BOUNDS FOR MARCINKIEWICZ INTEGRALS ALONG SURFACES AND EXTRAPOLATION

Let n ≥ 2 and Sn−1 denote the unit sphere in Rn which is equipped with the normalized Lebesgue surface measure dσ = dσ(·). Also, let x′ = x/|x| for x ∈ Rn \ {0} and p′ denote the exponent conjugate to p; that is 1/p+1/p′ = 1. For ρ = a+ib (a, b ∈ R with a > 0), let KΩ,h(u) = Ω(u ′)h(|u|) |u|, where h : [0,∞) → C is a measurable function and Ω is a homogeneous function of degree zero on Rn with Ω ∈ L1(Sn−1) and ∫


Introduction
Let n ≥ 2 and S n−1 denote the unit sphere in R n which is equipped with the normalized Lebesgue surface measure dσ = dσ(•).Also, let x ′ = x/|x| for x ∈ R n \ {0} and p ′ denote the exponent conjugate to p; that is 1/p + 1/p ′ = 1.
For ρ = a+ib (a, b ∈ R with a > 0), let K Ω,h (u) = Ω(u ′ )h(|u|) |u| ρ−n , where h : [0, ∞) → C is a measurable function and Ω is a homogeneous function of degree zero on R n with Ω ∈ L 1 (S n−1 ) and Let d = 0 and H d be the class of all functions φ : (0, ∞) → R which are smooth and satisfy the following growth conditions: for t ∈ (0, ∞), where C 1 , C 2 , C 3 and C 4 are positive constants independent of t.
For φ ∈ H d , h and Ω as above, we define the Marcinkiewicz integral operator M ρ Ω,φ,h , initially for C ∞ 0 on R n , by .
If φ(t) = t, we denote M ρ Ω,φ,h by M ρ Ω,h .The operators M ρ Ω,φ,h have their roots in the classical Marcinkiewicz integral operators which were introduced by Stein in [16].When h = 1 and ρ = 1, Stein established the The Marcinkiewicz integral operators have attracted the attention of many authors for along time due to the powerful role they play in many significant problems arisings in mathematics such as Poisson integrals, singular integrals and singular Radon transforms.
The study of parametric Marcinkiewicz integral operator M ρ Ω,h was initiated by Hörmander in [13] in which he satisfied the Ω,1 under the conditions ρ > 0 and Ω ∈ Lip α (S n−1 ) with α > 0. However, the authors of [14] showed that M ρ Ω,1 is still bounded on L p (R n ) for 1 < p < ∞ when Re(ρ) > 0 and Ω ∈ Lip α (S n−1 ) with 0 < α ≤ 1.These results were improved in [11].In fact, the authors proved that On the other hand, Al-Qassem and Al-Salman obtained in [1] that (S n−1 ) with q > 1.Further, they established the optimality of the condition Ω ∈ B (0,−1/2) q (S n−1 ) in the sense that −1/2 in B (0,−1/2) q (S n−1 ) cannot be replaced by any smaller number.Subsequently, the study of the L p boundedness of M 1 Ω,1 under various conditions on the kernels has received a large amount of attention of many authors.For example, Walsh in [18] found that M 1 . Moreover, he pointed out that the condition Ω ∈ L(log L) 1/2 (S n−1 ) is optimal in the sense that the operator M 1 Ω,1 may lose the L 2 boundedness if Ω is assumed to be in the space L(log L) ε (S n−1 ) for some ε < 1/2.Later on, under the same above conditions, Al-Salman et al. in [4] improve the result of [18].Precisely, he showed the same result for any 1 < p < ∞.
Recently, it was proved in [5] In view of the results in [5] and [6], a question arises naturally.Does the L p boundedness of the operators M ρ Ω,φ,h holds under the conditions φ ∈ H d , Ω belongs to the space Ω ∈ L(log L) 1/2 (S n−1 ) ∪ B (0,−1/2) q (S n−1 ) for some q > 1 and h ∈ ∆ γ (R + , dt t ) for some γ > 1.We shall obtain an answer to this question in the affirmative as described in the following theorems.
for some γ > 1, Ω ∈ L q S n−1 for some 1 < q ≤ 2 and φ ∈ H d for some d = 0. 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 conclusion from Theorem 1 and applying an extrapolation method as in [3] and [15] lead to the following theorem.
Here and henceforth, the letter C denotes a bounded positive constant that may vary at each occurrence but independent of the essential variables

Preparation
In this section, we present some definitions and establish some lemmas used in the sequel.Let us start this section by introducing the following definition.Definition 3. Let θ ≥ 2. For a suitable function φ defined on R + , a measurable function h : R + → C and Ω : S n−1 → R, we define the family of measures {σ Ω,φ,h,t : t ∈ R + } and the corresponding maximal operators σ * Ω,φ,h and M Ω,φ,h,θ on R n by where |σ Ω,φ,h,t | is defined in the same way as σ Ω,φ,h,t , but with replacing Ωh by |Ωh|.We write t ±α = inf{t +α , t −α } and σ for the total variation of σ.
In order to prove Theorem 1, it suffices to satisfy the following lemmas.
Proof.We prove this lemma only for the case d > 0 because the proof for d < 0 is essentially the same and requires only minor modifications.Also, we prove this lemma for 1 < q ≤ 2, since L q S n−1 ⊆ L 2 S n−1 for q ≥ 2. We reach (3) directly by using the definition of σ Ω,φ,h,t .By Hölder's inequality, we obtain that On one hand, if 1 < γ ≤ 2, then by a change of variable we get that e −iφ(ts)ξ•(x−y) ds s .As in [7]; write J(ξ, x, y) = where By Van der Corput's lemma, the assumptions on φ and integration by parts, we conclude which when combined with the trivial estimate |J(ξ, x, y)| ≤ ln 2 gives for any 0 < α < 1.This yields to .
By choosing 0 < 2αq ′ < 1, we get that the last integral is finite, and hence .
On the other hand, if γ > 2, then by using Hölder's inequality, we get e −iφ(st)ξ•x e iφ(st)ξ•y ds s Using Van der Corput's lemma and the above procedure gives and therefore, To prove the other estimate in (4), we use the cancelation property of Ω; By this, and since |σ Ω,φ,h,t (ξ)| ≤ (ln 2), we obtain that Therefore, by ( 6) and ( 7), we get (4).The proof is complete.