EUROPEAN JOURNAL OF PURE AND APPLIED MATHEMATICS 2025, Vol. 18, Issue 4, Article Number 6732 ISSN 1307-5543 – ejpam.com Published by New York Business Global Some New Boundedness Results for Variable Marcinkiewicz Fractional Integral Operator on Herz-Morrey-Hardy Spaces Babar Sultan1, Amjad Hussain1,∗, Mehvish Sultan2, Ioan-Lucian Popa3,4,∗ 1 Department of Mathematics, Quaid-I-Azam University, Islamabad 45320, Pakistan 2 Department of Mathematics, Capital University of Science and Technology, Islamabad, Pakistan 3 Department of Computing, Mathematics and Electronics, “1 Decembrie 1918” University of Alba Iulia, 510009 Alba Iulia, Romania 4 Faculty of Mathematics and Computer Science, Transilvania University of Brasov, Iuliu Maniu Street 50, 500091 Brasov, Romania Abstract. In this paper, we define the idea of Herz-Morrey-Hardy spaces by using variable Herz- Morrey spaces and Hardy spaces. Then we give the atomic characterization of these spaces by using the grand maximal function. Then our main objective is to prove the boundedness of higher order commutators of variable Marcinkiewicz fractional integral operator on Herz-Morrey-Hardy spaces where the exponents defining these spaces are variable. These results also hold for variable Herz-Hardy spaces. The higher order commutators of variable Marcinkiewicz fractional integral operator is the generalization of Marcinkiewicz integral operators, variable Marcinkiewicz fractional integral operator and commutators on Marcinkiewicz fractional integral operators, so these proofs generalize some previous results. 2020 Mathematics Subject Classifications: 42B20, 47B38 Key Words and Phrases: BMO spaces, Marcinkiewicz fractional integral, Herz-Morrey-Hardy spaces 1. Introduction and preliminaries Let E be an open set in Rn, consider a measurable function p(·) : E → [1,∞). The conjugate exponent denoted by p′(·), is defined as p′(·) = p(·)/(p(·)− 1). The set P(E) comprises all functions p(·) : E → [1,∞). We suppose that 1 ≤ p−(E) ≤ p(x) ≤ p+(E) < ∞, (1.1) ∗Corresponding author. ∗Corresponding author. DOI: https://doi.org/10.29020/nybg.ejpam.v18i4.6732 Email addresses: babarsultan40@yahoo.com (B. Sultan), a.hussain@qau.edu.pk (A. Hussain), mehvishsultanbaz@gmail.com (M. Sultan), lucian.popa@uab.ro (I.-L. Popa) https://www.ejpam.com 1 Copyright: © 2025 The Author(s). (CC BY-NC 4.0) B. Sultan et al. / Eur. J. Pure Appl. Math, 18 (4) (2025), 6732 2 of 20 such that p− = ess inf { p(x) : x ∈ E } > 1, p+ = ess sup { p(x) : x ∈ E } < ∞. We use the notation Lp(·)(E) to represent the space of all measurable functions f defined on E, such that, for a certain η > 0, ∫ E ( |f(x)| η )p(y) dy < ∞. Its norm is given as ∥f∥Lp(·)(E) = inf { η > 0 : ∫ E ( |f(y)| η )p(y) dy ≤ 1 } . Herz spaces with variable exponents have emerged as a generalization of Lebesgue spaces with variable exponents. Boundedness of sublinear operators on Herz spaces with variable exponents, K̇α,q p(·) and Kα,q p(·), was shown by Izuki in [1] in 2010. Boundedness results for a wide class of classical operators on Herz spaces were later developed by Almeida and Drihem in 2012 [2]. These spaces were represented as K̇ α(·)q p(·) and K α(·),q p(·) . Grand variable Herz spaces are the generalization of Herz spaces, for boundedness results in these spaces see [3–8]. For more results in variable exponent function spaces see [9–27]. In [28], the authors introduced Herz-Morrey-Hardy spaces with variable exponents and established the characterization of these spaces in terms of atom. The authors were able to determine the boundedness of certain singular integral operators on these spaces by applying the characterization. In this paper, we define the idea of variable Herz-Morrey-Hardy spaces and using the characterization we obtain boundedness results for some new operator in these spaces. Many classical function spaces, alongside Hardy-type spaces linked with operators, exhibit atomic and molecular decompositions. These decompositions simplify the action of linear operators on these spaces significantly; see [29–31]. Let Sn−1 is denoting the unit sphere in Rn (n ≥ 2) with the normalized Lebesgue measure. Let Φ ∈ Lr(Sn−1) be a homogeneous function of degree zero such that∫ Sn−1 Φ(y′)dΦ(y′) = 0, (1.2) where y′ = y/|y| and y is not zero. The Marcinkiewicz integral is define as µΦ(g)(z1) =  ∞∫ 0 |RΦ,s(g)(z1)|2 ds s3  1 2 , where RΦ,s(g)(z1) = ∫ |z1−z2|≤s Φ(z1 − z2) |z1 − z2|n−β(z1)−1 g(z2)dz2. B. Sultan et al. / Eur. J. Pure Appl. Math, 18 (4) (2025), 6732 3 of 20 Let h ∈ BMO (Rn), then the commutators on variable Marcinkiewicz fractional integral operator are given as [h, µΦ] m β (g)(z1) =  ∞∫ 0 ∣∣∣∣∣∣∣ ∫ |z1−z2|≤s Φ(z1 − z2)[h(z1)− h(z2)] m |z1 − z2|n−1−β(z1) g(z2)dz2 ∣∣∣∣∣∣∣ 2 ds s3  1 2 . Assume g ∈ L1 loc(Rn), the definition of the Hardy-Littlewood maximal operator is expressed as Mg(z1) = sup r>0 1 |Br(z1)| ∫ Br(z1) ∣∣g(z2)∣∣ dz2, where Br(z1) = {z2 ∈ Rn : |z1 − z2| < r}. The set B(Rn) is comprised of p(·) ∈ P(Rn) that fulfill the requirement that M is bounded on Lp(·)(Rn). Now we will define the well known log-condition |p(h1)− p(h2)| ≤ C(p) − ln |h1 − h2| , |h1 − h2| ≤ 1 2 , h1, h2 ∈ E, (1.3) where C(p) > 0. And the decay condition: there exists a number p∞ ∈ (1,∞), such that |p(h)− p∞| ≤ C ln(e+ |h|) , (1.4) and also decay condition |p(h)− p0| ≤ C ln |h| , |h| ≤ 1 2 , (1.5) holds for some p0 ∈ (1,∞). We use these notations in this article: (i) The set P(E) consists of all measurable functions p(·) satisfying p− > 1 and p+ < ∞. (ii) P log = P log(E) consists of all functions p ∈ P(E) satisfying (1.1) and (1.3). (iii) P∞(E) and P0,∞(E) are the subsets of P(E) and values of these subsets lies in [1,∞) which satisfy the condition (1.4) and both conditions (1.4) and (1.5) respectively. (iv) χi = χFi , Fi = Bi \Bi−1, Bi = B(0, 2i) = {x ∈ Rn : |x| < 2i} for all i ∈ Z. (v) Let E be a measurable subset in Rn, then |E| denotes the Lebesgue measure and χE is the characteristic function. B. Sultan et al. / Eur. J. Pure Appl. Math, 18 (4) (2025), 6732 4 of 20 (vi) Let β = (β1, β2, · · · , βn) then |β| is defined as |β| = β1, β2 + · · ·+ βn. (vii) The symbol N0 denotes the set of all nonnegative integers. For m ∈ N0, we denote χ̃m := χFm if m ≥ 1 and χ̃0 := χB0 . (viii) C is a positive constant. (ix) By a ≲ b, we mean a ≤ Cb. Lemma 1. [32] Let D > 1 and p ∈ P0,∞(Rn). Then 1 r0 o n p(0) ≤ ∥χFo,Do ∥p(·) ≤ r0o n p(0) , for 0 < o ≤ 1 (1.6) and 1 r∞ o n p∞ ≤ ∥χFo,Do ∥p(·) ≤ r∞o n p∞ , for o ≥ 1, (1.7) respectively, where r0 ≥ 1 and r∞ ≥ 1 and depending on D but independent of o. Lemma 2. [33] Let p(·) be a function within the class B(Rn). For any ball B in Rn, there exists a positive constant C such that the inequality 1 |B| ∥χB∥p(·)∥χB∥p′(·) ≤ C, holds. Lemma 3. [33] Assuming that p(·) is a function in the class B(Rn), there exists a positive constant C such that, for every ball B in Rn and every measurable subset S within B, the following inequalities hold: ∥χB∥p(·) ∥χS∥p(·) ≤ C |B| |S| , ∥χS∥p(·) ∥χB∥p(·) ≤ C ( |S| |B| )δ1 , ∥χS∥p′(·) ∥χB∥p′(·) ≤ C ( |S| |B| )δ2 , where δ1 and δ2 are constants satisfying 0 < δ1, δ2 < 1. Lemma 4. [34] Let f ∈ Lp(·)(E), g ∈ Lq(·)(E) where E ⊆ Rn, and 1 ≤ p−(E) ≤ p+(E) ≤ ∞. Then ∥fg∥r(·) ≤ ∥f∥p(·)∥g∥q(·) where 1 r(z) = 1 p(z) + 1 q(z) . Definition 5 (BMO space). Let h is a locally integrable function then a BMO function is consist of those functions whose mean oscillation given by 1 |B| ∫ B |h(i)−hB|di is bounded. A Mathematically, ∥h∥BMO = sup B 1 |B| ∫ B |h(i)− hB|di < ∞. B. Sultan et al. / Eur. J. Pure Appl. Math, 18 (4) (2025), 6732 5 of 20 Lemma 6. [33] Let j, i ∈ Z for i < ℓ, and h ∈ BMO(Rn), then 1 C ∥h∥nBMO ≤ sup B:ball 1 ∥χB∥p(·) ∥(h− hB) nχB∥p(·) (1.8) ≤C∥h∥nBMO, (1.9) ||(h− hBi) nχBk ||p(·) ≤ C(k − i)n||b||nBMO||χBk ||p(·). (1.10) Lemma 7 ([35]). If a > 0, s ∈ [1,∞], 0 < d ≤ s and −m+ (m− 1)ds < u < ∞, then ∫ |z2|≤a|z1| |z2|u|Φ(z1 − z2)|ddz2  1/d ≤ |z1|(u+m)/d ∥Φ∥Ls(Sm−1) . Definition 8. Let 0 < p ≤ ∞, q ∈ P(Rn), α : Rn → R with α(·) ∈ L∞(Rn). The inhomogeneous Herz space K α(·) p,q(·)(R n) consists of all f ∈ L q(·) loc (R n \ {0}) such that ∥f∥ K α(·) p,q(·)(R n) := ∥fχB0∥Lq(·) + ∑ k≥1 ∥∥∥2α(·)fχk ∥∥∥p q(·) 1/p < ∞. The homogeneous Herz space K̇ α(·) p,q(·)(R n) consists of all f ∈ L q(·) loc (R n \ {0}) such that ∥f∥ K̇ α(·) p,q(·)(R n) := (∑ k∈Z ∥∥∥2α(·)fχk ∥∥∥p q(·) )1/p < ∞. Next we give the definition of variable Herz-Morrey spaces. Definition 9. Let p : Rn → [1,∞), α(·) ∈ L∞(Rn), u ∈ [1,∞), and 0 ≤ Γ < ∞. The norm of variable Herz-Morrey spaces are defined as: MK̇ α(·),u Γ,p(·) (R n) = { g ∈ L p(·) loc (R n \ {0}) : ∥g∥ MK̇ α(·),u Γ,p(·) (R n) < ∞ } , where ∥g∥ MK̇ α(·),u Γ,p(·) (R n) = sup m0∈Z 2−m0Γ ( m0∑ k=−∞ ∥2kα(·)gχk∥up(·) ) 1 u . For Γ = 0, variable Herz-Morrey spaces becomes variable Herz spaces. The next proposition is the generalization of variable exponents Herz spaces in [2]. Proposition 10. Let α, u, p are as defined in definition 9, then ∥f∥ MK̇ α(·),u Γ,p(·) (R n) = sup m0∈Z 2−m0Γ ( m0∑ k=−∞ ∥2kα(·)fχk∥up(·) ) 1 u B. Sultan et al. / Eur. J. Pure Appl. Math, 18 (4) (2025), 6732 6 of 20 ≈max  sup m0<0,m0∈Z 2−m0Γ ( −1∑ k=−∞ 2kα(0)u∥fχk∥up(·) ) 1 u , sup m0≥0,m0∈Z 2−m0Γ ( −1∑ k=−∞ 2kα(0)u∥fχk∥up(·) ) 1 u + sup m0≥0,m0∈Z 2−m0Γ ( m0∑ k=0 2kα∞u∥fχk∥up(·) ) 1 u  2. The Atomic Characterization Let S (Rn) denotes the Schwartz space of all rapidly decreasing infinitely differentiable functions on Rn, and S ′ (Rn) denotes the dual space of S (Rn). Let GNg be the grand maximal function of g defined by GNg(x) := sup ϕ∈AN |ϕ∗ ∇(g)(x)| , x ∈ Rn where AN := { ϕ ∈ S (Rn) : sup |α|,|β|≤N,∀x∈Rn ∣∣xαDβϕ(x) ∣∣ ≤ 1 } and N > n+1 and ϕ∗ ∇ is the nontangential maximal operator defined by ϕ∗ ∇(g)(x) := sup |y−x| n+1. The Herz-Morrey-Hardy space with variable exponents HMK̇ α(·),u Γ,p(·) (R n) is defined by HMK̇ α(·),u Γ,p(·) (R n) := { g ∈ S ′ (Rn) : ∥g∥ MK̇ α(·),u Γ,p(·) (R n) := ∥GNg∥ MK̇ α(·),u Γ,p(·) (R n) < ∞ } . Definition 12. Let p(·) ∈ P (Rn) and α(·) ∈ L∞ (Rn) be log-Hölder continuous both at the origin and infinity, and nonnegative integer s ⩾ [αr − nδ2]; here αr = α(0), if r < 1, and αr = α∞, if r ⩾ 1, nδ2 ≤ αr < ∞ and δ2 as in Lemma 3. (i) A function a on Rn is called a central (α(·), p(·)) atom, if it satisfies (1) supp a ⊂ B(0, r), (2) ∥a∥p(·) ≤ |B(0, r)|−αr/n, (3) ∫ Rn a(x)x βdx = 0, |β| ≤ s. B. Sultan et al. / Eur. J. Pure Appl. Math, 18 (4) (2025), 6732 7 of 20 (ii) A function a on Rn is called a central (α(·), p(·))-atom of restricted type, if it satisfies 2, 3 and condition given below (a) supp α ⊂ B(0, r), r ⩾ 1. Theorem 13. [28] Let 0 < u < ∞, p(·) ∈ B (Rn) , 0 ≤ Γ < ∞, and α(·) ∈ L∞ (Rn) be log-Hölder continu- ous both at the origin and infinity, 2λ ≤ α(·), nδ2 ≤ α(0), α∞ < ∞, and δ2 as in Lemma 3. Then f ∈ HMK̇ α(·),u Γ,p(·) (R n) iff f = ∑∞ k=−∞ λkak in the sense of S ′ (Rn), where each ak is a central (α(·), p(·))-atom with support contained in Bk and sup ϑ>0 sup m0∈Z 2−m0Γ ∑m0 k=−∞ |λk|u < ∞. Moreover, ∥f∥ HMK̇ α(·),u Γ,p(·) (R n) ≈ inf  sup m0∈Z 2−m0Γ ( m0∑ k=−∞ |λk|u )1/u  . Theorem 14. Let 0 ≤ Γ < ∞, 0 < u < ∞, , q1(·) ∈ B (Rn), and α(·) ∈ L∞ (Rn) be log-Hölder continuous both at the origin and infinity. Let α be such that : (i) − n q1(0) − v − n s < α(0) < n q′1(0) − v − n s (ii) − n q1∞ − v − n s < α∞ < n q′1∞ − v − n s . Then ∥∥∥(|z1|+ 1)−λ(z1) [b, µΦ] m β f ∥∥∥ MK̇ α(·),u Γ,q2(·) (Rn) ≤ C ∥f∥ HMK̇ α(·),u Γ,q1(·) (Rn) , for f ∈ HMK̇ α(·),u Γ,q1(·)(R n) . Proof. Suppose that f ∈ HMK̇ α(·),u Γ,q1(·)(R n). By using Theorem 13, f = ∑∞ i=−∞ λibi converges in S ′ (Rn), where each bi is a central (α(·), q1(·))-atom with support contained in Bi and ∥f∥ HMK̇ α(·),u Γ,q1(·) (Rn) ≈ inf  sup m0∈Z 2−m0Γ ( m0∑ i=−∞ |λi|u ) 1 u  . To keep things simple, we denote Λ = supm0∈Z 2 −m0Γu ∑m0 i=−∞ |λi|u. By Proposition 10, we have ∥∥∥(|z1|+ 1)−λ(z1) [b, µΦ] m β f ∥∥∥u MK̇ α(·),u Γ,q2(·) (Rn) B. Sultan et al. / Eur. J. Pure Appl. Math, 18 (4) (2025), 6732 8 of 20 ≈ max { sup m0<0,m0∈Z 2−m0Γu ( m0∑ k=−∞ 2kα(0)u ∥∥∥((|z1|+ 1)−λ(z1) [b, µΦ] m β f ) χk ∥∥∥u q2(·) ) , sup m0≥0,m0∈Z 2−m0Γu ( −1∑ k=−∞ 2kα(0)u ∥∥∥((|z1|+ 1)−λ(z1) [b, µΦ] m β f ) χk ∥∥∥u q2(·) + m0∑ k=0 2kα∞u ∥∥∥((|z1|+ 1)−λ(z1) [b, µΦ] m β f ) χk ∥∥∥u q2(·) )} ≲ max{I, II + III}. We will find the estimated for I and III and estimate of II can be obtained similarly. We just need to demonstrate that there is a positive constant C such that I, II, III ≤ CΛ in order to finish our proof. Firstly, we will find the estimate of I : I = sup m0<0,m0∈Z 2−m0Γu ( m0∑ k=−∞ 2kα(0)u ∥∥∥((|z1|+ 1)−λ(z1) [b, µΦ] m β f ) χk ∥∥∥u q2(·) ) ≲ sup m0<0,m0∈Z 2−m0Γu m0∑ k=−∞ 2kα(0)u ( ∞∑ i=k |λi| ∥∥∥((|z1|+ 1)−λ(z1) [b, µΦ] m β bi ) χk ∥∥∥ q2(·) )u + sup m0<0,m0∈Z 2−m0Γu m0∑ k=−∞ 2kα(0)u ( k−1∑ i=−∞ |λi| ∥∥∥((|z1|+ 1)−λ(z1) [b, µΦ] m β bi ) χk ∥∥∥ q2(·) )u :=I1 + I2. Let k ∈ Z and i ≤ k and a.e. z1 ∈ Fk, z2 ∈ Fi, it is easy to check that |z1 − z2| ≈ |z1| ≈ 2k, ∣∣∣([b, µΦ] m β bi ) (z1) ∣∣∣ ≤  |z1|∫ o ∣∣∣∣∣∣∣ ∫ |z1−z2|≤t Φ(z1 − z2) [b(z1)− b(z2)] m |z1 − z2|n−1−β(z1) bi(z2)dz2 ∣∣∣∣∣∣∣ 2 dt t3  1/2 +  ∞∫ |z1| ∣∣∣∣∣∣∣ ∫ |z1−z2|≤t Φ(z1 − z2) [b(z1)− b(z2)] m |z1 − z2|n−1−β(z1) bi(z2)dz2 ∣∣∣∣∣∣∣ 2 dt t3  1/2 =: I11 + I12. Mean value theorem yields ∣∣∣∣ 1 |z1 − z2|2 − 1 |z1|2 ∣∣∣∣ ≤ |z2| |z1 − z2|3 . (2.1) B. Sultan et al. / Eur. J. Pure Appl. Math, 18 (4) (2025), 6732 9 of 20 For I11, we get I11 ≤ ∫ Rn |Φ(z1 − z2)| [b(z1)− b(z2)] m |z1 − z2|n−1−β(z1) |bi(z2)|  |z1|∫ |z1−z2| dt t3  1/2 dz2 ≤ ∫ Rn |Φ(z1 − z2)| [b(z1)− b(z2)] m |z1 − z2|n−1−β(z1) |bi(z2)| ∣∣∣∣ 1 |z1 − z2|2 − 1 |z1|2 ∣∣∣∣1/2 dz2 ≤ ∫ Rn |Φ(z1 − z2)| [b(z1)− b(z2)] m |z1 − z2|n−1−β(z1) |bi(z2)| ∣∣∣∣ |z2 |z1 − z2|3 ∣∣∣∣1/2 dz2 ≤ 2l/2 |z1|n+ 1 2 . |z1|−β(z1) ∫ Fi |Φ(z1 − z2)| [b(z1)− b(z2)] m |bi(z2)| dz2 ≤2(i−k)/22−kn |z1|β(z1) ∥bi∥q1(·) ∥Φ(z1 − ·)χi(·)∥q′1(·) . ≤2(i−k)/22−kn |z1|β(z1) { |b(z1)− bBi | m ∫ Fi |Φ(z1 − z2)| |bi(z2)| dz2 + ∫ Fi |b(z2)− bBi |m |Φ(z1 − z2)| |bi(z2)| dz2 } ≤2(i−k)/22−kn |z1|β(z1) ∥bi(z2)∥q1(·) ( |b(z1)− bBi | m ∥Φ(z1 − ·)χi(·)∥q′1(·) + ∥(b(·)− bBi) m(Φ(z1 − ·)χi(·)∥q′1(·) ) . Similarly, we can consider I12, we have I12 ≤ ∫ Rn |Φ(z − 1− z2)| |z1 − z2|n−1−β(z1) |bi(z2)|  ∞∫ |z1| dt t3  1/2 dz2 ≤ ∫ Rn |Φ(z1 − z2)| |z1 − z2|n−1−β(z1) |bi(z2)| dz2 ≤|z1|−n |z1|β(z1) ∫ Fi |Φ(z1 − z2)| |bi(z2)| dz2 ≤2−kn |z1|β(z1) ∥bi(z2)∥q1(·) { |b(z1)− bBi | m ∥Φ(z1 − ·)χi(·)∥q′1(·) + ∥(b(·)− bBi) m (Φ(z1 − ·)χi(·)∥q′1(·) } . We define q1(·) by the relation 1 q′1(x) = 1 q1(x) + 1 s . By using Lemma (7) and generalized Hölder’s inequality we have ∥Φ(z1 − ·)χi(·)∥q′1(·) ≤∥Φ(z1 − ·)χi(·)∥Ls(Sn−1) ∥χi(·)∥q1(·) B. Sultan et al. / Eur. J. Pure Appl. Math, 18 (4) (2025), 6732 10 of 20 ≤2−iv  ∫ 2l−1<|z2|<2i |Φ(z1 − z2)|s|z2|svdz2  1/s ∥χBi∥q1(·) ≤2−iv2k(v+ n s ) ∥Φ∥Ls(Sn−1) ∥χBi∥q1(·) . Similarly, by using Lemma (6) we have ∥(b(·)− bBi) m(Φ(z1 − ·)χi(·)∥q′1(·) ≤∥Φ(z1 − ·)χi(·)∥s ∥(b(·)− bBi) mχi(·)∥q(·) ≤C ∥f∥mBMO ∥χBi∥q(·) ∥Φ(z1 − ·)χi(·)∥s ≤C ∥f∥mBMO 2−iv2k(v+ n s ) ∥Φ∥Ls(Sn−1) ∥χBi∥q1(·) . It is known, see e.g. [36] that Iβ(·) ((b(z1)− bBi) mχBk ) (z1) ≥ Iβ(·)(χBk )(z1).(χBk )(z1) = ∫ Bk |b(z1)− bBi | m |z1 − z2|β(z1)−n dz2.χBk (z1) ≥ C |b(z1)− bBi | m |z1|β(z1) .χBk (z1) ≥ C |b(z1)− bBi | m |z1|β(z1) .χk(z1). Consequently, by using weighted Sobolev estimates [37] we have∥∥∥(b(z1)− bBi) m |z1|β(z1) χk(z1)(1 + |z1|)−λ(z1) ∥∥∥ q2(·) ≤ ∥∥∥(1 + |z1|)−λ(z1)(Iβ(·)((b(z1)− bBi) mχBk )(z1)) ∥∥∥ q2(·) ≤ ∥(b(z1)− bBi) mχBk )(z1)∥q1(·) . Thus we have∥∥∥χk(1 + |z1|)−λ(z1) [b, µΦ] m β bi ∥∥∥ q2(·) ≤ C2−kn ∥bi∥q1(·) {∥∥∥(b(z1)− bBi) m |z1|β(z1) χk(z1)(1 + |z1|)−λ(z1) ∥∥∥ q2(·) 2−iv2k(v+ n s ) ∥Φ∥Ls(Sn−1) ∥χBi∥q1(·) + ∥f∥mBMO 2−iv2k(v+ n s ) ∥Φ∥Ls(Sn−1) ∥χBi∥q1(·) ∥∥∥|z1|β(z1) χk(z1)(1 + |z1|)−λ(z1) ∥∥∥ q2(·) } ≤ C2−kn ∥bi∥q1(·) { (k − i)m ∥f∥mBMO ∥χBk ∥q1(·) 2 −iv2k(v+ n s ) ∥Φ∥Ls(Sn−1) ∥χBi∥q1(·) + ∥f∥mBMO 2−iv2k(v+ n s ) ∥Φ∥Ls(Sn−1) ∥χBi∥q1(·) ∥χBk ∥q1(·) } ≤ C2−kn ∥bi∥q1(·) (k − i)m ∥f∥mBMO ∥χBk ∥q1(·) 2 −iv2k(v+ n s ) ∥Φ∥Ls(Sn−1) ∥χBi∥q1(·) ≤ C(k − i)m ∥Φ∥Ls(Sn−1) ∥f∥ m BMO 2−kn2−iv2k(v+ n s ) ∥χBk ∥q1(·) ∥χBi∥q1(·) ∥bi∥q1(·) . B. Sultan et al. / Eur. J. Pure Appl. Math, 18 (4) (2025), 6732 11 of 20 Therefore, when 0 < u ≤ 1 and v1 = n/q′1(0)− v − n s − α(0), we get I1 = sup m0<0,m0∈Z 2−m0Γu m0∑ k=−∞ 2kα(0)u ( ∞∑ i=k |λi| ∥∥∥((|z1|+ 1)−λ(z1) [b, µΦ] m β bi ) χk ∥∥∥ q2(·) )u ≲ ∥f∥mBMO sup m0<0,m0∈Z 2−m0Γu m0∑ k=−∞ 2α(0)ku ( ∞∑ i=k |λi| 2−kn2−iv2k(v+ n s ) ∥χBk ∥q1(·) ∥χBi∥q1(·) (k − i)m2−αii )u ≲ ∥f∥mBMO sup m0<0,m0∈Z 2−m0Γu m0∑ k=−∞ 2kα(0)u × ( −1∑ i=k |λi|u 2−α(0)iu2u(i−k)(n/q′1(0)−v−n s )(k − i)mu + ∞∑ i=0 |λi|u 2−α∞iu2−ui(n/q1∞+v+n s )+uk(n/q1(0)+v+n s )(k − i)mu ) ≲ ∥f∥mBMO sup m0<0,m0∈Z 2−m0Γu m0∑ k=−∞ −1∑ i=k |λi|u 2v1(i−k)u(k − i)mu + ∥f∥mBMO sup m0<0,m0∈Z 2−m0Γu m0∑ k=−∞ 2(α(0)+(n/q1(0)+v+n s ))ku ∞∑ i=0 |λi|u 2((n/q1∞+v+n s )−α∞)iu(k − i)mu ≲ ∥f∥mBMO sup m0<0,m0∈Z 2−m0Γu −1∑ i=−∞ |λi|u i∑ k=−∞ 2v1(i−k)u(k − i)mu + ∥f∥mBMO sup m0<0,m0∈Z 2−m0Γu ∞∑ i=0 i∑ j=−∞ |λj |u 2(−ui(n/q1∞+v+n s +α∞))(k − i)mu ≲ ∥f∥mBMO sup m0<0,m0∈Z 2−m0Γu m0∑ i=−∞ |λi|u + Λ sup m0<0,m0∈Z 2−m0Γu ∞∑ i=0 2(−ui(n/q1∞+v+n s +α∞))(k − i)mu ≲ ∥f∥mBMO Λ. Now we will the estimate for the second case when 1 < u < ∞. Let 1 u + 1 u′ = 1, we obtain I1 = sup m0<0,m0∈Z 2−m0Γu m0∑ k=−∞ 2kα(0)u ( ∞∑ i=k |λi| ∥∥∥((|z1|+ 1)−λ(z1) [b, µΦ] m β bi ) χk ∥∥∥ q2(·) )u ≲ ∥f∥mBMO sup m0<0,m0∈Z 2−m0Γu m0∑ k=−∞ 2α(0)ku ( ∞∑ i=k |λi| 2−kn2−iv2k(v+ n s ) ∥χBk ∥q1(·) ∥χBi∥q1(·) (k − i)m2−αii )u B. Sultan et al. / Eur. J. Pure Appl. Math, 18 (4) (2025), 6732 12 of 20 ≲ ∥f∥mBMO sup m0<0,m0∈Z 2−m0Γu × m0∑ k=−∞ 2kα(0)u ( −1∑ i=k |λi|u 2−α(0)iu2u(i−k)(n/q′1(0)−v−n s )(k − i)mu + ∞∑ i=0 |λi|u 2−α∞iu2−ui(n/q1∞+v+n s )+uk(n/q1(0)+v+n s )(k − i)mu ) ≲ ∥f∥mBMO sup m0<0,m0∈Z 2−m0Γu m0∑ k=−∞ ( −1∑ i=k |λi|u 2v1(i−k)u/2 ) × ( −1∑ i=k 2α(0)(k−i)u′/2(k − i)mu′/2 )u/u′ + sup m0<0,m0∈Z 2−m0Γu m0∑ k=−∞ 2(α(0)n/q1(0)+v+n s )ku ( ∞∑ i=0 |λi|u 2−iu(α∞+n/q1∞+v+n s )u/2 ) × ( ∞∑ i=0 2−iu(α∞+n/q1∞+v+n s )u′/2(k − i)mu′/2 )u/u′ ≲ ∥f∥mBMO ∥f∥mBMO sup m0<0,m0∈Z 2−m0Γu m0∑ k=−∞ ( −1∑ i=k |λi|u 2v1(i−k)u/2 ) + ∥f∥mBMO sup m0<0,m0∈Z 2−m0Γu m0∑ k=−∞ 2(α(0)n/q1(0)+v+n s )ku ( ∞∑ i=0 |λi|u 2−iu(α∞+n/q1∞+v+n s )u/2 ) ≲ ∥f∥mBMO sup m0<0,m0∈Z 2−m0Γu −1∑ i=−∞ |λi|u i∑ k=−∞ 2v1(i−k)u/2 + ∥f∥mBMO sup m0<0,m0∈Z 2−m0Γu ( ∞∑ i=0 |λi|u 2−iu(α∞+n/q1∞+v+n s )u/2 ) ≲ ∥f∥mBMO sup m0<0,m0∈Z 2−m0Γu −1∑ i=−∞ |λi|u i∑ k=−∞ 2v1(i−k)u/2 + ∥f∥mBMO sup m0<0,m0∈Z 2−m0Γu ∞∑ i=0 i∑ j=−∞ |λj |u 2(−ui(n/q1∞+v+n s +α∞))/2 ≲ ∥f∥mBMO sup m0<0,m0∈Z 2−m0Γu m0∑ i=−∞ |λi|u + ∥f∥mBMO Λ sup m0<0,m0∈Z 2−m0Γu ∞∑ i=0 2(−ui(n/q1∞+v+n s +α∞))/2 ≲ ∥f∥mBMO Λ. Second, we estimate I2. Therefore, when 0 < u ≤ 1, we get I2 = sup m0<0,m0∈Z 2−m0Γu m0∑ k=−∞ 2kα(0)u ( k−1∑ i=−∞ |λi| ∥∥∥((|z1|+ 1)−λ(z1) [b, µΦ] m β bi ) χk ∥∥∥ q2(·) )u B. Sultan et al. / Eur. J. Pure Appl. Math, 18 (4) (2025), 6732 13 of 20 ≲ ∥f∥mBMO sup m0<0,m0∈Z 2−m0Γu m0∑ k=−∞ 2α(0)ku ( k−1∑ i=−∞ |λi| 2−kn2−iv2k(v+ n s ) ∥χBk ∥q1(·) ∥χBi∥q1(·) (k − i)m2−αii )u ≲ ∥f∥mBMO sup m0<0,m0∈Z 2−m0Γu m0∑ k=−∞ 2kα(0)u ( k−1∑ i=−∞ |λi|u 2−α(0)iu2u(i−k)(n/q′1(0)−v−n s )(k − i)mu ) ≲ ∥f∥mBMO sup m0<0,m0∈Z 2−m0Γu m0∑ k=−∞ k−1∑ i=−∞ |λi|u 2v1(i−k)u(k − i)mu ≲ ∥f∥mBMO sup m0<0,m0∈Z 2−m0Γu −1∑ i=−∞ |λi|u i∑ k=−∞ 2v1(i−k)u(k − i)mu ≲ ∥f∥mBMO sup m0<0,m0∈Z 2−m0Γu m0∑ i=−∞ |λi|u ≲ ∥f∥mBMO Λ. Now we will the estimate for the second case when 1 < u < ∞. Let 1 u + 1 u′ = 1, we obtain I2 = sup m0<0,m0∈Z 2−m0Γu m0∑ k=−∞ 2kα(0)u ( k−1∑ i=−∞ |λi| ∥∥∥((|z1|+ 1)−λ(z1) [b, µΦ] m β bi ) χk ∥∥∥ q2(·) )u ≲ ∥f∥mBMO sup m0<0,m0∈Z 2−m0Γu m0∑ k=−∞ 2α(0)ku ( k−1∑ i=−∞ |λi| 2−kn2−iv2k(v+ n s ) ∥χBk ∥q1(·) ∥χBi∥q1(·) (k − i)m2−αii )u ≲ ∥f∥mBMO sup m0<0,m0∈Z 2−m0Γu m0∑ k=−∞ 2kα(0)u ( −1∑ i=k |λi|u 2−α(0)iu2u(i−k)(n/q′1(0)−v−n s )(k − i)mu ) ≲ ∥f∥mBMO sup m0<0,m0∈Z 2−m0Γu m0∑ k=−∞ ( −1∑ i=k |λi|u 2v1(i−k)u/2 ) × ( −1∑ i=k 2α(0)(k−i)u′/2(k − i)mu′/2 )u/u′ ≲ ∥f∥mBMO sup m0<0,m0∈Z 2−m0Γu m0∑ k=−∞ ( −1∑ i=k |λi|u 2v1(i−k)u/2 ) ≲ ∥f∥mBMO sup m0<0,m0∈Z 2−m0Γu −1∑ i=−∞ |λi|u i∑ k=−∞ 2v1(i−k)u/2 ≲ ∥f∥mBMO sup m0<0,m0∈Z 2−m0Γu −1∑ i=−∞ |λi|u i∑ k=−∞ 2v1(i−k)u/2 ≲ ∥f∥mBMO sup m0<0,m0∈Z 2−m0Γu m0∑ i=−∞ |λi|u B. Sultan et al. / Eur. J. Pure Appl. Math, 18 (4) (2025), 6732 14 of 20 ≲ ∥f∥mBMO Λ. Finally, we estimate III : III = sup m0≥0,m0∈Z 2−m0Γu m0∑ k=0 2kα∞u ∥∥∥((|z1|+ 1)−λ(z1) [b, µΦ] m β f ) χk ∥∥∥u q2(·) ≲ sup m0≥0,m0∈Z 2−m0Γu m0∑ k=0 2kα∞u ( ∞∑ i=k |λi| ∥∥∥((|z1|+ 1)−λ(z1) [b, µΦ] m β bi ) χk ∥∥∥ q2(·) )u + sup m0≥0,m0∈Z 2−m0Γu m0∑ k=0 2kα∞u ( k−1∑ i=−∞ |λi| ∥∥∥((|z1|+ 1)−λ(z1) [b, µΦ] m β bi ) χk ∥∥∥ q2(·) )u := III1 + III2. If k ∈ Z and i ≥ k + 1 and a.e. z1 ∈ Fk, z2 ∈ Fi, then |z1 − z2| ≈ |z2| ≈ 2i, |µΦ(bi)(z1)| ≤  |z2|∫ o ∣∣∣∣∣∣∣ ∫ |z1−z2|≤t Φ(z1 − z2) |z1 − z2|n−1−β(z1) bi(z2)dz2 ∣∣∣∣∣∣∣ 2 dt t3  1/2 +  ∞∫ |z2| ∣∣∣∣∣∣∣ ∫ |z1−z2|≤t Φ(z1 − z2) |z1 − z2|n−1−β(z1) bi(z2)dz2 ∣∣∣∣∣∣∣ 2 dt t3  1/2 =: I31 + I32. It is easy to find that I31 ≤2(i−k)/22−in |z1|β(z1) ∥bi(z2)∥q1(·) ( |b(z1)− bBi | m ∥Φ(z1 − ·)χi(·)∥q′1(·) + ∥(b(·)− bBi) m(Φ(z1 − ·)χi(·)∥q′1(·) ) . Similarly we have I32 ≤2−in |z1|β(z1) ∥bi(z2)∥q1(·) { |b(z1)− bBi | m ∥Φ(z1 − ·)χi(·)∥q′1(·) + ∥(b(·)− bBi) m(Φ(z1 − ·)χi(·)∥q′1(·) } . ∥∥∥χk(1 + |z1|)−λ(z1) [b, µΦ] m β (gχi) ∥∥∥ q2(·) ≤ C2−in ∥bi∥q1(·) {∥∥∥(b(z1)− bBi) m |z1|β(z1) χk(z1)(1 + |z1|)−λ(z1) ∥∥∥ q2(·) 2−iv2k(v+ n s ) ∥Φ∥Ls(Sn−1) ∥χBi∥q1(·) + ∥f∥mBMO 2−iv2k(v+ n s ) ∥Φ∥Ls(Sn−1) ∥χBi∥q1(·) ∥∥∥|z1|β(z1) χk(z1)(1 + |z1|)−λ(z1) ∥∥∥ q2(·) } B. Sultan et al. / Eur. J. Pure Appl. Math, 18 (4) (2025), 6732 15 of 20 ≤ C2−in ∥bi∥q1(·) { (k − i)m ∥f∥mBMO ∥χBk ∥q1(·) 2 −iv2k(v+ n s ) ∥Φ∥Ls(Sn−1) ∥χBi∥q1(·) + ∥f∥mBMO 2−iv2k(v+ n s ) ∥Φ∥Ls(Sn−1) ∥ ∥χBi∥q1(·) ∥χBk ∥q1(·) } ≤ C2−in ∥bi∥q1(·) (k − i)m ∥f∥mBMO ∥χBk ∥q1(·) 2 −iv2k(v+ n s ) ∥Φ∥Ls(Sn−1) ∥χBi∥q1(·) ≤ C(k − i)m ∥Φ∥Ls(Sn−1) ∥f∥ m BMO 2−in2−iv2k(v+ n s ) ∥χBk ∥q1(·) ∥χBi∥q1(·) ∥bi∥q1(·) . Therefore, when 0 < u ≤ 1 and v2 = v + n s + n q1∞ + α∞, we get III1 = sup m0≥0,m0∈Z 2−m0Γu m0∑ k=0 2kα∞u ( ∞∑ i=k |λi| ∥∥∥((|z1|+ 1)−λ(z1) [b, µΦ] m β bi ) χk ∥∥∥ q2(·) )u ≲ ∥f∥mBMO sup m0≥0,m0∈Z 2−m0Γu m0∑ k=0 2kα∞u ( ∞∑ i=k |λi| 2−kn2−iv2k(v+ n s ) ∥χBk ∥q1(·) ∥χBi∥q1(·) (k − i)m2−αii )u ≲ ∥f∥mBMO sup m0≥0,m0∈Z 2−m0Γu m0∑ k=0 ∞∑ i=k |λi|u 2v2(k−i)u(k − i)mu ≲ ∥f∥mBMO sup m0≥0,m0∈Z 2−m0Γu ∞∑ i=0 |λi|u i∑ k=0 2v2(k−i)u(k − i)mu ≲ ∥f∥mBMO Λ. Now we will the estimate for the second case when 1 < u < ∞. Let 1 u + 1 u′ = 1, we obtain III1 = sup m0≥0,m0∈Z 2−m0Γu m0∑ k=0 2kα∞u ( ∞∑ i=k |λi| ∥∥∥((|z1|+ 1)−λ(z1) [b, µΦ] m β bi ) χk ∥∥∥ q2(·) )u ≲ ∥f∥mBMO sup m0≥0,m0∈Z 2−m0Γu m0∑ k=0 2kα∞u ( ∞∑ i=k |λi| 2−kn2−iv2k(v+ n s ) ∥χBk ∥q1(·) ∥χBi∥q1(·) (k − i)m2−αii )u ≲ ∥f∥mBMO sup m0≥0,m0∈Z 2−m0Γu m0∑ k=0 ∞∑ i=k |λi|u 2v2(k−i)u(k − i)mu ≲ ∥f∥mBMO sup m0≥0,m0∈Z 2−m0Γu m0∑ k=0 ( ∞∑ i=k |λi|u 2v2(i−k)u/2 ) × ( ∞∑ i=k 2α(0)(k−i)u′/2(k − i)mu′/2 )u/u′ ≲ ∥f∥mBMO sup m0≥0,m0∈Z 2−m0Γu m0∑ k=0 ( ∞∑ i=k |λi|u 2v2(i−k)u/2 ) ≲ ∥f∥mBMO sup m0≥0,m0∈Z 2−m0Γu −1∑ i=−∞ |λi|u i∑ k=−∞ 2v2(i−k)u/2 B. Sultan et al. / Eur. J. Pure Appl. Math, 18 (4) (2025), 6732 16 of 20 ≲ ∥f∥mBMO sup m0≥0,m0∈Z 2−m0Γu ∞∑ i=0 |λi|u i∑ k=0 2v2(k−i)u/2 ≲ ∥f∥mBMO Λ. Next we have III2 = sup m0≥0,m0∈Z 2−m0Γu m0∑ k=0 2kα∞u ( k−1∑ i=−∞ |λi| ∥∥∥((|z1|+ 1)−λ(z1) [b, µΦ] m β bi ) χk ∥∥∥ q2(·) )u ≲ sup m0≥0,m0∈Z 2−m0Γu m0∑ k=0 2kα∞u ( −1∑ i=−∞ |λi| ∥∥∥((|z1|+ 1)−λ(z1) [b, µΦ] m β bi ) χk ∥∥∥ q2(·) )u + sup m0≥0,m0∈Z 2−m0Γu m0∑ k=0 2kα∞u ( k−1∑ i=0 |λi| ∥∥∥((|z1|+ 1)−λ(z1) [b, µΦ] m β bi ) χk ∥∥∥ q2(·) )u ≲III12 + III22 . Estimate of second term is essentially similar to III1. For III 1 2 , we have ∥(1 + |z1|)−λ(z1) [b, µΦ] m β biχk∥q2(·) ≤C(k − i)m ∥Φ∥Ls(Sn−1) ∥f∥ m BMO 2−kn2−iv2k(v+ n s ) ∥χBk ∥q1(·) ∥χDl ∥q1(·) ∥bi∥q1(·) ≤C(k − i)m ∥Φ∥Ls(Sn−1) ∥f∥ m BMO 2 i( n q1(0) −v) 2 k(v+n s − n q′1∞ ) ∥bi∥q1(·) . When 0 < u ≤ 1, we have III12 = sup m0≥0,m0∈Z 2−m0Γu m0∑ k=0 2α∞ku ( −1∑ i=−∞ |λi| ∥∥∥((|z1|+ 1)−λ(z1) [b, µΦ] m β bi ) χk ∥∥∥ q2(·) ∥bi∥q1(·) )u ≲ ∥f∥mBMO sup m0≥0,m0∈Z 2−m0Γu m0∑ k=0 2α∞ku ( −1∑ i=−∞ |λi|u (k − i)mu2 lu( n q1(0) −v) 2 ku(v+n s − n q′1∞ ) ∥bi∥uq1(·) ) = ∥f∥mBMO sup m0≥0,m0∈Z 2−m0Γu × m0∑ k=0 2α∞ku ( −1∑ i=−∞ |λi|u (k − i)mu2 lu( n q1(0) −v−α∞) 2 ku(v+n s − n q′1∞ ) ) ≲ ∥f∥mBMO sup m0≥0,m0∈Z 2−m0Γu m0∑ k=0 2 ku(v+n s − n q′1∞ ) (k − i)mu −1∑ i=−∞ |λi|u 2 lu( n q1(0) −v−α∞) ≲ ∥f∥mBMO sup m0≥0,m0∈Z 2−m0Γu −1∑ i=−∞ |λi|u B. Sultan et al. / Eur. J. Pure Appl. Math, 18 (4) (2025), 6732 17 of 20 ≲ ∥f∥mBMO Λ. When 1 < u < ∞, we have III12 = sup m0≥0,m0∈Z 2−m0Γu m0∑ k=0 2α∞ku ( −1∑ i=−∞ |λi| ∥∥∥((|z1|+ 1)−λ(z1) [b, µΦ] m β bi ) χk ∥∥∥ q2(·) ∥bi∥q1(·) )u ≲ ∥f∥mBMO sup m0≥0,m0∈Z 2−m0Γu m0∑ k=0 2α∞ku ( −1∑ i=−∞ |λi|u (k − i)mu2 lu( n q1(0) −v) 2 ku(v+n s − n q′1∞ ) ∥bi∥uq1(·) ) ≲ ∥f∥mBMO sup m0≥0,m0∈Z 2−m0Γu m0∑ k=0 2α∞ku2 ku(v+n s − n q′1∞ ) ( −1∑ i=−∞ |λi|u (k − i)mu2 lu( n q1(0) −v−α(0)) ) ≲ ∥f∥mBMO sup m0≥0,m0∈Z 2−m0Γu m0∑ k=0 ( −1∑ i=−∞ |λi|u (k − i)mu2 lu( n q1(0) −v−α(0)) ) ≲ ∥f∥mBMO sup m0≥0,m0∈Z 2−m0Γu m0∑ k=0 ( −1∑ i=0 |λi|u 2 i( n q1(0) −v−α(0))u/2 ) × ( −1∑ i=0 (k − i)mu′/22 i( n q1(0) −v−α(0))u′/2 )u/(u)′ ≲ ∥f∥mBMO sup m0≥0,m0∈Z 2−m0Γu m0∑ k=0 ( −1∑ i=0 |λi|u 2 i( n q1(0) −v−α(0))u/2 ) ≲ ∥f∥mBMO sup m0≥0,m0∈Z 2−m0Γu −1∑ i=−∞ |λi|u ≲ ∥f∥mBMO Λ. Thus proof of the Theorem is completed. Conflict of interest The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper. 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