Parametric Marcinkiewicz Integral and Its Higher-Order Commutators on Variable Exponents Morrey-Herz Spaces

. In this article, we prove the boundedness of the parametric Marcinkiewicz integral and its higher-order commutators generated by BMO spaces on the variable Morrey-Herz space. All the results are new even when α ð · Þ is a constant.


Introduction
Throughout the entirety of this article, we assume that n ≥ 2, ℝ n is the n-dimensional Euclidean space, and S n−1 is the unit sphere in ℝ n equipped with the normalized Lebesgue measure dρ.The function Ω is assumed to be homogeneous of degree zero on ℝ n with Ω ∈ L 1 ðS n−1 Þ and where x ′ = x/jxj for any x ∈ ℝ n \ f0g.For ρ ∈ ð0, nÞ, the parametric Marcinkiewicz integral M ϱ Ω of higher dimensions is defined as follows: Let B be a ball with a radius τ > 0, and a center x ∈ ℝ n .A locally integrable function Λ is said to be in the BMO space, if it satisfies where Λ B = jBj −1 Ð B ΛðtÞdt and jEj denotes the Lebesgue measure of the set E in ℝ n .For Λ ∈ BMO, i ∈ ℕ, the i -order commutator for the parametric Marcinkiewicz integral M ϱ Ω,Λ i is defined as follows: If ρ = 1 in (2), then the operator M ϱ Ω is equivalent to the classical Marcinkiewicz function M 1 Ω , which was initially introduced by Stein [1] in 1958.When Ω ∈ Lip β ðS n−1 Þ, β ∈ð0, 1, Stein [1] demonstrated that M 1 Ω is bounded on L p for p ∈ ð1, 2. Subsequently, the authors of [2] established the L p -boundedness of M 1 Ω for every p ∈ ð1,∞Þ when Ω ∈ ℂ 1 ðS n−1 Þ.On the other hand, Calderón [3] proved that the commutator the Hilbert transform H generated by Λ ∈ BMO, defined by ½Λ, Tf ≔ ΛTð f Þ − TðΛf Þ, is bounded on L 2 ðℝ n Þ. Coifman et al. [4] arrived at the conclusion that the commutator, which was generated by the Calderón-Zygmund operator T and the Λ ∈ BMO, is bounded on L p for p ∈ ð1,∞Þ.Since then, the commutators of the Calderón-Zygmund operator have played an essential role in the study of the regularity of solutions to second-order elliptic, parabolic, and ultraparabolic partial differential equations, see for example [5][6][7][8][9][10][11].Moreover, the boundedness of the commutators of various operators generated by a BMO function has been widely studied.Particularly, Torchinsky and Wang [12] studied the weighted L p -boundedness of M 1 Ω,Λ i , where M 1 Ω,Λ i is the i-order commutator of Marcinkiewicz integral.The authors of [13] studied the behaviour of the Hardy-Littlewood maximal operator and the action of commutators in generalized local Morrey spaces and generalized Morrey spaces.For further research works studying the commutators on different function spaces, we refer to [9,[14][15][16][17][18][19][20][21] and references therein.
The parametric Marcinkiewicz integral M ϱ Ω was originally introduced by Hörmander in [22] where the author established the boundeness of M ϱ Ω on L p for p ∈ ð1,∞Þ under the condition Ω ∈ Lip β ðS n−1 Þ, ðβ ∈ ð0, 1Þ and ϱ > 0. Shi and Jiang [23] investigated the weighted L p -boundedness of M ϱ Ω and M ϱ Ω,Λ i .Since that time, the boundedness of the parametric Marcinkiewicz integral, as well as its related commutator, in several types of function spaces have attracted the attention of many researchers.Deringoz and Hasanov [24] considered the boundedness of the operator M ϱ Ω on generalized Orlicz-Morrey spaces.On generalized weighted Morrey spaces, Deringoz [25] investigated the boundedness of rough parametric Marcinkiewicz integral M ϱ Ω and its higher-order commutator M ϱ Ω,Λ i .For more applications and recent developments on the research of the parametric Marcinkiewicz function, see [26][27][28][29][30][31].
Inspired by the research mentioned above, the main goal of this article is to prove the boundedness of the rough parametric Marcinkiewicz integral and its higher-order commutators on the variable exponents Morrey-Herz spaces.
Henceforth, wherever the symbol C appears, it represents a positive constant whose value may vary but is independent of the basic variables.The expression f ≲ g denotes the existence of constant C such that f ≤ Cg, and f ≍g means that f ≲ g ≲ f .If no further instructions are provided, the symbol for any space denoted by Xðℝ n Þ is represented by X.For instance, L p ðℝ n Þ is abbreviated as L p .

Definitions and Preliminaries
In this section, we review some notations, definitions, and properties related to our work.
A variable exponent is a measurable function pð•Þ: ℝ n ⟶ ð0,∞.For any variable exponent pð•Þ, we set p − ≔ essinf fpðxÞ: x ∈ ℝ n g and p + ≔ esssup fpðxÞ: x ∈ ℝ n g.Define the sets P by where It is obvious that the variable exponent Lebesgue norm has the following property kjf j where M HL stands for the Hardy-Littlewood maximal function, which is defined as follows: Definition 1 (see [46]).Let Θð•Þ be real function on ℝ n .
(i) If there exists a constant C log > 0 such that then the function Θð•Þ is said to be a log-Hölder continuous at the origin (or has a log decay at the origin).
2 Journal of Function Spaces (ii) If there exist Θ ∞ ∈ ð0,∞Þ and a constant C log > 0 such that then the function Θð•Þ is said to be a log-Hölder continuous at the infinity (or has a log decay at the infinity).
Here and hereafter, p [56]). For for q < ∞, and the usual modification should be made when q = ∞.

Proofs of Theorems 10 and 11
Proof of Theorem 10.

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Journal of Function Spaces Case 3. 0 < q ≤ 1, combining the above inequalities and using (38), we can obtain Now, let us deal with Case 4. 1 < q < ∞, we have For T 1 ⋄ 3 , by the fact that αð0Þ + nδ 1 > 0 and using Hölder inequality, we infer that For T 1 ⋄⋄ 3 , from the inequality (36) and using the method as for T 1′′ 3 , we obtain By combining T 1′ 3 , T 1′′ 3 , T 1 ⋄ 3 , and T 1 ⋄⋄ 3 estimates, we arrive at By the similar method used in the estimate for T 1 3 , it is not difficult to show that Thus, we have The proof for Theorem 10 is finished.
Proof of Theorem 11.
Let us first estimate Y 1 .From Proposition 6 and the boundedness of M ρ Ω on L pð•Þ (see [30]), and using the similar methods as that for T 1 , it is not difficult to see that , sup κ≥0 κ∈ℤ , sup κ≥0 κ∈ℤ Now, let us turn to the estimates of Y 2 .We consider 2 ℓαðxÞ jM ϱ Ω,Λ ð f j ÞðxÞj It is clear that if x ∈ R ℓ , j + 2 ≤ ℓ, andy ∈ R j , then jx − yj ≍jxj.Thus, for ρ ∈ ð0, nÞ, we use the Minkowski's inequality to get Using Lemma 7 and inequality (12), it follows that Applying Hölder's inequality (12), the inequality (35), and Lemmas 4-8, we obtain Hence, combining the above estimate and using the same approach as the one used for estimating T 2 , we conclude that Finally, we estimate Y 3 .It is clear that if x ∈ R ℓ , j + 2 ≤ ℓ, andy ∈ R j , then jx − yj≍jyj≍2 jn .By (31), the Minkowski's inequality, and the inequality (12), we deduce, for ρ ∈ ð0, nÞ, From this, Lemmas 4-8 and (38), we deduce Thus, combining the above estimates and using the same approach as for the T 2 estimate, we deduce that Summing up the estimates of Y 1 , Y 2 , and Y 3 , we conclude that The proof for Theorem 11 is finished.