Generalized parabolic Marcinkiewicz integrals associated with polynomial compound curves with rough kernels

Abstract In this article, we study the generalized parabolic parametric Marcinkiewicz integral operators ℳ Ω , h , Φ , λ ( r ) { {\mathcal M} }_{{\Omega },h,{\Phi },\lambda }^{(r)} related to polynomial compound curves. Under some weak conditions on the kernels, we establish appropriate estimates of these operators. By the virtue of the obtained estimates along with an extrapolation argument, we give the boundedness of the aforementioned operators from Triebel-Lizorkin spaces to L p spaces under weaker conditions on Ω and h. Our results represent significant improvements and natural extensions of what was known previously.


Introduction
Throughout this article, let R n (n ≥ 2) be the n-dimensional Euclidean space and S n−1 be the unit sphere in R n equipped with the normalized Lebesgue surface measure dσ = dσ(·). Let α 1 , α 2 ,…,α n be fixed real numbers in the interval [1, ∞). Define the function H: R n × R + → R by ( ) = ∑ = H x ρ , with x = (x 1 , x 2 ,…,x n ) ∈ R n . Then, for each fixed x ∈ R n , the function H(x, ρ) is a strictly decreasing function in ρ > 0. We denote the unique solution of the equation H(x, ρ) = 1 by ρ = ρ(x). Fabes and Riviére showed in ref. [1] that (R n , ρ) is a metric space, which is known by the mixed homogeneity space related to { } = α i i n 1 . For ρ > 0, let A ρ be the diagonal n × n matrix: The change of variables related to the space (R n , ρ) is given by the transformation x ρ x ρ cos ϑ cos ϑ cos ϑ , cos ϑ cos ϑ sin ϑ , cos ϑ sin ϑ , sin ϑ . The authors of ref. [1] showed that J(x′) is a ( ) ∞ − S n 1 function, and there exists a constant L, such that 1 ≤ J(x′) ≤ L.
For a suitable mapping Φ: R n → R n , we define the generalized parabolic parametric Marcinkiewicz integral operators where r > 1; λ = τ + σi (τ, σ ∈ R with τ > 0); h: R + → C is a measurable function; and Ω is a real valued function on R n , integrable on S n−1 and satisfies the conditions.
Conversely, there has been a considerable amount of mathematicians with respect to the study of the boundedness of the generalized parametric Marcinkiewicz integrals . This operator was first introduced by Chen et al. [17] and showed that whenever Φ(u) = u, h ≡ 1, and Ω ∈ L q (S n−1 ) for some q > 1, then a positive constant C exists such that holds for all 1 < p, r < ∞, where f belongs to the homogeneous Triebel-Lizorkin space . Afterward, Le [18] improved the aforementioned result. Precisely, he established the inequality (1.3) for all p, r ∈ (1,∞) under the conditions that Φ(u) = u, Ω ∈ L(log L)(S n−1 ) and ∈ ( ) . For the significance and recent advances on the study of such operators, readers may refer to [16,[19][20][21][22]23].
Although many problems concerning the boundedness of the operator Ω ∈ L q (S n−1 ) for q > 1. Subsequently, the study of the L p boundedness of ( ) under various conditions on the kernel functions has been carried out by many researchers (see, for example, refs. [25][26][27][28][29][30]).
Let us recall the definition of the Triebel-Lizorkin spaces. For 1 < p, r < ∞ and α ∈ R, the homogeneous Triebel-Lizorkin space is a radial function satisfying the following conditions: The following properties of the Triebel-Lizorkin space are well known (for more details, see ref. [31]).
denote the set of all measurable functions h: R + → C that satisfy the condition γ any 0 and 1 . In this study, the class F denoted the set of all positive, increasing 1 functions ϕ:(0,∞) → R + satisfying the following conditions: (i) tϕ′(t) ≥ C ϕ ϕ(t) for all t > 0; and (ii) ϕ(2t) ≤ c ϕ ϕ(t) for all t > 0,where C ϕ , c ϕ are independent of t. There are many model examples for the class F such as t d with d > 0, t ι (ln(1 + t) κ ) with ι, κ > 0, real-valued polynomials P on R with positive coefficients and P(0) = 0, and so on.
for some 1 in Theorems 1.1-1.5 is independent of Ω, h, γ, q, and the coefficients of P j for 1 ≤ j ≤ n.
It is worth mentioning to the following remark related to our results and their optimality.
Moreover, they established the optimality of the condition . Furthermore, he showed that the condition Ω ∈ L(log L) 1/2 (S n−1 ) is optimal in the sense that the operator Ω,1 may lose the L 2 boundedness if Ω is assumed to be in the space Ω ∈ L(log L) ε (S n−1 ) for some 0 < ε < 1/2.  , Ω ∈ L(log L) 1/2 (S n−1 ), and Φ is given as in Theorem 1.1.
Here and henceforth, the letter C denotes a positive constant that may be different at different occurrences and independent of the essential variables.

Some notations and lemmas
In this section, we give some lemmas, which we shall need in the proof of the main results. Let = ( ) We shall need the following lemma from ref. [29].
By using Lemma 2.2 from [29], we directly obtain the following lemma.

Lemma 2.2.
Let Ω, ϕ be given as in Lemma 2.1, and let h ∈ Δ γ (R + ) for some γ > 1. Then, for any 1 ≤ s ≤ N, t > 0 and ξ ∈ R n , there exists a constant C > 0, such that . The constant C is independent of Ω, h, γ, and q, but depends on ϕ.
To prove Theorem 1.1, we employ the next lemmas with arguments similar to those in refs. [20] and [29].
Lemma 2.3. Let h ∈ Δ γ (R + ) for some 1 < γ ≤ 2 and Ω ∈ L q (S n−1 ) for some 1 < q ≤ 2 and θ = 2 q′γ′ . Assume that ϕ is given as in Lemma 2.1, and r is a real number with r > 1. Then, for 0 ≤ s ≤ N, there exists a constant C > 0, such that hold for arbitrary functions {g k (·), k ∈ Z} on R n . The constant C = C n,p,ϕ is independent of Ω, h, γ, q, and the coefficients of {P j } for all 1 ≤ j ≤ n.
Proof. First, we prove (2.6). For fixed p with r ≤ p < ∞, by duality, there is a nonnegative function  Hence, by (2.8) and (2.9) and Hölder's inequality, we have that which shows that (2.6) is satisfied for the case p = r.
Next, we prove (2.7). Let 1 < p < r. By the duality, there exist functions Following the same above argument, we obtain where ( ) = (− ) b x b x . Therefore, the inequality (2.7) follows from (2.12) and (2.14). This completes the proof of Lemma 2.3. □ In the same manner, we establish the following: Lemma 2.4. Let h ∈ Δ γ (R + ) for some 2 ≤ γ < ∞ and Ω ∈ L q (S n−1 ) for some 1 < q ≤ 2 and θ = 2 q′ . Suppose that ϕ is given as in Lemma 2.1, and r is a real number with r ≤ γ′. Then, for 0 ≤ s ≤ N and 1 < p < r, a positive constant C exists such that the inequality holds for arbitrary functions {g k (·), k ∈ Z} on R n . The constant C = C n,p,ϕ is independent of Ω, h, γ, q, and the coefficients of {P j } for all 1 ≤ j ≤ n.

(2.16)
Notice that for any b ∈ L p (R n ) with 1 < p < ∞, we have So, by using Lemma 2.2 from [35], we obtain Since p′ > r′, there is a nonnegative function b ∈ L (p′/r′) (R n ), such that for arbitrary functions {g k (·), k ∈ Z} on R n . The constant C = C n,p,ϕ is independent of Ω, h, γ, q, and the coefficients of {P j } for all 1 ≤ j ≤ n.
Proof. We follow the same aforementioned procedure as in (2.9); by a change of variable and Hölder's inequality, we obtain Since γ′ < p < ∞ with γ′ < r, then by duality, there exists a nonnegative function Consequently, by interpolation (2.22) with (2.23), and using the fact Let ∈ ∞ ψ 0 be supported in {|t| ≤ 1} and ψ(t) ≡ 1 for |t| ≤ 1/2. For 1 ≤ s ≤ N, t > 0 and ξ ∈ R n , define the family of measures {ω t,s } by where ν j = rank(L j ); → R R R : j ν ν j j and Q j : R n → R n are two nonsingular linear transformations satisfying ( ) ≤ ( ) ≤ ( ) R π Q ξ L ξ C R π Q ξ j ν n j j j ν n j j j (3.3) and π ν n j is a projection operator from R n to R ν j . It is easy to check that