A Note on a Class of Generalized Parabolic Marcinkiewicz Integrals along Surfaces of Revolution

: In this article, certain sharp L p estimates for a speciﬁc class of generalized Marcinkiewicz operators with mixed homogeneity associated to surfaces of revolution are established. By virtue of Yano’s extrapolation argument, beside these estimates, the L p boundedness of the aforementioned operators under weaker assumptions on the kernels is conﬁrmed. The obtained results in this article are fundamental extensions and improvements of numerous previously known results on parabolic generalized Marcinkiewicz integrals.


Introduction
Let s ≥ 2 be an integer, and let S s−1 denote the unit sphere in the Euclidean space R s , which is equipped with the normalized Lebesgue surface measure dσ.
For λ = λ 1 + iλ 2 (λ 1 ∈ R and λ 2 ∈ R + ), we define the kernel K ,h on R s by where h : R + → C is a measurable function and is a function belonging to L 1 (S s−1 ) that satisfies the conditions and For a suitable function Φ : R + → R, we consider the generalized parabolic Marcinkiewicz operator where g ∈ S(R s+1 ) and τ > 1.
It is obvious that when α 1 = α 2 = · · · = α s = 1, then we have α = s and κ(ω) = |ω|. In this instance, we denote M ,h,Φ , denoted by M , reduces to the classical parametric Marcinkiewicz integral operator. Historically, the integral operator M was introduced by Stein in [2] where he established the L p boundedness of M for all p ∈ (1, 2] whenever λ = 1 and belongs to the space Lip γ (S s−1 ) for some 0 < γ ≤ 1. Afterwards, the above result was improved by Hörmander in [3]. Indeed, he confirmed the L p boundedness of M for all p ∈ (1, ∞) under the conditions λ > 0 and ∈ Lip γ (S s−1 ) for some 0 < γ ≤ 1. Later on, the authors of [4] extended and improved the result of Stein. In fact, they proved that M is bounded on L p (R s ) for all p ∈ (1, ∞) if λ = 1 and ∈ C 1 (S s−1 ). Subsequently, the L p boundedness of the M (τ),c ,h,Φ under various assumptions on the kernels has attracted a considerable amount of attention from many mathematicians. For instance, Walsh in [5] obtained the L 2 boundedness of the operator M provided that ∈ L(logL) 1/2 (S s−1 ) and λ = 1. Furthermore, he found that M will lose the L 2 boundedness when the assumption ∈ L(log L) 1/2 (S s−1 ) is replaced by ∈ L(log L) υ (S s−1 ) for any υ ∈ (0, 1/2). On the other hand, the L p (1 < p < ∞) boundedness for M was proved in [6] whenever λ = 1 and belongs to the block space B (0,−1/2) q (S s−1 ) for some q > 1. In the same article, the authors verified that −1/2 in B (0,−1/2) q (S s−1 ) cannot be substituted by any number in (−1, −1/2), meaning that M is still bounded on L 2 (R s ).
Recently, the authors of [19] confirmed that if ∈ B ,h,Φ is satisfied for all p ∈ [τ, ∞). Further, they proved that if the condition µ ∈ (1, 2] is replaced by µ > 2, then the boundedness of M (τ),c ,h,Φ is satisfied for all p ∈ (1, τ) if 2 < µ < ∞ and µ ≤ τ , and also for all p ∈ (µ , ∞) if 2 < µ ≤ ∞ and µ > τ . Very recently, the authors of [20] extended the results in [19]. In fact, they confirmed that the above results are true not only for the case Φ(κ) = κ, but also when Φ belongs to the class I or the class D, which were introduced in [21]. Precisely, the class I is the collection of all C 1 functions Φ : R + → R that are non-negative and satisfy the following: (i) Φ is monotone and Φ(κ) > 0 for all κ ∈ R + ; . In addition, D is the class of all C 1 functions Φ : R + → R that are non-negative and satisfy the following: For recent advances in the study of the operator M (τ),c ,h,Φ , we refer the readers to consult [19,20,[22][23][24][25] and the references therein.
Let us recall the definitions of some spaces that are related to this work. For m > 0, we let ∆ m (R + ) denote the collection of all functions h that are measurable on R + such that It is clear that Υ m (R + ) ⊂ ∆ ν (R + ) for any m ≥ 1 and ν > 0.
In this work, let L(log L) m (S s−1 ) (for m > 0) denote the class of all functions that are measurable on S s−1 such that In addition, let B (0,υ) q (S s−1 ) (with υ > −1 and q > 1) denote the block space that was introduced in [26]. Furthermore, H τ,p ε denotes the homogeneous Triebel-Lizorkin space, which is defined as follows: assume that θ ∈ R s and W ∈ C ∞ 0 (R s ) is a radial function satisfying the following: ,h,Φ,κ was recently introduced in [27] under some weak conditions on the kernels. This result was studied in [28] under some weaker conditions on the kernels only for the case τ = 2. In view of the result in [19] on the boundedness of classical generalized parametric Marcinkiewicz M ,h,Φ,κ satisfied under the same conditions assumed in [19] while replacing the condition τ = 2 by a weaker condition τ > 1 and when Φ belongs to I or D?
In the next section, we shall give an affirmative answer to this question.

Statement of results
We devote this section to presenting the main results of this article. Indeed, they are formulated as follows. Theorem 1. Let belong to the space L q S s−1 and h belong to the space and The constant C p is independent of , h, Φ, q, and µ.
By utilizing the conclusions of Theorems 1 and 2, Yano's extrapolation argument, and the same method used in [12,29,30], we obtain the following results.
Here and henceforward, the letter C refers to a positive number whose value does not depend on the primary variables and also that is not necessary the same at each occurrence.

Some Auxiliary Lemmas
We devote this section to establishing some preliminary lemmas that are needed to prove our main results. Let us begin by introducing some notations. Let b ≥ 2. Define the family of measures {σ K ,h ,Φ,t := σ t : t ∈ R + } and the corresponding maximal operators σ * h and M h,b on R s+1 by R s+1 where |σ t | is defined in the same way as σ t , but replacing h by | h|.
The following two lemmas play a great role in the proofs of Theorems 1 and 2. They can be established by following the exact procedure utilized in [31] (with a very simple minor modification).

Lemma 1.
Let b ≥ 2, ∈ L q S s−1 and h ∈ Υ µ (R + ) for some q, µ > 1. Suppose that Φ belongs to I or D. Then, there are constants δ and C with 0 < δ ≤ min{ 1 2 , m 2q , m 2α } such that for all j ∈ Z, where σ t is the total variation of σ t and m is denoted to be the distinct numbers of α j .

Lemma 2.
Suppose that b, , h, and Φ are given as in Lemma 1. Then, for 1 < p < ∞, there exists C p > 0 such that inequalities and hold for all g ∈ L p (R s+1 ).
One of the key tools in proving our main results is Plancherel's Theorem, which states that g L 2 (R s+1 ) = C ĝ L 2 (R s+1 ) . A significant step towards handling our main results is to prove the following: Lemma 3. Let b ≥ 2 and Φ be in I or D. Assume that ∈ L q S s−1 for some 1 < q ≤ 2 and h ∈ Υ µ (R + ) for some 1 < µ ≤ 2. Then, there is C > 0 such that for arbitrary functions {A j (·), j ∈ Z} on R s+1 , nd we have for 1 < τ ≤ p < ∞, and for 1 < p < τ.
In the same manner, we get the following lemma.
In the same manner, except invoking Lemma 4 with b = 2 q instead of Lemma 3, we can prove Theorem 2.

Conclusions
In this work, we found appropriate L p bounds for the generalized parabolic Marcinkiewicz operators M (τ) ,h,Φ,κ whenever belongs to the space L q (S s−1 ). By employing these bounds along with Yano's extrapolation argument, we establish the L p boundedness of M (τ) ,h,Φ,κ under very weak conditions assumed on the integral kernels. The results in this article represent substantial extensions and improvements to previously known results. In fact, our results improve and extend the results in [2][3][4][5][6][7][8][9]19,22,25]. We notice that the range of p, |1/p − 1/2| < {1/µ − 1/2} becomes a tiny open interval when µ → 1 + . In future work, we aim to prove the L p boundedness of the operator M (τ) ,h,Φ,κ for the full range of p ∈ (1, ∞) and also whenever ∈ B