ON CERTAIN ESTIMATES FOR PARABOLIC MARCINKIEWICZ INTEGRAL AND EXTRAPOLATION

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


Introduction
Throughout this article, let R n , n ≥ 2, be the n-dimensional Euclidean space, and let S n−1 be the unit sphere in R n which is equipped with the normalized Lebesgue surface measure dσ = dσ(•).Also, let p ′ denote to the exponent conjugate to p; that is 1/p + 1/p ′ = 1.
Let α i ≥ 1, (i = 1, 2, • • • , n), be fixed real numbers.For fixed x ∈ R n and ρ > 0, let F (x, ρ) = n i=1 x 2 i ρ 2α i .Then it is easy to see that for any fixed x ∈ R n \ {0}, F (x, ρ) is strictly decreasing function in ρ > 0. The unique solution of the equation F (x, ρ) = 1 is denoted by ρ(x).It was proved in [13] that ρ(x) is a metric on R n , and (R n , ρ) is the mixed homogeneity space.
For λ > 0, let For τ = a + ib (a, b ∈ R with a > 0), let K Ω,h (u) = Ω(u)h(ρ(u))ρ(u) τ −α , where h : [0, ∞) → C is a measurable function and Ω is a real valued and measurable function on R n with Ω ∈ L 1 (S n−1 ) that satisfying the conditions Ω(A λ x) = Ω(x), and where J(x ′ ) is a function will be defined later.Define the parabolic Marcinkiewicz integral operator M τ Ω,h for f ∈ S(R n ) by When τ = 1 and h = 1, we denote M τ Ω,h by µ Ω .The parabolic Littlewood-Paley operator µ Ω was introduced by Xue, Ding and Yabuta in [20] in which they proved that µ Ω is bounded for p ∈ (1, ∞) provided that Ω ∈ L q (S n−1 ) for q > 1. Subsequently, the study of the L p boundedness of µ Ω under various conditions on the function Ω has been studied by many authors.For example, Cheng and Ding improved the above result in [8]; they obtained the L p boundedness of µ Ω when Ω belongs to the Hardy space H 1 (S n−1 ) for 1 < p < ∞.However, the authors of [6] established that µ Ω is bounded under the condition Ω ∈ L(log L) 1/2 (S n−1 ) for 1 < p < ∞.In addition, the authors of [10] found that µ Ω is bounded when Ω belongs to the block space B (0,−1/2) q (S n−1 ) for 1 < p < ∞ and q > 1.Recently, Wang, Chen and Yu verified in [19] that if we replace n by n + 1, and y by (y, φ(ρ(y))), where φ is a polynomial of degree m, then µ Ω is bounded on L p (R n+1 ) for p ∈ ( 2+2ν 1+2ν , 2 + 2ν) provided that Ω ∈ F (ν, S n−1 ) for some ν > 0, where F (ν, S n−1 ) denotes the set of all Ω which are integrable over S n−1 and satisfying sup We point out that the class of the operators µ Ω is related to the class of the parabolic singular integral operators The class of the operators T Ω belongs to the class of singular Radon transforms, which has considered to study by many mathematicians (we refer the readers, in particular, to [13], [15] and [16]). If In this case, µ Ω is just the classical Marcinkiewicz integral, which was introduced by Stein in [18].For more information about the importance and the recent advances on the study of such operators, the readers are refereed (for instance to [2], [3], [5], [9], [11], [12], [14], and the references therein).
Our main interest in this paper is to study the L p boundedness of the parabolic Marcinkiewicz integral under weak conditions on Ω and h, and then apply an extrapolation method to establish new improved results.In this work, we let ∆ γ (R + ) ( for γ > 1) denote the collection of all measurable functions h : [0, ∞) → C satisfying In this article, we extend and improve some known results in the parabolic Marcinkiewicz operators (see [1], [6], [10], and [20]).Our main result is formulated as follows: Theorem 1.Let Ω ∈ L q S n−1 for some 1 < q ≤ 2 and h ∈ △ γ (R + ) for some γ > 1.Then for any f ∈ L p (R n ) with p satisfying |1/p − 1/2| < min{1/2, 1/γ ′ }, a constant C p (independent of Ω, h, γ, and q) exists such that where The fruit of our result is earned by using its conclusion and the extrapolation method (see [4]).In particular, Theorem 1 and extrapolation lead to the following theorem.
Throughout this paper, the letter C denotes a bounded positive constant that may vary at each occurrence but independent of the essential variables.

Some Lemmas
In this section, we give some auxiliary lemmas used in the sequel.Let us first recall the polar coordinates transform in the mixed homogeneity space (R n , ρ).
is the Jacobian of the above transforms.It was shown in [13] In order to prove Theorem 1, we need the following lemmas.Lemma 3. [17].Suppose that λ ′ i s and α ′ i s are fixed real numbers, and where A λ is defined as above and m denotes the distinct numbers of {α i }.
We shall recall the following lemma due to Chen and Ding.
Let θ ≥ 2. For a measurable function h : R + → C and Ω : S n−1 → R, we define the family of measures {σ Ω,h,t : t ∈ R + } and its corresponding maximal operators σ * Ω,h,t and M h,θ on R n by and where |σ Ω,h,t | is defined in the same way as σ Ω,h,t , but with replacing h by |h| and Ω by |Ω|.We write ν t,s for the total variation of ν t,s .
Lemma 6.Let θ ≥ 2, h ∈ ∆ γ (R + ) for some γ > 1, Ω ∈ L q S n−1 for some q > 1.Then there exist constants C and β with 0 hold for all k ∈ Z, where . The constant C is independent of k and ξ.
Proof.We prove this lemma only for the case 1 < q ≤ 2, since L q S n−1 ⊆ L 2 S n−1 for q ≥ 2. By Hölder's inequality, we obtain that On one hand, if 1 < γ ≤ 2, then by a change of variable we get that ξ| .Since β < m/α, then by lemma 4 we get that As 0 < β < m 2q ′ , we get that the last integral is finite, and hence Thus, we derive On the other hand, if γ > 2, then by using Hölder's inequality, we get that Following the above procedure give and therefore, To prove the other estimate in Lemma 6, we use the cancelation property of Ω.
Since γ > 1 and 1 2 < ρ < 1, we obtain that which when combine with the trivial estimate provides The proof is complete.
Lemma 7. Let Ω ∈ L q S n−1 for some 1 < q ≤ 2 and h ∈ ∆ γ (R + ) for some γ > 1.Then for any f ∈ L p (R n ) with γ ′ < p ≤ ∞, there exists a constant C p (independent of Ω, h and f ) such that Proof.By Hölder's inequality, we have Using Minkowski's inequality for integrals gives Consequently, by using Lemma 3, we finish the proof of lemma 7.
Proof.We employ some ideas from [1], [4] and [14].By Schwarz's inequality, we obtain Let us first prove this lemma for the case 2 ≤ p < 2γ 2−γ .By duality, there is a non-negative function By this, equation ( 10) and a change of variable, we derive , then by Lemma 7, Hölder's inequality and the same arguments used in [4], we achieve that .
In the same manner, we achieve the following lemma.
holds for arbitrary functions {g k (•), k ∈ Z} on R n .