Marcinkiewicz Integrals along Subvarieties on Product Domains

We study the L p mapping properties of a class of Marcinkiewicz integral operators on product domains with rough kernels supported by subvarieties.


Introduction.
Let R d (d = n or d = m), d ≥ 2 be the d-dimensional Euclidean space and S d−1 be the unit sphere in R d equipped with the induced Lebesgue measure dσ . Suppose that Ω is a homogeneous function of degree zero on R n which is integrable on S n−1 and S n−1 Ω(y)dσ (y) = 0. Then the Marcinkiewicz integral operator µ Ω which was introduced by Stein in [18] is defined by Stein proved that if Ω ∈ Lip α (S n−1 ), (0 < α ≤ 1), then µ Ω is bounded on L p for all 1 < p ≤ 2 [18]. Since then, the study of the L p boundedness of µ Ω under various conditions on the function Ω has attracted the attention of many authors ( [1,4,5,7,10,13], among others). In particular, Chen et al. in [8] studied the L p boundedness of µ Ω under the following condition on the function Ω which was introduced by Grafakos and Stefanov in their study of singular integral operators [17]: for some α > 0. Chen et al. [8] showed that if Ω satisfies (1.2) for some α > 0, then µ Ω is bounded on L p for p ∈ ((2+2α)/(1+2α), 2+2α). It should be pointed out here that Grafakos and Stefanov showed that for any α > 0, the following relations hold: F α, S n−1 L log + L S n−1 , L log + L S n−1 F α, S n−1 , (1.3) where F(α,S n−1 ) is the space of all integrable functions Ω on S n−1 which satisfy S n−1 Ω(y)dσ (y) = 0 and (1.2). For conditions similar to (1.2), we refer the readers to consult [1,6].
Recently, a number of authors started to study the analog of the operator µ Ω on product domains. More precisely, let Ω ∈ L 1 (S n−1 × S m−1 ) be such that Ω(tx, sy) = Ω(x, y) for any t, s > 0, Then, the Marcinkiewicz integral operator on product domains ᏹ Ω,c is given by It has been known for quite some time that the operator ᏹ Ω,c is bounded on L p for all 1 < p < ∞ under the condition that Ω ∈ L(log + L) 2 (S n−1 ×S m−1 ) [9,12]. Recently, the L p boundedness of ᏹ Ω,c was established under the weaker condition Ω ∈ L(log + L)(S n−1 × S m−1 ); see Choi [11] for p = 2 and Al-Qassem et al. [2] for all 1 < p < ∞. Motivated by [1,6], the main purpose of this note is to investigate the L p boundedness of Marcinkiewicz integral operators on product domains with kernels satisfying conditions similar to (5) in [6] (see also [1]) and supported by subvarieties determined by polynomial mappings. To be more specific, let ᐂ(d, l) be the set of real-valued polynomials in R d which have degrees at most l. For ᏼ = (P 1 ,...,P N ) ∈ (ᐂ(d, l)) N and ᏽ = (Q 1 ,...,Q M ) ∈ (ᐂ(d, r )) M , consider the Marcinkiewicz integral operator In order to formulate our main results regarding the operators (1.7), we let Ᏼ(d, l) be the collection of all homogeneous polynomials of degree l in ᐂ(d, l). We will associate with ᐂ(d, l) the norm · given by P = ( |β|≤d |a β | 2 ) 1/2 , where P (y) = |β|≤d a β y β . Let where α > 0 and It is clear that L q (S n−1 × S m−1 ) F α (S n−1 , S m−1 ,l,r ) for all q > 1. Moreover, by (1.3) it can be easily shown that for any α > 0. Therefore, by (1.12) and the results of Choi [11] and Al-Qassem et al. [2] when Ω ∈ L(log + L)(S n−1 × S m−1 ), it is natural to investigate the L p boundedness of ᏹ Ω,ᏼ,ᏽ under the conditions (1.10).
Our main result is the following.  (1.4) and for some α > 0, then for all p ∈ ((2+2α)/(1+2α), 2+2α) and f ∈ L p (R N ×R M ). The constant C p is independent of the coefficients of the polynomials {P j ,Q k : In Section 4, we will show that Therefore, we obtain the following.

General tools.
For a nonnegative C ∞ radial function Φ on R and a linear transformation L : and 0 ≤ Φ(t) ≤ 1, and positive real numbers a and b, define the family of operators {Z We also, let M σ denote the maximal function corresponding to σ , that is, θ,γ,Φ,L,G be the operator given by Then by a well-known argument (see [16,19]), we obtain for all 1 < p < ∞. Now for p > 2, by an argument similar to that used in the proof of a lemma in [14, page 544], choose a nonnegative function (2.14) Thus by condition (a), condition (f), and (2.13), we get is bounded on L p (R N × R M ) for all p ∈ ((2 + 2α)/(1 + 2α), 2 + 2α). Moreover the L p bounds are independent of the linear transformations L and G.
Then, it is easy to see that

(2.23)
Hence the proof is complete by Lemma 2.1. Now, we have the following lemma which can be proved by a proper modification of the arguments in [3,15].
Finally, we turn to the proof of (viii). It can be easily verified that  S 1 , 1, 1).
A proof of Lemma 4.1 can be obtained by adapting the one parameter argument in [6]. For readers convenience, details are presented below.