EUROPEAN JOURNAL OF PURE AND APPLIED MATHEMATICS 2025, Vol. 18, Issue 3, Article Number 6274 ISSN 1307-5543 – ejpam.com Published by New York Business Global Grand Variable Herz-Morrey Type Besov Spaces and Triebel-Lizorkin Spaces Mehvish Sultan1, Babar Sultan2,∗, Ioan-Lucian Popa3,4,∗ 1 Department of Mathematics, Capital University of Science and Technology, Islamabad, Pakistan 2 Department of Mathematics, Quaid-I-Azam University, Islamabad 45320, 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 the article, the boundedness of vector-valued sublinear operators in grand variable Herz-Morrey spaces MK̇ η(·),q),θ λ,p(·) (Rn) are obtained. Then grand variable Herz-Morrey type Besov and Triebel-Lizorkin spaces are defined. We will also prove the equivalent quasi-norms by Peetre’s maximal operators in these spaces. 2020 Mathematics Subject Classifications: 46E35, 42B25, 42B35 Key Words and Phrases: Grand variable Herz-Morrry space, Besov space, Triebel Lizorkin space, maximal operator 1. Introduction In variable exponent spaces, the Hardy-Littlewood maximal operator’s boundedness is crucial. For instance, it is well known that if the Hardy-Littlewood maximal operator is bounded in variable exponent Lebesgue space, then numerous conclusions from classical harmonic analysis and function theory also apply for the variable exponent case; see [1–4]. Moreover, a variety of variable exponent spaces are presented, including: Bessel potential spaces, Besov and Trieble-Lizorkin spaces, Hardy spaces, Herz spaces, grand variable Herz spaces, grand variable weighted Herz spaces, Herz-Morrey spaces, grand variable Herz- Morrey spaces, Morrey spaces, Morrey type Besov and Trieble-Lizorkin spaces, Trieble- Lizorkin-Morrey spaces, Trieble-Lizorkin-Morrey spaces, and so forth; see [5–13, 13, 14, ∗Corresponding author. ∗Corresponding author. DOI: https://doi.org/10.29020/nybg.ejpam.v18i3.6274 Email addresses: mehvishsultanbaz@gmail.com (M. Sultan), babarsultan40@yahoo.com (B. Sultan), lucian.popa@uab.ro (I-L. Popaa) https://www.ejpam.com 1 Copyright: © 2025 The Author(s). (CC BY-NC 4.0) M. Sultan, B. Sultan, I-L. Popa / Eur. J. Pure Appl. Math, 18 (3) (2025), 6274 2 of 33 14, 14–35] and references therein. As you can see from [36–40], numerous conclusions about the boundedness of sublinear operators in these spaces have been established. A sublinear operator T satisfies the size condition |Tg(x)| ⩽ C ∫ Rn |x− y|−n|g(y)|dy for all g ∈ L1 loc (Rn) with compact support and a.e. x /∈ supp g. Then, T is bounded on the grand variable Herz-Morrey spaces and the homogeneous and non-homogeneous Herz spaces (see the monographs [41, 42]). L. Tang and D. Yang then expand these conclusions for the weighted vector-valued situation in [43]. Herz type Besov and Triebel- Lizorkin spaces with variable exponent K̇η,q p(·) (R n) were introduced by C. Shi and the second author in [18]. . For the Besov and Triebel-Lizorkin spaces of constant exponent Herz type, we direct the reader to [44–47]. M. Izuki [48] used variable exponent MK̇η,λ q,p(·) (R n) to get the vector-valued boundedness for certain sublinear operators that meet the size requirement on Herz-Morrey spaces. Sultan et al. [49] presented the concept of grand variable Herz-Morrey spaces; see [50–52] for additional findings on these spaces. The boundedness of vector-valued Hardy-Littlewood maximal operators in Herz spaces with variable exponents was established by the authors in [53]. Peetre’s maximal operators were used to characterize Herz type Besov and Triebel-Lizorkin spaces with variable exponents. The current research examines the boundedness of vector-valued Hardy-Littlewood maximal operators on grand variable Herz-Morrey type Besov and Triebel-Lizorkin spaces, drawing inspiration from the aforementioned publications. The paper is organised as follows. We provide some conventions in the remainder of the section. Our primary findings, which are a generalization of related findings in [48] and [53, 54], will be presented in Section 2. We provide evidence for our findings in Section 3. 2. Main results The n-dimensional real Euclidean space is denoted as Rn, as is customary. Let A be a measurable subset of Rn. Throughout this paper, C and c are always positive constants which may vary from line to line. For a measurable set S, |S| denote its Lebesgue measure and 1S denote its characteristic function. We write u ⩽ v, if u ≤ cv, and if u ⩽ v and v ⩽ u, then u ∼ v. By variable exponents we mean a measurable function on Rn. For every l ∈ Z, we denote Bt = B(0, 2t) = {x ∈ Rn : |x| < 2t}. After deducting Bt−1 from Rt, 1Rt = 1t. By P(Rn) we denote the subset of variable exponents with range [1,∞]. Let q− := ess inf y∈A q(y) > 1 and q+ := ess sup y∈A q(y) < ∞, then we have 1 ≤ q−(A) ≤ q(y) ≤ q+(A) < ∞. (2.1) The notation B is the ball such that B(z, r) := {y ∈ A : |z − y| < r}. Now variable Lebesgue space Lq(·)(A) is given as Lp(·)(A) = { g is measurable : ∫ A ( |g(z)| λ )q(z) dz < ∞ where λ is a constant } . M. Sultan, B. Sultan, I-L. Popa / Eur. J. Pure Appl. Math, 18 (3) (2025), 6274 3 of 33 Lebesgue space is equipped with the norm ∥g∥Lq(·)(A) = inf { λ > 0 : ∫ A ( |g(z)| λ )q(z) dz ≤ 1 } . The Hardy-Littlewood maximal operator M for g ∈ L1 loc(A) is defined as Mg(z) := sup 0 0, q(·) satisfiy the decay condition if lim |x|→∞ q(x) = q∞ := q(∞) such that C∞ ln(e+ |h|) ≥ |q(h)− q∞|. (2.3) Let C0 > 0, q(·) satisfy the log Hölder continuity condition at 0 for |h| ≤ 1 2 , such that C0 ln |h| ≥ |q(h)− q(0)|. (2.4) P log = P log(A) consists of all functions q(·) ∈ P(A) satisfying (2.1) and (2.2). P∞(A) (resp. P0,∞(A)) is the subset of P(A) consisting of functions which satisfy condition (2.3) (resp. both conditions (2.3) and (2.4)). The set of positive integers is denoted by Z+. For each t0 ∈ Z,1At0 = 1t0 ; for t0 ∈ Z+, 1B0 = 1̃0; and so on. 1At0 denote the characteristic function of At0 . By a ⩽ b we denote a ≤ Cb. Moreover, we define P0 (Rn) to be the set of measurable functions p on Rn with the range in (0,∞) such that p− > 0 and p+ < ∞. Given p(·) ∈ P0 (Rn), one can define the space Lp(·) (Rn). This is equivalent to defining it to be the set of all functions g such that |g|p0 ∈ Lp(·)/p0 (Rn) where 0 < p0 < p−, p(·) p0 ∈ P (Rn) . Then the quasi-norm is given as ∥g∥Lp(·)(Rn) = ∥|g|p0∥1/p0 Lp(·)/p0 (Rn) . M. Sultan, B. Sultan, I-L. Popa / Eur. J. Pure Appl. Math, 18 (3) (2025), 6274 4 of 33 Lemma 1. [55] Let f belong to the Lebesgue space Lp(·)(Rn) and g belong to Lp′(·)(Rn), where p(·) ∈ P(Rn). Then, the product function fg is integrable over Rn, and the following inequality holds: ∫ Rn ∣∣f(x)g(x)∣∣ dx ≤ rp∥f∥p(·)∥g∥p′(·), where rp is defined as rp = 1 + 1 p− − 1 p+ . Lemma 2. [56] Let q̃(·) be a variable exponent defined as 1 p(z) − 1 q = 1 q̃(z) where (z ∈ Rn). Then for all measurable functions f and g we have ∥fg∥p(·) ≤ C ∥g∥q̄(·) ∥g∥q. Lemma 3. [57] Let p(·) be a function within the class B(Rn). For any ball B in Rn, we get 1 |B| ∥1B∥p′(·)∥1B∥p(·) ⩽ C, where C > 0 . Lemma 4. [57] 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: ∥1B∥p(·) ∥1S∥p(·) ⩽ |B| |S| , ∥1S∥p(·) ∥1B∥p(·) ⩽ ( |S| |B| )ω1 , ∥1S∥p′(·) ∥1B∥p′(·) ⩽ ( |S| |B| )ω2 , here 0 < ω1, ω2 < 1. Lemma 5. [1] Let {gj}∞j=1 be the sequences of locally integrable functions, 1 < r < ∞ and p(·) ∈ B (Rn). Then ∥∥∥∥∥∥∥  ∞∑ j=1 |Mgj |r 1/r ∥∥∥∥∥∥∥ p(·) ⩽ C ∥∥∥∥∥∥∥  ∞∑ j=1 |gj |r 1/r ∥∥∥∥∥∥∥ p(·) . Definition 6. Let 0 < q < ∞, p(·) ∈ P (Rn) , 0 ≤ λ < ∞, θ > 0 and η(·) : Rn → R with η ∈ L∞ (Rn). (i) The homogeneous grand variable Herz-Morrey space MK̇ η(·),q),θ λ,p(·) (Rn) is defined by MK̇ η(·),q),θ λ,p(·) (Rn) := { g ∈ L p(·) loc (Rn\{0}) : ∥g∥ MK̇ η(·),q),θ λ,p(·) (Rn) < ∞ } where M. Sultan, B. Sultan, I-L. Popa / Eur. J. Pure Appl. Math, 18 (3) (2025), 6274 5 of 33 ∥g∥ MK̇ η(·),q),θ λ,p(·) (Rn) := sup ϵ>0 sup L∈Z 2−Lλ ( ϵθ L∑ k=−∞ ∥∥∥2kη(·)g1k∥∥∥q(1+ϵ) p(·) ) 1 q(1+ϵ) . (ii) The non-homogeneous grand variable Herz-Morrey spaceMK η(·),q),θ λ,p(·) (Rn) is defined by MK η(·),q),θ λ,p(·) (Rn) := { g ∈ L p(·) loc (Rn\{0}) : ∥g∥ MK η(·),q),θ λ,p(·) (Rn) < ∞ } where ∥g∥ MK η(·),q),θ λ,p(·) (Rn) := sup ϵ>0 sup L∈N0 2−Lλ ( ϵθ L∑ k=0 ∥∥∥2kη(·)g1k∥∥∥q(1+ϵ) p(·) ) 1 q(1+ϵ) . Lemma 7. [17] Let D > 1 and q ∈ P0,∞(Rn). Then 1 c0 r n q(0) ≤ ∥1B(0,Dr)\B(0,r)∥Lq(·) ≤ c0r n q(0) , for 0 < r ≤ 1 (2.5) and 1 c∞ r n q∞ ≤ ∥1B(0,Dr)\B(0,r)∥Lq(·) ≤ c∞r n q∞ , for r ≥ 1, (2.6) respectively, where c0 ≥ 1 and c∞ ≥ 1 depend on D but not depend on r. Theorem 8. If 1 < r < ∞ and η(·) ∈ L∞ (Rn)∩ P log 0 (Rn) ∩ P log ∞ (Rn) with η(0), η∞ ∈ (−nω1, nω2), where ω1, ω2 ∈ (0, 1) are constants appearing in (2.1). Let p(·) ∈ B (Rn) , 0 < q < ∞, and 0 ⩽ λ < min {(nω1 + η(0)) /2, (nω1 + η∞) /2} . Suppose that T is a sublinear operator satisfying vector-valued inequality on Lp(·) (Rn)∥∥∥∥∥∥∥  ∞∑ j=1 |Tgj |r  1 r ∥∥∥∥∥∥∥ p(·) ⩽ C ∥∥∥∥∥∥∥  ∞∑ j=1 |gj |r  1 r ∥∥∥∥∥∥∥ p(·) , (2.7) for all sequences {gj}∞j=1 of locally integrable functions on Rn. Then we have the vector-valued inequality on MK̇ η(·),q),θ λ,p(·) (Rn)∥∥∥∥∥∥ ( ∞∑ k=1 |Tgk|r ) 1 r ∥∥∥∥∥∥ MK̇ η(·),q),θ λ,p(·) (Rn) ⩽ C ∥∥∥∥∥∥ ( ∞∑ k=1 |gk|r ) 1 r ∥∥∥∥∥∥ MK̇ η(·),q),θ λ,p(·) (Rn) . Remark 9. Here and below, we only declare our main results in the homogeneous grand variable Herz-Morrey space because the proof for the non-homogeneous case can be treated by the similar way and is much more easier. M. Sultan, B. Sultan, I-L. Popa / Eur. J. Pure Appl. Math, 18 (3) (2025), 6274 6 of 33 Lemma 10. (see [1]) Let p(·), r, and {gj}∞j=1 are given in Theorem 2.8, then∥∥∥∥∥∥∥  ∞∑ j=1 |Mgj |r  1 r ∥∥∥∥∥∥∥ Lp(·)(Rn) ⩽ C ∥∥∥∥∥∥∥  ∞∑ j=1 |gj |r  1 r ∥∥∥∥∥∥∥ Lp(·)(Rn) . From Theorem 2.5 and Lemma 2.7, we obtain the following result for the Hardy- Littlewood maximal operator. Corollary 1. Let η, r, q, p, are given in Theorem 2.8 and 0 ⩽ λ < min {(nω1 + η(0)) /2, (nω1 + η∞) /2} , then ∥∥∥∥∥∥ ( ∞∑ k=1 |Mgk|r ) 1 r ∥∥∥∥∥∥ MK̇ η(·),q),θ λ,p(·) (Rn) ⩽ C ∥∥∥∥∥∥ ( ∞∑ k=1 |gk|r ) 1 r ∥∥∥∥∥∥ MK̇ η(·),q),θ λ,p(·) (Rn) . Let S (Rn) denote the Schwartz functions and S ′ (Rn) the set of all tempered distribu- tions. We define the Fourier transform of a function g ∈ S (Rn) by φ̂(g)(y) = 2π−n/2 ∫ Rn e−ix·yg(x)dx, y ∈ Rn, while φ∨ is the inverse Fourier transform. Let φ0 ∈ S (Rn) with φ0(y) ⩾ 0 then we have φ0(y) = { 1, |y| ⩽ 1 0, |y| ⩾ 2. Let φ(y) = φ0(y)− φ0(2y) and define φℓ(y) = φ ( 2−ℓy ) , ℓ ∈ N. Then {φℓ}ℓ∈N0 be the resolution of unity, such that ∞∑ ℓ=0 φℓ(x) = 1, x ∈ Rn. Definition 11. Let {φj}j∈N0 be a resolution of unity as above, s ∈ R, 0 < κ, q ≤ ∞, p(·) ∈ P (Rn) and η(·) : Rn → R with η(·) ∈ L∞ (Rn). (i) Then the grand variable Herz-Morrey type Besov space is defined by M. Sultan, B. Sultan, I-L. Popa / Eur. J. Pure Appl. Math, 18 (3) (2025), 6274 7 of 33 MK̇ η(·),q),θ λ,p(·) Bs κ (Rn) := { g ∈ S ′ (Rn) : ∥g∥ ℓκ ( MK̇ η(·),q),θ λ,p(·) (Rn) ) < ∞ } , where ∥g∥ MK̇ η(·),q),θ λ,p(·) Bs κ := ∥∥∥{2sjφ∨ j ∗ g }∞ j=0 ∥∥∥ ℓκ ( MK̇ η(·),q),θ λ,p(·) ) . (ii) For p+ < ∞, the grand variable Herz-Morrey type Triebel-Lizorkin space is defined by MK̇ η(·),q),θ λ,p(·) F s κ (Rn) := { g ∈ S ′ (Rn) : ∥g∥ MK̇ η(·),q),θ λ,p(·) (ℓκ) < ∞ } , where ∥g∥ MK̇ η(·),q),θ λ,p(·) F s κ := ∥∥∥{2sjφ∨ j ∗ g }∞ j=0 ∥∥∥ MK̇ η(·),q),θ λ,p(·) (ℓκ) . Here we denote respectively by ℓκ ( MK̇ η(·),q),θ λ,p(·) ) and MK̇ η(·),q),θ λ,p(·) (ℓκ) the spaces of all sequences {gj} of measurable functions on Rn with finite quasi-norms ∥∥∥{gj}∞j=0 ∥∥∥ ℓκ ( MK̇ η(·),q),θ λ,p(·) ) :=  ∞∑ j=0 ∥gj∥κMK̇ η(·),q),θ λ,p(·)  1 κ , and ∥∥∥{gj}∞j=0 ∥∥∥ MK̇ η(·),q),θ λ,p(·) (ℓκ) := ∥∥∥∥∥∥∥  ∞∑ j=0 |gj |κ 1/κ ∥∥∥∥∥∥∥ MK̇ η(·),q),θ λ,p(·) . Let S ⩾ 0, ε > 0 and Ψ0, Ψ ∈ S (Rn) such that ∣∣∣Ψ̂0(ϱ) ∣∣∣ > 0 on {|ϱ| < 2ε} (2.8) |Ψ̂(ϱ)| > 0 on {ε 2 < |ϱ| < 2ε } (2.9) and Dτ Ψ̂(0) = 0, for all |τ | ⩽ S . (2.10) Here, (2.8) and (2.9) are Tauberian conditions, while (2.10) expresses the vanishing moment conditions in Ψ . In [58], J. Peetre introduced the classical Peetre’s maximal operator: M. Sultan, B. Sultan, I-L. Popa / Eur. J. Pure Appl. Math, 18 (3) (2025), 6274 8 of 33 Let a tempered distribution g ∈ S ′ (Rn), a > 0, and {Ξℓ}ℓ∈Z ⊂ S (Rn). Then system of maximal functions are given as (Ξ∗ ℓ )a g(x) := sup y∈Rn |Ξℓ ∗ g(x+ y)| (1 + 2k|y|)a , x ∈ Rn, ℓ ∈ Z. Since Ξℓ ∗ g(y) makes sense pointwise, everything is well defined. We will often use dilates Ξ′ ℓ(x) = 2knΞ ( 2kx ) of a fixed function Ξ ∈ S (Rn), where Ξ0(x) might be given by a separate function. Continuous dilates are also needed. If Ξt := t−nΞ ( t−1· ) . Then Ψ∗ t,ag(x) := sup y∈Rn |Ξt ∗ g(x+ y)|( 1 + |y| t )a x ∈ Rn, t > 0. Theorem 12. If κ, q ∈ (0,∞], ω > 0, s ∈ R with s < S + 1, and η, q, p, are the same as given in Lemma 2.9. Let p(·)/p0 ∈ B (Rn) with p0 < min (p−, 1). Let Θ0,Θ ∈ S (Rn) be given by (2.8) and (2.9), respectively. Then (i) For a > n/p0, then the space MK̇ η(·),q),θ λ,p(·) Bs κ (Rn) can be characterized by MK̇ η(·),q),θ λ,p(·) Bs κ (Rn) = { g ∈ S ′ (Rn) : ∥g∥(i) MK̇ η(·),q),θ λ,p(·) Bs κ < ∞ } , i = 1, · · · , 4 where ∥g∥(1) MK̇ η(·),q),θ λ,p(·) Bs κ := ∥Φ0 ∗ g∥MK̇ η(·),q),θ λ,p(·) + (∫ 1 0 t−sκ ∥Φt ∗ g∥κMK̇ η(·),q),θ λ,p(·) dt t )1/κ ∥g∥(2) MK̇ η(·),q),θ λ,p(·) Bs κ := ∥(Φ∗ 0g)a∥MK̇ η(·),λ q,p + (∫ 1 0 t−sκ ∥(Φ∗ t g)a∥ κ MK̇ η(·),q),θ λ,p(·) dt t )1/κ ∥g∥(3) MK̇ η(·),q),θ λ,p(·) Bs κ := ( ∞∑ k=0 2skκ ∥(Φ∗ kg)a∥ κ MK̇ η(·),q),θ λ,p(·) )1/κ ∥g∥(4) MK̇ η(·),q),θ λ,p(·) Bs κ := ( ∞∑ k=0 2skκ ∥Φk ∗ g∥κMK̇ η(·),q),θ λ,p(·) )1/κ . Then, { ∥ · ∥(i) MK̇ η(·),q),θ λ,p(·) Bs κ }4 i=1 are equivalent. M. Sultan, B. Sultan, I-L. Popa / Eur. J. Pure Appl. Math, 18 (3) (2025), 6274 9 of 33 (ii) If p0 < κ, then for a > n/p0 the space MK̇ η(·),q),θ λ,p(·) F s κ (Rn) can be characterized by MK̇ η(·),q),θ λ,p(·) F s κ (Rn) = { g ∈ S ′ (Rn) : ∥g∥(i) MK̇ η(·),q),θ λ,p(·) F s κ < ∞ } , i = 1, . . . , 5 where ∥g∥(1) MK̇ η(·),q),θ λ,p(·) F s κ := ∥Φ0 ∗ g∥MK̇ η(·),q),θ λ,p(·) + ∥∥∥∥∥ (∫ 1 0 t−sκ |Φt ∗ g|κ dt t )1/κ ∥∥∥∥∥ MK̇ η(·),q),θ λ,p(·) (2.11) ∥g∥(2) MK̇ η(·),q),θ λ,p(·) F s κ := ∥(Φ∗ 0g)a∥MK̇ η(·),q),θ λ,p(·) + ∥∥∥∥∥ (∫ 1 0 [ t−s (Φ∗ t g)a ]κ dt t )1/κ ∥∥∥∥∥ MK̇ η(·),q),θ λ,p(·) (2.12) ∥g∥(3) MK̇ η(·),q),θ λ,p(·) F s κ := ∥Φ0 ∗ g∥MK̇ η(·),q),θ λ,p(·) ∥∥∥∥(∫ 1 0 t−sκ × ∫ |z| 0, then ∥g∥ MK̇ η(·),q),θ λ,p(·) ≈ max sup ϵ>0 sup L≤0,L∈Z 2−Lλ ( ϵθ L∑ k=−∞ 2kη(0)q(1+ϵ) ∥g1k∥ q(1+ϵ) p(·) ) 1 q(1+ϵ) M. Sultan, B. Sultan, I-L. Popa / Eur. J. Pure Appl. Math, 18 (3) (2025), 6274 10 of 33 sup L>0,L∈Z sup ϵ>0 2−Lλ ( ϵθ −1∑ k=−∞ 2kη(0)q(1+ϵ) ∥g1k∥ q(1+ϵ) p(·) ) 1 q(1+ϵ) + 2−Lλ ( ϵθ L∑ k=0 2kη∞q(1+ϵ) ∥g1k∥ q(1+ϵ) p(·) ) 1 q(1+ϵ)  Lemma 3.1 is similar to Proposition 3.8 in [5]. Indeed, when η(·) ∈ L∞ (Rn)∩ P log 0 (Rn) ∩ P log ∞ (Rn), there exist positive constants C1, C2 such that if k ≤ 0 and x ∈ Dk then C12 kη(0) ≤ 2kη(x) ≤ C22 kη(0); if k > 1 and x ∈ Dk then C12 kη∞ ≤ 2kη(x) ≤ C22 kη∞ , where Dk := Bk \Bk−1. Thus, we obtain Lemma 3.1. Proof. Now we will give the proof of Theorem 2.8. Let ( ∑∞ k=1 |gk| r) 1 r ∈ MK̇ η(·),q),θ λ,p(·) (Rn). Using the Lemma 3.1 we get ∥∥∥∥∥∥∥  ∞∑ j=1 |Tgj |r  1 r ∥∥∥∥∥∥∥ MK̇ η(·),q),θ λ,p(·) ≈ max sup ϵ>0 sup L≤0,L∈Z 2−Lλ ϵθ L∑ k=−∞ 2kη(0)q(1+ϵ) ∥∥∥∥∥∥∥  ∞∑ j=1 |Tgj |r  1 r 1k ∥∥∥∥∥∥∥ q(1+ϵ) p(·)  1 q(1+ϵ) sup ϵ>0 sup L>0,L∈Z 2−Lλ ϵθ −1∑ k=−∞ 2kη(0)q(1+ϵ) ∥∥∥∥∥∥∥  ∞∑ j=1 |Tgj |r  1 r 1k ∥∥∥∥∥∥∥ q(1+ϵ) p(·)  1 q(1+ϵ) +2−Lλ ϵθ L∑ k=0 2kη∞q(1+ϵ) ∥∥∥∥∥∥∥  ∞∑ j=1 |Tgj |r  1 r 1k ∥∥∥∥∥∥∥ q(1+ϵ) p(·)  1 q(1+ϵ)   =: max {ET , FT } . We also denote FT by FT := sup ϵ>0 sup L>0,L∈Z [GT +HT ] with GT := 2−Lλ ϵθ −1∑ k=−∞ 2kη(0)q(1+ϵ) ∥∥∥∥∥∥∥  ∞∑ j=1 |Tgj |r  1 r 1k ∥∥∥∥∥∥∥ q(1+ϵ) p(·)  1 q(1+ϵ) HT := 2−Lλ ϵθ L∑ k=0 2kη∞q(1+ϵ) ∥∥∥∥∥∥∥  ∞∑ j=1 |Tgj |r  1 r 1k ∥∥∥∥∥∥∥ q(1+ϵ) p(·)  1 q(1+ϵ) . M. Sultan, B. Sultan, I-L. Popa / Eur. J. Pure Appl. Math, 18 (3) (2025), 6274 11 of 33 We need to prove that ET ≲ Ef , GT ≲ Gf and HT ≲ Hf respectively, where Ef := sup ϵ>0 sup L⩽0,L∈Z 2−Lλ ϵθ L∑ k=−∞ 2η(0)q(1+ϵ)k ∥∥∥∥∥∥∥1k  ∞∑ j=1 |gj |r  1 r ∥∥∥∥∥∥∥ q(1+ϵ) p(·)  1 q(1+ϵ) Gf := 2−Lλ ϵθ −1∑ k=−∞ 2kη(0)q(1+ϵ) ∥∥∥∥∥∥∥  ∞∑ j=1 |gj |r  1 r 1k ∥∥∥∥∥∥∥ q(1+ϵ) p(·)  1 q(1+ϵ) Hf := 2−Lλ ϵθ L∑ k=0 2kη∞q(1+ϵ) ∥∥∥∥∥∥∥  ∞∑ j=1 |gj |r  1 r 1k ∥∥∥∥∥∥∥ q(1+ϵ) p(·)  1 q(1+ϵ) . Hence we get ET ≲ Ef and FT ≲ Fg where Fg denote sup ϵ>0 sup L>0,L∈Z [Gf +Hf ]. From above all and using Lemma 3.1 again, we have ∥∥∥∥∥∥∥  ∞∑ j=1 |Tgj |r  1 r ∥∥∥∥∥∥∥ MK̇ η(·),q),θ λ,p(·) ≈ max {ET , FT } ≲ max {Eg, Fg} ≈ ∥∥∥∥∥∥∥  ∞∑ j=1 |gj |r  1 r ∥∥∥∥∥∥∥ MK̇ η(·),q),θ λ,p(·) . Estimate of GT ≲ Gf is similar to EM ≲ Ef so omit the details. By using the size condition and Minkowski’s inequality, for ET we get ET =sup ϵ>0 sup L⩽0,L∈Z 2−Lλ ϵθ L∑ k=−∞ 2η(0)q(1+ϵ)k ∥∥∥∥∥∥∥1k  ∞∑ j=1 |Tgj |r  1 r ∥∥∥∥∥∥∥ q(1+ϵ) p(·)  1 q(1+ϵ) =sup ϵ>0 sup L⩽0,L∈Z 2−Lλ ϵθ L∑ k=−∞ 2η(0)q(1+ϵ)k ∥∥∥∥∥∥∥1k  ∞∑ j=1 ∣∣∣∣∣T ∞∑ i=−∞ gij ∣∣∣∣∣ r  1 r ∥∥∥∥∥∥∥ q(1+ϵ) p(·)  1 q(1+ϵ) M. Sultan, B. Sultan, I-L. Popa / Eur. J. Pure Appl. Math, 18 (3) (2025), 6274 12 of 33 ⩽ sup ϵ>0 sup L⩽0,L∈Z 2−Lλ ϵθ L∑ k=−∞ 2η(0)q(1+ϵ)k ∥∥∥∥∥∥∥1k ∞∑ i=−∞  ∞∑ j=1 ∣∣Tgij∣∣r  1 r ∥∥∥∥∥∥∥ q(1+ϵ)  1 q(1+ϵ) p(·) ≲ sup ϵ>0 sup L⩽0,L∈Z 2−Lλ ϵθ L∑ k=−∞ 2η(0)q(1+ϵ)k ∥∥∥∥∥∥∥1k k−2∑ i=−∞  ∞∑ j=1 ∣∣Tgij∣∣r  1 r ∥∥∥∥∥∥∥ q(1+ϵ) p(·)  1 q(1+ϵ) + sup ϵ>0 sup L⩽0,L∈Z 2−Lλ ϵθ L∑ k=−∞ 2η(0)q(1+ϵ)k ∥∥∥∥∥∥∥1k k+1∑ i=k−1  ∞∑ j=1 ∣∣Tgij∣∣r  1 r ∥∥∥∥∥∥∥ q(1+ϵ) p(·)  1 q(1+ϵ) + sup ϵ>0 sup L⩽0,L∈Z 2−Lλ ϵθ L∑ k=−∞ 2η(0)q(1+ϵ)k ∥∥∥∥∥∥∥1k ∞∑ i=k+2  ∞∑ j=1 ∣∣Tgij∣∣r  1 r ∥∥∥∥∥∥∥ q(1+ϵ) p(·)  1 q(1+ϵ) =:E1 T + E2 T + E3 T . By the same way we consider HT . HT ≲2−Lλ ϵθ L∑ k=0 2η∞q(1+ϵ)k ∥∥∥∥∥∥∥1k k−2∑ i=−∞  ∞∑ j=1 ∣∣Tgij∣∣r  1 r ∥∥∥∥∥∥∥ q p(·)  1 q(1+ϵ) + 2−Lλ ϵθ L∑ k=0 2η∞q(1+ϵ)k ∥∥∥∥∥∥∥1k k+1∑ i=k−1  ∞∑ j=1 ∣∣Tgij∣∣r  1 r ∥∥∥∥∥∥∥ q(1+ϵ) p(·)  1 q(1+ϵ) + 2−Lλ ϵθ L∑ k=0 2η∞q(1+ϵ)k ∥∥∥∥∥∥∥1k ∞∑ i=k+2  ∞∑ j=1 ∣∣Tgij∣∣r  1 r ∥∥∥∥∥∥∥ q(1+ϵ) p(·)  1 q(1+ϵ) =:H1 T +H2 T +H3 T . Secondly, we will prove Ei T and H i T , i = 1, 2, 3. Step 1. For E2 T , we have M. Sultan, B. Sultan, I-L. Popa / Eur. J. Pure Appl. Math, 18 (3) (2025), 6274 13 of 33 E2 T ≲ sup ϵ>0 sup L⩽0,L∈Z 2−Lλ ϵθ L∑ k=−∞ 2η(0)q(1+ϵ)k k+1∑ i=k−1 ∥∥∥∥∥∥∥  ∞∑ j=1 ∣∣gij∣∣r  1 r ∥∥∥∥∥∥∥ q(1+ϵ) p(·)  1 q(1+ϵ) = sup ϵ>0 sup L⩽0,L∈Z 2−Lλ ϵθ L∑ k=−∞ 2η(0)q(1+ϵ)k k+1∑ i=k−1 ∥∥∥∥∥∥∥1i  ∞∑ j=1 |gj |r  1 r ∥∥∥∥∥∥∥ q(1+ϵ) p(·)  1 q(1+ϵ) ≲ sup ϵ>0 sup L⩽0,L∈Z 2−Lλ ϵθ L∑ k=−∞ 2η(0)q(1+ϵ)k ∥∥∥∥∥∥∥1k−1  ∞∑ j=1 |gj |r  1 r ∥∥∥∥∥∥∥ q(1+ϵ) p(·)  1 q(1+ϵ) + sup ϵ>0 sup L⩽0,L∈Z 2−Lλ ϵθ L∑ k=−∞ 2η(0)q(1+ϵ)k ∥∥∥∥∥∥∥1k  ∞∑ j=1 |gj |r  1 r ∥∥∥∥∥∥∥ q(1+ϵ) p(·)  1 q(1+ϵ) + sup ϵ>0 sup L⩽0,L∈Z 2−Lλ ϵθ L∑ k=−∞ 2η(0)q(1+ϵ)k ∥∥∥∥∥∥∥1k+1  ∞∑ j=1 |gj |r  1 r ∥∥∥∥∥∥∥ q(1+ϵ) p(·)  1 q(1+ϵ) ≲ sup ϵ>0 sup L⩽0,L∈Z 2−Lλ ϵθ L∑ k=−∞ 2η(0)q(1+ϵ)k ∥∥∥∥∥∥∥1k  ∞∑ j=1 |gj |r  1 r ∥∥∥∥∥∥∥ q(1+ϵ) p(·)  1 q(1+ϵ) =: Ef . Then turn to H2 T , similarly, we have H2 T ≲ 2−Lλ ϵθ L∑ k=0 2η∞q(1+ϵ)k ∥∥∥∥∥∥∥1k−1  ∞∑ j=1 |gj |r  1 r ∥∥∥∥∥∥∥ q(1+ϵ) p(·)  1 q(1+ϵ) + 2−Lλ ϵθ L∑ k=0 2η∞q(1+ϵ)k ∥∥∥∥∥∥∥1k  ∞∑ j=1 |gj |r  1 r ∥∥∥∥∥∥∥ q(1+ϵ) p(·)  1 q(1+ϵ) M. Sultan, B. Sultan, I-L. Popa / Eur. J. Pure Appl. Math, 18 (3) (2025), 6274 14 of 33 + 2−Lλ ϵθ L∑ k=0 2η∞q(1+ϵ)k ∥∥∥∥∥∥∥1k+1  ∞∑ j=1 |gj |r  1 r ∥∥∥∥∥∥∥ q(1+ϵ) p(·)  1 q(1+ϵ) ≲ Hf . Step 2. Let ∀i ⩽ k − 2, x ∈ Rk, 1 < r < ∞, by the size condition and the generalized Minkowski’s inequality, we obtain  ∞∑ j=1 T r ( gij ) (x)  1 r ≲  ∞∑ j=1 ( 2−kn ∫ Rn ∣∣gij(y)∣∣dy)r  1 r ≲ 2−kn ∫ Rn  ∞∑ j=1 ∣∣gij∣∣r  1 r dy. By Hölder’s inequality we get E1 T ≲ sup ϵ>0 sup L⩽0,L∈Z 2−Lλ ϵθ L∑ k=−∞ 2η(0)q(1+ϵ)k ∥∥∥∥∥∥∥1k k−2∑ i=−∞ 2−kn ∫ Rn  ∞∑ j=1 ∣∣gij∣∣r  1 r dy ∥∥∥∥∥∥∥ q(1+ϵ) p(·)  1 q(1+ϵ) = sup ϵ>0 sup L⩽0,L∈Z 2−Lλ ϵθ L∑ k=−∞ 2η(0)q(1+ϵ)k ∥1k∥ q(1+ϵ) p(·)  k−2∑ i=−∞ 2−kn ∫ Rn  ∞∑ j=1 ∣∣gij∣∣r  1 r dy  q(1+ϵ)  1 q(1+ϵ) ≲ sup ϵ>0 sup L⩽0,L∈Z 2−Lλ ϵθ L∑ k=−∞ 2η(0)q(1+ϵ)k ∥1k∥ q(1+ϵ) p(·)  k−2∑ i=−∞ 2−kn ∥∥∥∥∥∥∥  ∞∑ j=1 |gj |r  1 r 1i ∥∥∥∥∥∥∥ p(·) ∥1i∥Lp′(·)  q(1+ϵ)  1 q(1+ϵ) . (3.1) On the other hand, by using Lemma 2.4, we have 2−kn ∥1k∥p(·) ∥1i∥Lp′(·) ≲ 2−kn2 kn p(0) 2 in p′(0) ≲ 2 (i−k)n p′(0) . (3.2) We put (3.2) into (3.1) and get M. Sultan, B. Sultan, I-L. Popa / Eur. J. Pure Appl. Math, 18 (3) (2025), 6274 15 of 33 E1 T ≲ sup ϵ>0 sup L⩽0,L∈Z 2−Lλ ϵθ L∑ k=−∞ 2η(0)kq(1+ϵ)  k−2∑ i=−∞ ∥∥∥∥∥∥∥  ∞∑ j=1 |gj |r  1 r 1i ∥∥∥∥∥∥∥ p(·) 2 (i−k)n p′(0)  q(1+ϵ)  1 q(1+ϵ) = sup ϵ>0 sup L⩽0,L∈Z 2−Lλ ϵθ L∑ k=−∞  k−2∑ i=−∞ 2η(0)k ∥∥∥∥∥∥∥  ∞∑ j=1 |gj |r  1 r 1i ∥∥∥∥∥∥∥ p(·) 2 (i−k)n p′(0)  q(1+ϵ)  1 q(1+ϵ) = sup ϵ>0 sup L⩽0,L∈Z 2−Lλ ϵθ L∑ k=−∞  k−2∑ i=−∞ 2η(0)i ∥∥∥∥∥∥∥  ∞∑ j=1 |gj |r  1 r 1i ∥∥∥∥∥∥∥ p(·) 2b(i−k)  q(1+ϵ)  1 q(1+ϵ) , (3.3) here b := n p′(0) − η(0) > 0. Let 1 < q(1 + ϵ) < ∞, then the Hölder’s inequality yields E1 T ≲ sup ϵ>0 sup L⩽0,L∈Z 2−Lλ ϵθ L∑ k=−∞  k−2∑ i=−∞ 2η(0)iq(1+ϵ) ∥∥∥∥∥∥∥  ∞∑ j=1 |gj |r  1 r 1i ∥∥∥∥∥∥∥ q(1+ϵ) p(·) 2 bq(1+ϵ)(i−k) 2  × ( k−2∑ i=−∞ 2 b(q(1+ϵ))′(i−k) 2 ) q(1+ϵ) (q(1+ϵ))′  1 q(1+ϵ) ≲ sup ϵ>0 sup L⩽0,L∈Z 2−Lλ ϵθ L∑ k=−∞ k−2∑ i=−∞ 2η(0)iq(1+ϵ) ∥∥∥∥∥∥∥  ∞∑ j=1 |gj |r  1 r 1i ∥∥∥∥∥∥∥ q(1+ϵ) p(·) 2 bq(1+ϵ)(i−k) 2  1 q = sup ϵ>0 sup L⩽0,L∈Z 2−Lλ ϵθ L−2∑ i=−∞ 2η(0)iq(1+ϵ) ∥∥∥∥∥∥∥  ∞∑ j=1 |gj |r  1 r 1i ∥∥∥∥∥∥∥ q(1+ϵ) p(·) L∑ k=i+2 2 bq(1+ϵ)(i−k) 2  1 q(1+ϵ) ≲ sup ϵ>0 sup L⩽0,L∈Z 2−Lλ ϵθ L−2∑ i=−∞ 2η(0)iq(1+ϵ) ∥∥∥∥∥∥∥  ∞∑ j=1 |gj |r  1 r 1i ∥∥∥∥∥∥∥ q(1+ϵ) p(·)  1 q(1+ϵ) ≲ Ef . M. Sultan, B. Sultan, I-L. Popa / Eur. J. Pure Appl. Math, 18 (3) (2025), 6274 16 of 33 If 0 < q(1 + ϵ) ⩽ 1, then we have( ∞∑ i=1 ai )q(1+ϵ) ⩽ ∞∑ i=1 a q(1+ϵ) i , i ∈ N, ai ⩾ 0, (3.4) and obtain E1 T ≲ sup ϵ>0 sup L⩽0,L∈Z 2−Lλ ϵθ L∑ k=−∞ k−2∑ i=−∞ 2η(0)q(1+ϵ)i ∥∥∥∥∥∥∥  ∞∑ j=1 |gj |r  1 r 1i ∥∥∥∥∥∥∥ q(1+ϵ) p(·) 2bq(1+ϵ)(i−k)  1 q(1+ϵ) = sup ϵ>0 sup L⩽0,L∈Z 2−Lλ ϵθ L−2∑ i=−∞ 2η(0)iq(1+ϵ) ∥∥∥∥∥∥∥  ∞∑ j=1 |gj |r  1 r 1i ∥∥∥∥∥∥∥ q(1+ϵ) p(·) L∑ k=i+2 2bq(1+ϵ)(i−k)  1 q(1+ϵ) ≲ sup ϵ>0 sup L⩽0,L∈Z 2−Lλ ϵθ L−2∑ i=−∞ 2η(0)iq(1+ϵ) ∥∥∥∥∥∥∥  ∞∑ j=1 |gj |r  1 r 1i ∥∥∥∥∥∥∥ q(1+ϵ) p(·)  1 q(1+ϵ) ≲ Ef . Similarly, by using Lemmas 2.5 and 3.2, we have 2−kn ∥1k∥p(·) ∥1i∥Lp′(·) ≲ 2−kn ∥1Bi∥Lp′(·) |Bk| ∥1Bk ∥−1 Lp′(·) ≲ 2nω2(i−k). H1 T ≲ 2−Lλ ϵθ ∞∑ k=0  k−2∑ i=−∞ 2η∞i ∥∥∥∥∥∥∥  ∞∑ j=1 |gj |r  1 r 1i ∥∥∥∥∥∥∥ p(·) 2b1(i−k)  q(1+ϵ)  1 q(1+ϵ) here b1 = nω2 − η∞ > 0. For 1 < q(1 + ϵ) < ∞, using Hölder’s inequality, we have M. Sultan, B. Sultan, I-L. Popa / Eur. J. Pure Appl. Math, 18 (3) (2025), 6274 17 of 33 H1 T ≲ 2−Lλ ϵθ L∑ k=0 k−2∑ i=−∞ 2η∞iq(1+ϵ) ∥∥∥∥∥∥∥  ∞∑ j=1 |gj |r  1 r 1i ∥∥∥∥∥∥∥ q(1+ϵ) p(·) 2 bq(i−k) 2 ( k−2∑ i=−∞ 2 b1(q(1+ϵ))′(i−k) 2 ) q(1+ϵ) q(1+ϵ)′  1 q(1+ϵ) ≲ 2−Lλ ϵθ L∑ k=0 k−2∑ i=−∞ 2η∞iq(1+ϵ) ∥∥∥∥∥∥∥  ∞∑ j=1 |gj |r  1 r 1i ∥∥∥∥∥∥∥ q(1+ϵ) p(·) 2 b1q(1+ϵ)(i−k) 2  1 q(1+ϵ) ⩽ 2−Lλ ϵθ L∑ k=0 k−2∑ i=−2 2η∞iq(1+ϵ) ∥∥∥∥∥∥∥  ∞∑ j=1 |gj |r  1 r 1i ∥∥∥∥∥∥∥ q p(·) 2 b1q(1+ϵ)(i−k) 2  1 q(1+ϵ) + 2−Lλ ϵθ L∑ k=0 −3∑ i=−∞ 2η∞iq(1+ϵ) ∥∥∥∥∥∥∥  ∞∑ j=1 |gj |r  1 r 1i ∥∥∥∥∥∥∥ q(1+ϵ) p(·) 2 b1q(1+ϵ)(i−k) 2  1 q(1+ϵ) =I1 + I2. Now we consider I1 and I2 respectively. Due to b1 > 0, we have I1 = 2−Lλ ϵθ L−2∑ i=−2 2η∞iq(1+ϵ) ∥∥∥∥∥∥∥  ∞∑ j=1 |gj |r  1 r 1i ∥∥∥∥∥∥∥ q(1+ϵ) p(·) L∑ k=i+2 2 b1q(1+ϵ)(i−k) 2  1 q(1+ϵ) ≲ 2−Lλ ϵθ L−2∑ i=−2 2η∞iq(1+ϵ) ∥∥∥∥∥∥∥  ∞∑ j=1 |gj |r  1 r 1i ∥∥∥∥∥∥∥ q(1+ϵ) p(·)  1 q(1+ϵ) ≲ Hf . Because L > 0, b1 > 0 and λ ⩾ 0, we obtain I2 ≲ 2−Lλ ϵθ L∑ k=0 −3∑ i=−∞ 2 b1q(1+ϵ)(i−k) 2  i∑ m=∞ 2η∞mq(1+ϵ) ∥∥∥∥∥∥∥  ∞∑ j=1 |gj |r  1 r 1m ∥∥∥∥∥∥∥ q(1+ϵ) p(·)   1 q(1+ϵ) = 2−Lλ { ϵθ L∑ k=0 −3∑ i=−∞ 2 b1q(1+ϵ)(i−k) 2 · 2iq(1+ϵ)λH q(1+ϵ) f } 1 q(1+ϵ) M. Sultan, B. Sultan, I-L. Popa / Eur. J. Pure Appl. Math, 18 (3) (2025), 6274 18 of 33 = 2−Lλ { ϵθ ( L∑ k=0 2−kb1q(1+ϵ)/2 )( −3∑ i=−∞ 2(b1/2+λ)q(1+ϵ)i ) H q(1+ϵ) f } 1 q(1+ϵ) ≲ 2−Lλ { ϵθ ( L∑ k=0 2−kb1q(1+ϵ)/2 )( L∑ i=−∞ 2(b1/2+λ)q(1+ϵ)i ) H q(1+ϵ) f } 1 q(1+ϵ) ≲ 2−Lλ2−Lb1/22(b1/2+λ)LHf = Hf . For 0 < q(1 + ϵ) ⩽ 1, using (3.4) H1 T ≲ 2−Lλ ϵθ ∞∑ k=0 k−2∑ i=−∞ 2η∞iq(1+ϵ) ∥∥∥∥∥∥∥  ∞∑ j=1 |gj |r  1 r 1i ∥∥∥∥∥∥∥ q(1+ϵ) p(·) 2b1(i−k)  1 q(1+ϵ) ≲ 2−Lλ ϵθ L∑ k=0 k−2∑ i=−2 2η∞iq(1+ϵ) ∥∥∥∥∥∥∥  ∞∑ j=1 |gj |r  1 r 1i ∥∥∥∥∥∥∥ q(1+ϵ) p(·) 2b1(i−k)  1 q(1+ϵ) + 2−Lλ ϵθ L∑ k=0 −3∑ i=−∞ 2η∞iq(1+ϵ) ∥∥∥∥∥∥∥  ∞∑ j=1 |gj |r  1 r 1i ∥∥∥∥∥∥∥ q(1+ϵ) p(·) 2b1(i−k)  1 q(1+ϵ) = : J1 + J2. Similar to Step 2 H2 T ≲ Hf is also true for 0 < q(1 + ϵ) ⩽ 1. Step 3. Let ∀i ≥ k + 2, x ∈ Rk, then we get  ∞∑ j=1 T r ( gij ) (x)  1 r ≲  ∞∑ j=1 ( 2−in ∫ Rn ∣∣gij(y)∣∣dy)r  1 r = 2−in  ∞∑ j=1 (∫ Rn ∣∣gij(y)∣∣ dy)r  1 r ≲ 2−in ∫ Rn  ∞∑ j=1 ∣∣gij∣∣r  1 r dy. By Hölder’s inequality we get M. Sultan, B. Sultan, I-L. Popa / Eur. J. Pure Appl. Math, 18 (3) (2025), 6274 19 of 33 E3 T ≲ sup ϵ>0 sup L⩽0,L∈Z 2−Lλ ϵθ L∑ k=−∞ 2η(0)q(1+ϵ)k ∥∥∥∥∥∥∥1k ∞∑ i=k+2 2−in ∫ Rn  ∞∑ j=1 ∣∣gij∣∣r  1 r dy ∥∥∥∥∥∥∥ q(1+ϵ) Lp(·)  1 q(1+ϵ) =sup ϵ>0 sup L⩽0,L∈Z 2−Lλ ϵθ L∑ k=−∞ 2η(0)q(1+ϵ)k ∥1k∥ q(1+ϵ) p(·)  ∞∑ i=k+2 2−in ∫ Rn  ∞∑ j=1 ∣∣gij∣∣r  1 r dy  q(1+ϵ)  1 q(1+ϵ) ≲ sup ϵ>0 sup L⩽0,L∈Z 2−Lλ ϵθ L∑ k=−∞ 2η(0)q(1+ϵ)k ∥1k∥ q(1+ϵ) p(·)  ∞∑ i=k+2 2−in ∥∥∥∥∥∥∥  ∞∑ j=1 |gj |r  1 r 1i ∥∥∥∥∥∥∥ p(·) ∥1i∥Lp(·)  q(1+ϵ)  1 q(1+ϵ) . (3.5) Using Lemmas 2.5 and 3.2 again, we obtain 2−in ∥1k∥p(·) ∥1i∥Lp′(·) ⩽ 2−in ∥1Bk ∥p(·) ∥1Bi∥Lp′(·) ≲ 2−in ∥1Bk ∥p(·) |Bi| ∥1Bi∥ −1 p(·) ≲ 2nω1(k−i). (3.6) We put (3.6) into (3.5) and get E3 T ≲ sup ϵ>0 sup L⩽0,L∈Z 2−Lλ ϵθ L∑ k=−∞ 2η(0)kq(1+ϵ)  ∞∑ i=k+2 ∥∥∥∥∥∥∥  ∞∑ j=1 |gj |r  1 r 1i ∥∥∥∥∥∥∥ p(·) 2nω1(k−i)  q(1+ϵ)  1 q(1+ϵ) = sup ϵ>0 sup L⩽0,L∈Z 2−Lλ ϵθ L∑ k=−∞  ∞∑ i=k+2 2η(0)k ∥∥∥∥∥∥∥  ∞∑ j=1 |gj |r  1 r 1i ∥∥∥∥∥∥∥ p(·) 2nω1(k−i)  q(1+ϵ)  1 q(1+ϵ) = sup ϵ>0 sup L⩽0,L∈Z 2−Lλ ϵθ L∑ k=−∞  ∞∑ i=k+2 2η(0)i ∥∥∥∥∥∥∥  ∞∑ j=1 |gj |r  1 r 1i ∥∥∥∥∥∥∥ p(·) 2d(k−i)  q(1+ϵ)  1 q(1+ϵ) . (3.7) where d := nω1 + η(0) > 0. Let 1 < q(1 + ϵ) < ∞, then we get M. Sultan, B. Sultan, I-L. Popa / Eur. J. Pure Appl. Math, 18 (3) (2025), 6274 20 of 33 E3 T ≲ sup ϵ>0 sup L⩽0,L∈Z 2−Lλ ϵθ L∑ k=−∞  ∞∑ i=k+2 2η(0)iq(1+ϵ) ∥∥∥∥∥∥∥  ∞∑ j=1 |gj |r  1 r 1i ∥∥∥∥∥∥∥ q(1+ϵ) p(·) 2 dq(1+ϵ)(k−i) 2  ( ∞∑ i=k+2 2 d(q(1+ϵ))′(k−i) 2 ) q(1+ϵ) (q(1+ϵ))′  1 q(1+ϵ) ≲ sup ϵ>0 sup L⩽0,L∈Z 2−Lλ ϵθ L∑ k=−∞ ∞∑ i=k+2 2η(0)iq(1+ϵ) ∥∥∥∥∥∥∥  ∞∑ j=1 |gj |r  1 r 1i ∥∥∥∥∥∥∥ q(1+ϵ) p(·) 2 dq(1+ϵ)(k−i) 2  1 q(1+ϵ) =sup ϵ>0 sup L⩽0,L∈Z 2−Lλ ϵθ L∑ k=−∞ L+2∑ i=k+2 2η(0)iq(1+ϵ) ∥∥∥∥∥∥∥  ∞∑ j=1 |gj |r  1 r 1i ∥∥∥∥∥∥∥ q(1+ϵ) p(·) 2 dq(1+ϵ)(k−i) 2  1 q(1+ϵ) + sup ϵ>0 sup L⩽0,L∈Z 2−Lλ ϵθ L∑ k=−∞ ∞∑ i=L+3 2η(0)iq(1+ϵ) ∥∥∥∥∥∥∥  ∞∑ j=1 |gj |r  1 r 1i ∥∥∥∥∥∥∥ q(1+ϵ) p(·) 2 dq(1+ϵ)(k−i) 2  1 q(1+ϵ) =:I3 + I4. Now we consider I3 and I4 respectively. For d > 0, we get I3 = sup ϵ>0 sup L⩽0,L∈Z 2−Lλ ϵθ L+2∑ i=−∞ 2η(0)iq(1+ϵ) ∥∥∥∥∥∥∥  ∞∑ j=1 |gj |r  1 r 1i ∥∥∥∥∥∥∥ q(1+ϵ) p(·) i−2∑ k=−∞ 2 dq(1+ϵ)(k−i) 2  1 q(1+ϵ) ≲ sup ϵ>0 sup L⩽0,L∈Z 2−Lλ ϵθ L+2∑ i=−∞ 2η(0)iq(1+ϵ) ∥∥∥∥∥∥∥  ∞∑ j=1 |gj |r  1 r 1i ∥∥∥∥∥∥∥ q(1+ϵ) p(·)  1 q(1+ϵ) ≲ Ef . If d > 0 and λ− d/2 < 0, then we get I4 ≲ sup ϵ>0 sup L⩽0,L∈Z 2−Lλ ϵθ L∑ k=−∞ ∞∑ i=L+3 2 dq(1+ϵ)(k−i) 2  i∑ m=−∞ 2η(0)mq(1+ϵ) ∥∥∥∥∥∥∥  ∞∑ j=1 |gj |r  1 r 1m ∥∥∥∥∥∥∥ q(1+ϵ) p(·)   1 q(1+ϵ) ≲ sup ϵ>0 sup L⩽0,L∈Z 2−Lλ { ϵθ L∑ k=−∞ ∞∑ i=L+3 2 dq(1+ϵ)(k−i) 2 · 2iq(1+ϵ)λE q(1+ϵ) f } 1 q(1+ϵ) M. Sultan, B. Sultan, I-L. Popa / Eur. J. Pure Appl. Math, 18 (3) (2025), 6274 21 of 33 = sup ϵ>0 sup L⩽0,L∈Z 2−Lλ { ϵθ ( L∑ k=−∞ 2dq(1+ϵ)k/2 )( ∞∑ i=L+3 2(λ−d/2)q(1+ϵ)i ) E q(1+ϵ) f } 1 q(1+ϵ) ≲ sup ϵ>0 sup L⩽0,L∈Z { 2−Lq(1+ϵ)λ2dq(1+ϵ)L/22(λ−d/2)q(1+ϵ)LE q(1+ϵ) f } 1 q(1+ϵ) = Ef . Hence we get E3 T ≲ Ef . For 0 < q(1 + ϵ) ⩽ 1, then using (3.4) in (3.7) we get E3 T ≲ sup ϵ>0 sup L⩽0,L∈Z 2−Lλ  L∑ k=−∞ ∞∑ i=k+2 2η(0)iq(1+ϵ) ∥∥∥∥∥∥∥  ∞∑ j=1 |gj |r  1 r 1i ∥∥∥∥∥∥∥ q(1+ϵ) p(·) 2dq(k−i)  1 q(1+ϵ) ≲ sup ϵ>0 sup L⩽0,L∈Z 2−Lλ  L∑ k=−∞ L+2∑ i=k+2 2η(0)iq(1+ϵ) ∥∥∥∥∥∥∥  ∞∑ j=1 |gj |r  1 r 1i ∥∥∥∥∥∥∥ q(1+ϵ) p(·) 2dq(1+ϵ)(k−i)  1 q(1+ϵ) + sup ϵ>0 sup L⩽0,L∈Z 2−Lλ  L∑ k=−∞ ∞∑ i=L+3 2η(0)iq(1+ϵ) ∥∥∥∥∥∥∥  ∞∑ j=1 |gj |r  1 r 1i ∥∥∥∥∥∥∥ q(1+ϵ) p(·) 2dq(1+ϵ)(k−i)  1 q(1+ϵ) = : J3 + J4. Similarly we conclude that E3 T ≲ Ef holds for 0 < q(1 + ϵ) ⩽ 1. Then we consider H3 T . Similarly, we have H3 T ≲ 2−Lλ ϵθ L∑ k=0  ∞∑ i=k+2 2η∞i ∥∥∥∥∥∥∥  ∞∑ j=1 |gj |r  1 r 1i ∥∥∥∥∥∥∥ p(·) 2d1(k−i)  q(1+ϵ)  1 q(1+ϵ) (3.8) here d1 = nω1 + η∞ > 0. If 1 < q(1 + ϵ) < ∞, then Hölder’s inequality yields H3 T ≲2−Lλ ϵθ L∑ k=0  ∞∑ i=k+2 2η∞q(1+ϵ)i ∥∥∥∥∥∥∥  ∞∑ j=1 |gj |r  1 r 1i ∥∥∥∥∥∥∥ q(1+ϵ) p(·) 2 d1q(1+ϵ)(k−i) 2  M. Sultan, B. Sultan, I-L. Popa / Eur. J. Pure Appl. Math, 18 (3) (2025), 6274 22 of 33 × ( ∞∑ i=k+2 2 d1(q(1+ϵ))′(k−i) 2 ) q(1+ϵ) (q(1+ϵ))′  1 q(1+ϵ) ≲2−Lλ ϵθ 1 q∑ k=0 ∞∑ i=k+2 2η∞q(1+ϵ)i ∥∥∥∥∥∥∥  ∞∑ j=1 |gj |r  1 r 1i ∥∥∥∥∥∥∥ q(1+ϵ) p(·) 2 d1q(1+ϵ)(k−i) 2  1 q(1+ϵ) ⩽2−Lλ ϵθ L∑ k=0 L+2∑ i=k+2 2η∞q(1+ϵ)i ∥∥∥∥∥∥∥  ∞∑ j=1 |gj |r  1 r 1i ∥∥∥∥∥∥∥ q(1+ϵ) p(·) 2 d1q(1+ϵ)(k−i) 2  1 q(1+ϵ) + 2−Lλ ϵθ L∑ k=0 ∞∑ i=L+3 2η∞q(1+ϵ)i ∥∥∥∥∥∥∥  ∞∑ j=1 |gj |r  1 r 1i ∥∥∥∥∥∥∥ q(1+ϵ) p(·) 2 d1q(1+ϵ)(k−i) 2  1 q(1+ϵ) = : I5 + I6. Because d1 > 0, we have I5 = 2−Lλ ϵθ L+2∑ i=2 2η∞q(1+ϵ)i ∥∥∥∥∥∥∥  ∞∑ j=1 |gj |r  1 r 1i ∥∥∥∥∥∥∥ q(1+ϵ) p(·) i−2∑ k=0 2 d1q(k−i) 2  1 q(1+ϵ) ≲ 2−Lλ ϵθ L+2∑ i=2 2η∞q(1+ϵ)i ∥∥∥∥∥∥∥  ∞∑ j=1 |gj |r  1 r 1i ∥∥∥∥∥∥∥ q(1+ϵ) p(·)  1 q(1+ϵ) ≲ Hf . Since d1 > 0 and λ− d1/2 < 0, we obtain I6 = 2−Lλ ϵθ L∑ k=0 ∞∑ i=L+3 2η∞q(1+ϵ)i ∥∥∥∥∥∥∥  ∞∑ j=1 |gj |r  1 r 1i ∥∥∥∥∥∥∥ q(1+ϵ) p(·) 2 d1q(1+ϵ)(k−i) 2  1 q(1+ϵ) ⩽ 2−Lλ { ϵθ ( L∑ k=0 2 d1q(1+ϵ)k 2 )( ∞∑ i=L+3 2(λ−d1/2)q(1+ϵ)i ) M. Sultan, B. Sultan, I-L. Popa / Eur. J. Pure Appl. Math, 18 (3) (2025), 6274 23 of 33 ×  i∑ m=0 2η∞q(1+ϵ)m ∥∥∥∥∥∥∥2−iλ  ∞∑ j=1 |gj |r  1 r 1m ∥∥∥∥∥∥∥ q(1+ϵ) p(·)   1 q(1+ϵ) ≲ 2−Lλ2d1/2L2(λ−d1/2)LHf = Hf . For 0 < q(1 + ϵ) ⩽ 1, we use (3.4) in (3.8) and have H3 T ≲ 2−Lλ ϵθ L∑ k=0 ∞∑ i=k+2 2η∞q(1+ϵ)i ∥∥∥∥∥∥∥  ∞∑ j=1 |gj |r  1 r 1i ∥∥∥∥∥∥∥ q(1+ϵ) p(·) 2d1q(1+ϵ)(k−i)  1 q(1+ϵ) ⩽ 2−Lλ ϵθ L∑ k=0 L+2∑ i=k+2 2η∞q(1+ϵ)i ∥∥∥∥∥∥∥  ∞∑ j=1 |gj |r  1 r 1i ∥∥∥∥∥∥∥ q(1+ϵ) p(·) 2d1q(1+ϵ)(k−i)  1 q(1+ϵ) + 2−Lλ ϵθ L∑ k=0 ∞∑ i=L+3 2η∞q(1+ϵ)i ∥∥∥∥∥∥∥  ∞∑ j=1 |gj |r  1 r 1i ∥∥∥∥∥∥∥ q(1+ϵ) p(·) 2d1q(1+ϵ)(k−i)  1 q(1+ϵ) = : IJ + J6. Similarly we can get H3 T ≲ Hf where 0 < q(1 + ϵ) ⩽ 1. Hence we completes our proof. Now we turn to prove Theorem 2.13. Because the proofs of B-parts and F -parts are similar, we only prove F -parts below. Our proof will use the idea that comes from [59]. To continue, we recall some lemmas. Lemma 14. [60] Let µ, ν ∈ S (Rn) ,−1 ⩽ M ∈ Z, Dτ µ̂(0) = 0 for all |τ | ⩽ M. Then for any N > 0 there is a constant CN such that sup z∈Rn |µt ∗ ν(z)| (1 + |z|)N ⩽ CN tM+1, where µt(x) = t−nµ ( x t ) for all 0 < t ⩽ 2. M. Sultan, B. Sultan, I-L. Popa / Eur. J. Pure Appl. Math, 18 (3) (2025), 6274 24 of 33 Lemma 15. [60] If ω > 0 and q ∈ (0,∞]. Then for a sequence {gj}∞0 , we have Gj = ∞∑ ℓ=0 2−|ℓ−j|ωgℓ. Then ∥∥{Gj}∞0 ∥∥ ℓq ⩽ C ∥∥{gj}∞0 ∥∥ℓq . (3.9) Lemma 16. If κ, q ∈ (0,∞], ω > 0, s ∈ R, and η, q, p, are same as given in Theorem 2.8. For a sequence {gj}∞0 , we have Gj(x) = ∞∑ k=0 2−|k−j|ωgk(x), x ∈ Rn. Then there are some constants C1 = C1(q, ω) and C2 = C2(p(·), q, ω) such that∥∥∥{Gj}∞j=0 ∥∥∥ MK̇ η(·),q),θ λ,p(·) (ℓκ) ⩽ C1 ∥∥∥{gj}∞j=0 ∥∥∥ MK̇ η(·),q),θ λ,p(·) (ℓκ) (3.10) and ∥∥∥{Gj}∞j=0 ∥∥∥ ℓκ ( MK̇ η(·),q),θ λ,p(·) ) ⩽ C2 ∥∥∥{gj}∞j=0 ∥∥∥ ℓκ ( MK̇ η(·),q),θ λ,p(·) ) . (3.11) Proof. Firstly, (3.10) follows immediately from Lemma 3.3. Next we prove (3.11) for p(·) ∈ P0 (Rn) and we separate it into two cases. Case 1. p− ⩾ 1, q ⩾ 1. Because ∥ · ∥ MK̇ η(·),q),θ λ,p(·) is a norm, we have ∥Gj∥MK̇ η(·),q),θ λ,p(·) ⩽ ∞∑ k=0 2−|k−j|ω ∥gk∥MK̇ η(·),q),θ λ,p(·) . Using Lemma 3.4, we get (3.11). Case 2. If q < 1, let p0 < min (p−, q) then we get ∥Gj∥p0 MK̇ η(·),q),θ λ,p(·) = ∥|Gj |p0∥MK̇ p0η(·), q/p0),p0θ p0λ,p(·)/p0 ⩽ ∥∥∥∥∥ ∞∑ k=0 2−|k−j|p0ω |gk|p0 ∥∥∥∥∥ MK̇ p0η(·), q/p0),p0θ p0λ,p(·)/p0 ⩽ ∞∑ k=0 2−|k−j|p0ω ∥|gk|p0∥MK̇ p0η(·), q/p0),p0θ p0λ,p(·)/p0 . Consequently we get M. Sultan, B. Sultan, I-L. Popa / Eur. J. Pure Appl. Math, 18 (3) (2025), 6274 25 of 33 ∥{Gj}∥p0 ℓκ ( MK̇ η(·),q),θ λ,p(·) ) = ∥{|Gj |p0}∥ℓκ/p0 ( MK̇ p0η(·), q/p0),p0θ p0λ,p(·)/p0 ) ≲ ∥{|gk|p0}∥ℓκ/p0 ( MK̇ p0η(·), q/p0),p0θ p0λ,p(·)/p0 ) = ∥{gk}∥p0 ℓκ ( MK̇ η(·),q),θ λ,p(·) ) . By using the power 1/p0, we get (3.11). Lemma 17 ([61], Theorem 6 ). Let {φj}j∈N0 is the resolution of unity, R ∈ N. Then there exists functions θ0, θ ∈ S (Rn) which satisfy supp θ, supp θ0 ⊆ {y ∈ Rn : |y| ⩽ 1} ,∣∣∣θ̂0(ϱ)∣∣∣ > 0 on {|ϱ| < 2ε}, |θ̂(ϱ)| > 0 on {ε 2 < |ϱ| < 2ε } ,∫ Rn yγθ(x)dy = 0, ∀γ, 0 < |γ| ⩽ R, such that θ̂0(ϱ)Ψ̂0(ϱ) + ∞∑ j=1 θ̂ ( p−jϱ ) Ψ̂ ( p−jϱ ) = 1, ∀ϱ ∈ Rn, and Ψ0, Ψ ∈ S (Rn) are given as Ψ̂0(ϱ) = φ0(ϱ) θ̂0(ϱ) , Ψ̂(ϱ) = φ1(2ϱ) θ̂(ϱ) . Proof. Now we will give the proof of Theorem 2.13. Step 1. Let g ∈ S ′ (Rn), then ∥g∥(2) MK̇ η(·),q),θ λ,p(·) F s κ ≲ ∥g∥(1) MK̇ η(·),q),θ λ,p(·) F s κ ≲ ∥g∥(2) MK̇ η(·),q),θ λ,p(·) F s κ . Using Lemmas 3.2 and 3.5, and the fact r < min {p−, κ} for N ∈ N. Then for g ∈ S (Rn), then (∫ 2 1 ∣∣∣2ls (Φ∗ 2−ltg ) a (x) ∣∣∣κ dt t )r/κ ≲ ∑ k∈l+N0 2(l−k)(Nr−n+rs)2krs ×M [(∫ 2 1 |((Φk)t ∗ g) (·)| κdt t )r/κ ] (x). M. Sultan, B. Sultan, I-L. Popa / Eur. J. Pure Appl. Math, 18 (3) (2025), 6274 26 of 33 If l ∈ N, p0 = r ∈ (n/a,< min {p−, κ}) , N > max{0,−s}+a and ω := N+s−d/r > 0, then we get (∫ 2 1 ∣∣∣2ls (Φ∗ 2−lg ) a (x) ∣∣∣κ dt t )r/κ ≲ ∑ k∈l+N0 2−ωr|l−k|2krsM [(∫ 2 1 |((Φk)t ∗ g) (·)| κ dt t )r/κ ] (x). Using Lemma 3.5 in MK̇ rη(·), q/r),rθ rλ,p(·)/r ( ℓκ/r ) , we obtain ∥∥∥∥∥ {(∫ 2 1 ∣∣∣2ls (Φ∗ 2−ltg ) a (x) ∣∣∣κ dt t )r/κ } l∈N ∥∥∥∥∥ MK̇ rη(·), q/r),rθ rλ,p(·)/r (ℓκ/r) ≲ ∥∥∥∥∥ { M [(∫ 2 1 ∣∣∣2ks ((Φl)t ∗ g) (·) ∣∣∣κ dt t )r/κ ]} l∈N ∥∥∥∥∥ MK̇ rη(·), q/r),rθ rλ,p(·)/r (ℓκ/r) Theorem 2.8 yields ∥∥∥∥∥ {(∫ 2 1 ∣∣∣2ls (Φ∗ 2−lg ) a (x) ∣∣∣κ dt t )r/κ } l∈N ∥∥∥∥∥ MK̇ rη(·), q/r),rθ rλ,p(·)/r (ℓκ/r) ≲ ∥∥∥∥∥ {(∫ 2 1 ∣∣∣2ks ((Φl)t ∗ g) (·) ∣∣∣κ dt t )r/κ } l∈N ∥∥∥∥∥ MK̇ rη(·), q/r),rθ rλ,p(·)/r (ℓκ/r) = ∥∥∥∥∥ {(∫ 2 1 ∣∣∣2ks ((Φl)t ∗ g) (·) ∣∣∣κ dt t )1/κ } l∈N ∥∥∥∥∥ r MK̇ η(·),q),θ λ,p(·) (ℓκ) . Hence, we have ∥∥∥∥∥ (∫ 1 0 ∣∣λ−s (Φ∗ λg)a (·) ∣∣κ dλ λ )1/κ ∥∥∥∥∥ MK̇ η(·),q),θ λ,p(·) ≈ ∥∥∥∥∥∥ ( ∞∑ l=1 ∫ 2 1 ∣∣∣2ls (Φ∗ 2−ltg ) a (·) ∣∣∣κ dt t )1/κ ∥∥∥∥∥∥ MK̇ η(·),q),θ λ,p(·) ≲ ∥∥∥∥∥ {(∫ 2 1 ∣∣∣2lsΦ2−lt ∗ g(·) ∣∣∣κ dt t )1/κ } l∈N ∥∥∥∥∥ MK̇ η(·),q),θ λ,p(·) (ℓκ) M. Sultan, B. Sultan, I-L. Popa / Eur. J. Pure Appl. Math, 18 (3) (2025), 6274 27 of 33 ≈ ∥∥∥∥∥ (∫ 1 0 ∣∣λ−sΦλ ∗ g(·) ∣∣κ dλ λ )1/κ ∥∥∥∥∥ MK̇ η(·),q),θ λ,p(·) . This proves ∥g∥(2) MK̇ η(·),q),θ λ,p(·) F s κ ≲ ∥g∥(1) MK̇ η(·),q),θ λ,p(·) F s κ . Step 2. Suppose that Ψ0,Ψ ∈ S ′ (Rn) and g ∈ S ′ (Rn). ∥g∥(4) MK̇ η(·),q),θ λ,p(·) F s κ(Rn,Ψ) ≲ ∥g∥(2) MK̇ η(·),q),θ λ,p(·) F s κ(Rn,Φ) . (3.11) By applying Lemmas 3.5 and 3.2, and the fact ω = min{1, S + 1− s},then for g ∈ S, we get 2ls (Ψ∗ l g)a (x) ⩽ C ∑ k∈N0 2−|k−l|ω2ks ( Φ∗ 2−ktg ) a (x), x ∈ Rn and t ∈ [1, 2]. (3.12) If κ ⩾ 1. By using the (∫ 2 1 | · |κdt/t )1/κ , we get 2ls (Ψ∗ l g)a (x) ≲ ∑ k∈N0 2−|k−l|ω2ks (∫ 2 1 ∣∣(Φ∗ 2−ktg ) a (x) ∣∣κ dt t )1/κ . Applying Lemma 3.5, we obtain ∥∥∥{2ls (Ψ∗ l g)a } l∈N ∥∥∥ MK̇ η(·),q),θ λ,p(·) (ℓκ) ≲ ∥∥∥∥∥∥ ( ∞∑ k=1 2ksκ ∫ 2 1 ∣∣(Φ∗ 2−ktg ) a (x) ∣∣κ dt t )1/κ ∥∥∥∥∥∥ MK̇ η(·),q),θ λ,p(·) . Hence we obtain the required result. If κ < 1, then (∫ 2 1 | · |κdt/t )1/κ is not the norm. Thus we get( 2ls (Ψ∗ l g)a (x) )κ ≲ ∑ k∈N0 2−κ|k−l|ω2ksκ ∫ 2 1 ∣∣(Φ∗ 2−ktg ) a (x) ∣∣κ dt t . Convolution (γ ∗ η)ℓ of the sequences yields γk = 2−|k|ωκ and τk = 2ksκ ∫ 2 1 ∣∣(Φ∗ 2−ktg ) a (x) ∣∣κ dt t For x ∈ Rn, using ℓ1-norm gives as∥∥∥2ls (Ψ∗ l g)a (x) ∥∥∥κ ℓκ ⩽ ∥γ∥ℓ1 · ∥τ∥ℓ1 ≲ ∞∑ k=1 2ksκ ∫ 2 1 ∣∣(Φ∗ 2−ktg ) a (x) ∣∣κ dt t . M. Sultan, B. Sultan, I-L. Popa / Eur. J. Pure Appl. Math, 18 (3) (2025), 6274 28 of 33 By taking (· · · )1/κ and using K̇ η(·),q),θ p(·) -norm. We obtain desired result (3.11). Similarly, for any g ∈ S ′ (Rn), we obtain ∥g∥(2) MK̇ η(·),q),θ λ,p(·) F s κ(Rn,Φ) ≲ ∥g∥(4) MK̇ η(·),q),θ λ,p(·) F s κ(Rn,Ψ) . Step 3. Using t = 1 in Step 1, we get ∥g∥(5) MK̇ η(·),q),θ λ,p(·) F s κ ≲ ∥g∥(4) MK̇ η(·),q),θ λ,p(·) F s κ ≲ ∥g∥(5) MK̇ η(·),q),θ λ,p(·) F s κ . Step 4. We show (2.15) is equivalent to the rest. First, we will show that for any g ∈ S ′ (Rn) ∥g∥(2) MK̇ η(·),q),θ λ,p(·) F s κ(Rn) ≲ ∥g∥(3) MK̇ η(·),q),θ λ,p(·) F s κ . (3.13) For 0 < r < min {p−, κ}, see [59], there exists a positive constant C such that for any g ∈ S ′ (Rn), (∫ 2 1 ∣∣(Ψ∗ 2−lg ) a (x) ∣∣κ dt t )r/κ ⩽ C ∑ k∈N0 2−kNs2(k+l)n ∫ Rn (∫ 2 1 ∫ |z|<2−(k+l)t |((Φk+l)t ∗ g) (z + y)|κ dz dt tn+1 )r/κ (1 + 2l|x− y|)ar dy. Let ar > n, we get gl(y) := 2nl (1 + 2l|y|)ar , ∀y ∈ Rn. Hence we get (∫ 2 1 ∣∣∣2ls (Φ∗ 2−llt g ) a (x) ∣∣∣κ dt t )r/κ ≲ ∑ k∈N0 2−kNr2kn2lsr gl ∗ (∫ 2 1 ∫ |z|<2−(k+l)t |((Φk+l)t ∗ g) (z + ·)|κ dz dt tn+1 )r/κ  (x). By applying the majorant property see [62] to obtain (∫ 2 1 ∣∣∣2ls (Φ∗ 2−ltg ) a (x) ∣∣∣κ dt t )r/κ M. Sultan, B. Sultan, I-L. Popa / Eur. J. Pure Appl. Math, 18 (3) (2025), 6274 29 of 33 ≲ ∑ k∈N0 2lsr2k(−Nr+n)M (∫ 2 1 ∫ |z|<2−(k+l)t |((Φk+l)t ∗ g) (z + ·)|κ dz dt tn+1 )r/κ  (x). An index shift on the right-hand side gives (∫ 2 1 ∣∣∣2ls (Φ∗ 2−lg ) a (x) ∣∣∣κ dt t )r/κ ≲ ∑ k∈l+N0 2lsr2(k−l)(−Nr+n)M (∫ 2 1 ∫ |z|<2−kt |((Φk)t ∗ g) (z + ·)|κ dz dt tn+1 )r/κ  (x) = ∑ k∈l+N0 2(l−k)(Nr−n+rs)2krsM (∫ 2 1 ∫ |z|<2−kt |((Φk)t ∗ g) (z + ·)|κ dz dt tn+1 )r/κ  (x). It is simple to note that ∥g∥(3) MK̇ η(·),q),θ λ,p(·) F s κ ≲ ∥g∥(2) MK̇ η(·),q),θ λ,p(·) F s κ , since for any t > 0 1 tn ∫ |z|