant C p > 0 such that R d ) (1.4) for any f ∈ L p (R d ).
It should be pointed out here that the condition Ω ∈ B 0,0 q (S n−1 ) in Theorem 1.1 was recently proved to be nearly optimal.In fact, Al-Qassem et al. [2] showed that if the condition Ω ∈ B 0,0 q (S n−1 ) is replaced by Ω ∈ B 0,ν q (S n−1 ) for some ν < 0, then the corresponding classical Calderón-Zygmund operator R n f (x − y)K Ω (y)dy (1.5) may fail to be bounded on L p at any 1 < p < ∞.
Fefferman [11] and Fefferman and Stein [12] studied singular integrals on product domains.Namely, they studied operators of the form where n, m ≥ 2, In [12], it was shown that P Ω is bounded on L p (R n+m ) for 1 < p < ∞ if Ω satisfies some regularity conditions.Subsequently, the L p (1 < p < ∞) boundedness of P Ω was established under weaker conditions on Ω, first in [7] for Ω ∈ L q (S n−1 ×S m−1 ) with q > 1 and then in [9] for Ω ∈ q>1 B 0,1 q (S n−1 × S m−1 ) which contains q>1 L q (S n−1 × S m−1 ) as a proper subspace, where B 0,1 q represents a special class of block space on S n−1 × S m−1 ; for p = 2, it was proved by Jiang and Lu in [13]).The definition of block spaces will be recalled in Section 2 (see Definitions 2.2 and 2.3).
The analogue of the operators T Φ,Ω in (1.3) on product domains is defined as follows.
For N, M ∈ N, let Φ : B n (0,r ) → R N and Ψ : B m (0,r ) → R M be Ꮿ ∞ mappings.Define the singular integral operator P Ω,Φ,Ψ by (1.8) Using the ideas developed in [4,8], we can easily show that P Ω,Φ,Ψ is bounded on L p (1 < p < ∞) provided that Φ and Ψ are of finite type at 0 and Ω ∈ L q (S n−1 × S m−1 ) for some q > 1.However, the natural question that arises here is as follows.
In this paper, we will answer this question in the affirmative.In fact, we have the following theorem.
Theorem 1.3.Let P Ω,Φ,Ψ be given by (1.8).Suppose that Ω ∈ B 0,1 q (S n−1 × S m−1 ) for some q > 1.If Φ and Ψ are of finite type at 0, then for 1 < p < ∞ there exists a constant C p > 0 such that (1.9) Regarding the condition Ω ∈ B 0,1 q (S n−1 × S m−1 ) in Theorem 1.3, we should remark here that in a recent paper [5], Al-Salman was able to obtain a similar result to that in [2].More precisely, Al-Salman showed that the size condition Ω ∈ B 0,1 for some ε > 0, then the operator P Ω may fail to be bounded on L p for any p.
Also, in this paper we will give a similar result for the truncated singular integral operator where In fact, we have the following.
Theorem 1.4.Let P * Ω,Φ,Ψ be given by (1.1) with r = 1.Suppose that Ω ∈ B 0,1 q (S n−1 × S m−1 ) for some q > 1.If Φ and Ψ are of finite type at 0, then for 1 < p < ∞ there exists a constant C p > 0 such that It is worth pointing out that, as in the one-parameter setting, we can show that the L p boundedness of the operators P Ω,Φ,Ψ and P * Ω,Φ,Ψ may fail for any p if at least one of the mappings Φ and Ψ is not of finite type at 0.

Some definitions and lemmas.
We start by the following definition.Definition 2.1.Let U be an open set in R n and Θ : U → R d a smooth mapping.For x 0 ∈ U , it is said that Θ is of finite type at x 0 if, for each unit vector η in R d , there is a multi-index α so that (2.1) Definition 2.2.For 1 < q ≤ ∞, it is said that a measurable function b(x, y) on S n−1 × S m−1 is a q-block if it satisfies the following: (i) supp(b) ⊆ I, where I is a cap on S n−1 × S m−1 , that is, for some α, β > 0, x 0 ∈ S n−1 , and y 0 ∈ S m−1 ; (ii) b L q ≤ |I| −1/q , where 1/q + 1/q = 1.
Definition 2.3.The class B 0,1 where each c µ is a complex number; each b µ is a q-block supported on a cap I µ on S n−1 × S m−1 ; and In dealing with singular integrals along subvarieties with rough kernels, an approach well-established by now is to decompose the operator into an infinite sum of Borel measures then to seek certain Fourier transform estimates and certain L p estimates of Littlewood-Paley type.For more details, we advise the readers to consult [1,3,4,6,7,8,10], among others.A particular result that we will need to prove our results is the following result in [4] which is an extension of a result of Duoandikoetxea in [7].

.4)
Then for p 0 < p < p 0 , where p 0 is the conjugate exponent of p 0 , there exists a positive constant C p such that It is clear that inequality (2.4) is one of the key elements in Theorem 2.4.In particular, the range of the parameter p where (2.5) and (2.6) hold is completely determined by the largest p 0 where (2.4) holds.Clearly, if (2.4) holds for large p 0 → ∞, then (2.5) and (2.6) hold for all 1 < p < ∞.It turns out that to prove our results, we will indeed run into the case where we need to obtain (2.5) and (2.6) for all 1 < p < ∞.However, in our case this obstacle can be resolved.In fact, we will show that inequality (2.4) holds for all p 0 = 4, 8, 16,.... Our main tools to achieve this are Lemma 2.5 and Theorem 2.6.
By a quick investigation of the proof of [7, Lemma 1], we have the following.
for every f in L q (R n × R m ).Then the vector-valued inequality Clearly, if inequality (2.7) holds for all 1 < q < ∞, then inequality (2.8) holds for all p 0 = 4, 8, 16,... which is the case that we will need to prove our results.But in many applications including the ones in this paper inequality (2.7) is not always freely available for all 1 < q < ∞.However, this problem can be resolved by repeated use of Theorem 2.4 and Lemma 2.5 along with a certain bootstrapping argument (see (2.15)-(2.22)).To be more specific, we prove the following theorem.
. Suppose also that the maximal functions M (l,s) Then the inequality holds for all 1 < p ∞, and f in L p (R n × R m ).The constant C is independent of B and the linear transformations L and Q.
Proof.Let d be a fixed positive integer.For 1 ≤ u ≤ d, we let π d u : R d → R u be the projection operator.By a similar argument to that in [10], we may assume that (2.11) Then one can easily verify that (2.12) By (2.11) we have (1,2) f (x,y) (2,1) f (x,y) (1,1) f (x,y) , (2.13) (1,2) f (x,y) (2,1) f (x,y) where ᏹ R d is the classical Hardy-Littlewood maximal function on R d .By Plancherel's theorem and (2.12), we get which implies by (2.9) and (2.14) that (2.16) By applying Lemma 2.5 (for q = 2) along with the trivial estimate Γ k,j ≤ CB 2 , we get which when combined with (2.9) and (2.13) implies that For p = ∞, the inequality holds trivially.The proof of the theorem is complete.
For suitable mappings Γ : R n → R N , Λ : R m → R M , and b : S n−1 × S m−1 → R, we define the measures {∆b ,Γ ,Λ,k,j,ρ : k, j ∈ Z − } and the related maximal operator (2.23) For l ∈ N, let Ꮽ l denote the class of polynomials of l variables with real coefficients. (2.24) The following result can be found in [15].
Lemma 2.7.For 1 < p ≤ ∞, there exists a positive constant C p such that The constant C p may depend on the degrees of the polynomials 1 ,..., d , but it is independent of their coefficients.
Lemma 2.8.Let Φ : B n (0, 1) → R N and Ψ : B m (0, 1) → R M be C ∞ mappings and let, ᏼ = (P 1 ,...,P N ) : R n → R N and ᏽ = (Q 1 ,...,Q M ) : R m → R M be polynomial mappings.Let b(•, •) be a function on S n−1 × S m−1 satisfying the following conditions: −1/q for some q > 1 and for some cap , where [•] is the greatest integer function.If Φ and Ψ of finite type at 0, then for 1 < p ≤ ∞ and f ∈ L p (R N ×R M ) there exists a positive constant C p which is independent of b such that (2.27) Proof.We will only present the proof of (2.26).By the definition of ∆ * b,ᏼ,Ψ ,ρ we notice that ∆ * b,ᏼ,Ψ ,ρ f (x,y) is dominated by where (2.29) By Lemma 2.7 we immediately get

.30)
where and b0 is a function on S m−1 defined by b0 . By the arguments in the proof of the L p boundedness of the corresponding maximal function in the oneparameter setting in [1, Theorem 3.8], we obtain (2.26).This ends the proof of our theorem.
By Lemma 2.7 we immediately get the following.