EUROPEAN JOURNAL OF PURE AND APPLIED MATHEMATICS 2025, Vol. 18, Issue 4, Article Number 6641 ISSN 1307-5543 – ejpam.com Published by New York Business Global On the Boundedness of the Intrinsic Square Function on Continual Herz Spaces with Variable Exponents Ghada Ali Basendwah1, Mehvish Sultan2, Babar Sultan3,∗, Ioan-Lucian Popa4,5,∗ 1 Department of Mathematics, King Abdulaziz University, Jeddah 21589, Saudi Arabia 2 Department of Mathematics, Capital University of Science and Technology, Islamabad, Pakistan 3 Department of Mathematics, Quaid-I-Azam University, Islamabad 45320, Pakistan 4 Department of Computing, Mathematics and Electronics, “1 Decembrie 1918” University of Alba Iulia, 510009 Alba Iulia, Romania 5 Faculty of Mathematics and Computer Science, Transilvania University of Brasov, Iuliu Maniu Street 50, 500091 Brasov, Romania Abstract. Our aim in this paper is to prove the boundedness of the intrinsic square function on Herz spaces with variable exponents. Firstly we define the Lebesgue spaces with variable exponent, Herz spaces with variable exponents and some basic notations. Then we give some basic lemmas and definition of continual Herz spaces. Finally we obtain the boundedness of the intrinsic square function on continual Herz spaces under some proper assumptions. 2020 Mathematics Subject Classifications: 46E30, 47B38 Key Words and Phrases: Intrinsic square function, continual Herz spaces 1. Introduction Consider an open set H of Rn and a measurable function q(·) : H → [1,∞). Assume that the following condition holds, 1 ≤ q−(H) ≤ q+(H) <∞, (1.1) where i) q− := essinf h∈H q(H) ∗Corresponding author. ∗Corresponding author. DOI: https://doi.org/10.29020/nybg.ejpam.v18i4.6641 Email addresses: gbasendwah@kau.edu.sa (G. A. Basendwah), mehvishsultanbaz@gmail.com (M. Sultan), babarsultan40@yahoo.com (B. Sultan), lucian.popa@uab.ro (I.-L. Popa) https://www.ejpam.com 1 Copyright: © 2025 The Author(s). (CC BY-NC 4.0) G. A. Basendwah et al. / Eur. J. Pure Appl. Math, 18 (4) (2025), 6641 2 of 16 ii) q+ := esssup h∈H q(H). Let q′(h) = q(h) q(h)−1 denotes the conjugate exponent of q(h). We denote by P(Rn) the set of all measurable function satisfying (1.1). For measurable function f , Lebesgue space Lq(·)(H) is given by Lq(·)(f) = ∫ G |f(h|q(h)dh <∞, with the norm defined by, ∥f∥Lp(·)(H) = inf { η > 0 : Lq(·) ( f η ) ≤ 1 } . Note that Lp(·)(H) is the Banach function space. In the last two decades it was evident that classical function spaces are no longer appropriate for studying a number of modern problems arising in many mathematical models of applied sciences. It thus became necessary to introduce and study new function spaces. Such spaces are: variable exponent Lebesgue and Sobolev spaces, grand function spaces, Morrey-type spaces, amalgam spaces, Herz spaces, their hybrid variants, etc (see e.g., the monographs [1], [2], [3], [28] and references therein dedicated to new function spaces). Morrey spaces describe local regularity more precisely than Lebesgue spaces. For more results in Herz spaces see [45–48]. As a result, one can use Morrey spaces widely not only in Harmonic Analysis but also in the theory of PDEs. We refer to the recent monographs [4] for Morrey-type spaces and applications. For more results on variable exponent function spaces see [5–9, 39, 40]. Herz spaces are indeed important in the field of harmonic analysis and partial differ- ential equations, particularly as substitutes for Hardy spaces in certain contexts. This substitution is particularly useful when dealing with non-translation invariant singular in- tegral operators. Herz spaces continue to play a significant role in various mathematical contexts, including the characterization of multipliers on Hardy spaces and in the regu- larity theory for elliptic and parabolic equations in divergence form, see [10, 11]. Herz space has undergone significant advancements, proving to be highly valuable in the field of Harmonic analysis. For more results in variable exponent function spaces see [35–44]. The classical versions of the homogeneous and non-homogeneous Herz spaces are defined as: Izuki introduced the idea of Herz spaces K̇κ p(·),q(R n) and Kκ p(·),q(R n), where the expo- nent p was a variable and norms are defined as, for p ∈ [1,∞), q(·) ∈ P(Rn) and κ ∈ R. The Herz space K̇κ p,q(·)(R n) of homogeneous type is given by K̇κ p,q(·)(R n) = { g ∈ L q(·) loc (R n \ {0}) : ∥g∥K̇κ p,q(·)(R n) <∞ } , (1.2) G. A. Basendwah et al. / Eur. J. Pure Appl. Math, 18 (4) (2025), 6641 3 of 16 where ∥g∥K̇κ p,q(·)(R n) = ( i=∞∑ i=−∞ ∥2iκg1i∥pLq(·) ) 1 p . The non-homogeneous version of Herz spaces Kκ p(·),q(R n) are defined below. Let κ ∈ R, p ∈ [1,∞) and q(·) ∈ P(Rn). The Herz space Kκ p,q(·)(R n) of non- homogeneous type is given by Kκ p,q(·)(R n) = { g ∈ L q(·) loc (R n \ {0}) : ∥g∥Kκ p,q(·)(R n) <∞ } , (1.3) where ∥g∥Kκ p,q(·)(R n) = ∥g∥Lq(·)(O(0,1)) + ( ∞∑ τ=−∞ ∥2iκg1i∥pLq(·) ) 1 p . An important focus within Herz spaces involves investigating the boundedness of sub- linear operators. This exploration has led to studies motivated by this topic, culminating in the establishment of sublinear operator boundedness in Herz spaces with variable ex- ponents, as demonstrated in [12]. In [13], an alternative method was employed to define variable exponent Herz spaces. A notable aspect of this method involves substituting the discrete Lp-norm (Lebesgue- norm) with the continuous Lp-norm relative to Haar measure, streamlining and clarifying the proofs. These spaces are called variable continual Herz spaces. For the boundedness results on some operators on continual Herz spaces see [15–17]. In [18], authors defined the variable Herz-Morrey spaces MK̇κ,λ p,q(·)(R n), and obtained the estimates of sublinear oper- ators within these spaces. It’s worth mentioning that Herz-Morrey spaces with variable exponents serve as extensions of Herz spaces with variable exponents. Additionally, in [19], the boundedness of higher-order commutators of fractional integrals on Herz-Morrey spaces can be verified. Our findings not only consolidate and build on prior discoveries, but also offer novel applications to the regularity solutions of some elliptic PDEs with smooth boundaries. The investigation of the fractional Hardy operator, belongs to one of the hot topics in the area of PDEs because of its wide-ranging interest to various fields in Mathematics and Physics. For instance, it is motivated by physical models related to relativistic Schrödinger operator with Coulomb potential (see [22, 23]) and by the study of Hardy inequalities and Hardy-Lieb-Thirring inequalities (see, e.g., [20–26]). The intrinsic square function Sζ holds significant importance in function spaces. Izuki obtained the boundedness of Sζ on weighted variable Herz spaces under certain appropri- ate conditions, as detailed in [27]. This study investigates the boundedness of the intrinsic square function on continuous Herz spaces with variable exponents. The article is struc- tured into four sections: the first section serves as an introduction, the second section presents fundamental definitions and lemmas, the concept of continuous Herz spaces is defined in part three, and the final section delves into examining the boundedness of the intrinsic square function on variable continuous Herz spaces with variable exponents. For more results see [29–34]. We will use following notations in the paper: G. A. Basendwah et al. / Eur. J. Pure Appl. Math, 18 (4) (2025), 6641 4 of 16 Notations (i) O(y, s) denotes a ball with center at y and radius s ; (ii) T (τ, t) denotes the spherical layer such that (iii) T τ, t) := O(0, t) \ O(0, τ) = {y ∈ Rn : τ < |y| < t}; (iv) Tm := T (2m−1, 2m); (v) 1E(y) is the characteristic function of a set E; (vi) Rµ+ := (µ,∞), where µ ≥ 0; (vii) 1τ,t(y) = 1Tτ,t(y); (viii) dτ/τ represents the Haar measure on R+; (ix) N denotes the set of natural numbers; (x) N0 = N ∪ {0}; (xi) Z denotes the set of all integers; (xii) For two non-nagative functions f and g, f ≤ g we mean f ≤ Cg, (xiii) C denote the positive constant. 2. Preliminaries Next for compact subsets K ⊂ H, the space L q(·) loc (H) is given as L p(·) loc (H) := { K : K ∈ Lq(·)(K) } . Let x, y ∈ H with |x−y| ≤ 1 2 and C(q) is not depending on x, y. We have the log-condition, |q(x)− q(y)| ≤ C(q) − ln |x− y| . (2.1) We say that q(·) satisfies log decay condition at infinity if there exists q∞ ∈ (1,∞), such that |q(x)− q∞| ≤ C ln(e+ |x|) . (2.2) Similarly, we say that q(·) satisfies log decay condition at origin if there exists q0 ∈ (1,∞), such that |q(x)− q0| ≤ C ln |x| , |x| ≤ 1 2 . (2.3) With respect to classes of variable exponents used in this paper, we adopt the following notation: G. A. Basendwah et al. / Eur. J. Pure Appl. Math, 18 (4) (2025), 6641 5 of 16 (i) The set Plog = Plog(H) comprises all functions q ∈ L∞(H) that fulfill both (1.1) and (2.1). (ii) WhenH is unbounded, P∞(H) and P0,∞(H) are subsets of L∞(H). The functions in these sets take values in the interval [1,∞) and satisfy condition (2.2), and (2.2) and (2.3), respectively. (iii) Plog ∞ (H) is the set consists of exponents that satisfy the condition (2.1); (iv) The notation Rµ+ represents the set of all non-negative real numbers. M∞(Rµ+) is the class consists of functions defined on the domain Rµ+ that have certain proper- ties. Specifically, these functions are of the form g(t) = constant+ g0(t), where g0(t) is a function belonging to the class P∞(Rµ+), where H = Rµ+, µ ≥ 0. (v) When H = R+ (or when µ = 0), M0,∞(R+) represents the class of functions defined on the positive real line R+ that belong to the class M∞(R+) and meet a decay criterion at the origin (y = 0). The decay condition implies that these functions are bounded by a logarithmic term as y approaches zero. Specifically, for |y| ≤ 1 2 , the function satisfies the inequality |f(y)−f0| ≤ C ln |y| for some real numbers f0 and C. We also write f0 = f(0), f∞ = f(∞) in this case; (vi) P0,∞(R+) is a subclass of functions within the class M0,∞(R+) that have values in the interval [1,∞). In other words, these are the functions from M0,∞(R+) that satisfy the given conditions and have their values constrained within the specified range, with values in [1,∞). Hölder’s inequality in variable Lebesgue spaces are stated as ∥fg∥r(·) ≤ ∥f∥p(·)∥g∥q(·), where we define r as 1 r(i) = 1 p(i) + 1 q(i) for every i ∈ H, and p, q, r ∈ P(Rn) The set B(Rn) is comprised of p(·) ∈ P(Rn) that fulfill the requirement that M is bounded on Lp(·)(Rn). Lemma 1. [13] Suppose 0 < s ≤ 1, t0 ≥ 1, t∞ ≥ 1, p ∈ P0,∞(Rn) and D > 1 such that D but not on s. Then, for we have: 1 t0 s n p(0) ≤ ∥1Ts,Ds ∥p(·) ≤ t0s n p(0) . (2.4) Similarly, for s ≥ 1, we get: 1 t∞ s n p∞ ≤ ∥1Ts,Ds ∥p(·) ≤ t∞s n p∞ . (2.5) G. A. Basendwah et al. / Eur. J. Pure Appl. Math, 18 (4) (2025), 6641 6 of 16 Definition 1. If Rµ+, µ ≥ 0 and dt t denotes the Haar measure, norm of Lebesgue spaces with Haar measure is defined as, ∥g∥Lq(·)(Rµ+; dt t ) = inf γ > 0 : ∞∫ µ ∣∣∣∣g(t)γ ∣∣∣∣q(t) dtt ≤ 1  . Let Hω(j) = ∞∫ 0 K( j θ )ω(θ) dθ θ . (2.6) This is an integral operator referred to as the Mellin convolution operator, employing Haar measure dθ θ and featuring a kernel homogeneous of order 0. Lemma 2. (see [13]) Let p ∈ P0,∞(R+) and p(0) = p(∞). The operator H is bounded on Lp(·)(R+; dt t ) if ∞∫ 0 |H(t)|sdt t <∞ when s = 1 & s = s0, (2.7) for 1 s0 = 1− 1 p− + 1 p+ . Lemma 3. (see [13]) For every measurable function Ω, the following relations∫ 2b<|x| 2κ > 0, (2.8) and ∫ |x|≥2j |Ω(x)|dx = 1 ln 2 ∫ ∞ t dθ θ ∫ max(θ,2j)<|x|<2θ |Ω(x)|dx, j > 0. (2.9) These conditions are valid provided that the integrals on the left-hand side of the above expressions exist. 3. Continual Herz spaces with variable exponent Definition 2. We can define continual Herz spaces with variable exponents H p(·),q(·),κ(·) µ,δ (Rn) by its norm, ∥g∥ H p(·),q(·),κ(·) µ,δ (Rn) := ∥g∥Lp(·)(O(0,γµ+θ)) + ∥tκ(t)∥g1Rγt,δt ∥Lp(·)∥Lq(·)((γµ,∞); dt t <∞. (3.1) These lemmas are already proved in [13]. Lemma 4. Let 4 ≤ R <∞ and 0 < ρ < 2 ∥g∥Lp(·)(O(0,R)\O(0,2+θ)) ≤ C(ρ,R)∥jκ(j)∥g1j,2j∥Lp(·)∥Lq((2,∞); dj j ) . G. A. Basendwah et al. / Eur. J. Pure Appl. Math, 18 (4) (2025), 6641 7 of 16 Lemma 5. Let 1 ≤ p− ≤ p(x) ≤ p+ <∞ holds, ∥g∥ H p(·),q,κ(·) µ,δ ≈ ∥g∥Lp(·)(O(0,γµ+θ)) + ∥tκ∞∥g1Tγt,δt∥Lp(·)∥Lq(R+.µ; dt t ), µ > 0, (3.2) and ∥g∥ H p(·),q,κ(·) 0,δ ≈ ∥tκ(0)(1 + t)κ∞−κ(0)∥g1Rγt,δt ∥Lp(·)∥Lq(·)(R+; dt t ), (3.3) then equivalences of norms given above are valids, if κ ∈ M log ∞ (R+,µ) for (3.2) and κ ∈ M log 0,∞(R+) in the case of (3.3). 4. Boundedness result on continual Herz spaces with variable exponent In this section, we demonstrate the boundedness of an intrinsic square function on continuous Herz spaces with variable exponent. Initially, we will introduce the intrinsic square function Sζf(x). Definition 3. Let x ∈ Rn and Rn+1 + = Rn × (0,∞) we define a set, Γ(x) :≡ {(y, t) ∈ Rn+1 + : |x− y| < t}. Let 0 < ζ ≤ 1, then by Cζ we denote the functions ϕ defined on Rn satisfying the following conditions: (i) supp ϕ ⊂ {|x| ≤ 1}, (ii) ∫ Rn ϕ(x)dx = 0, (iii) |ϕ(x)− ϕ(x′)| ≤ |x− x′|ζ for x, x′ ∈ Rn. For each (y, t) ∈ Rn+1+, we denote ϕt(y) = t−nϕ(y/t). Let g ∈ L1 loc(Rn), then Aζg(y, t) := sup ϕ∈Cζ |g ∗ ϕt(y)|, where (y, t) ∈ Rn+1 + . Then the intrinsic square function with order ζ is given as Sζg(x) := ∫ ∫ Γ(x) Aζg(y, t) 2dydt tn+1  1/2 . The boundedness of Sζ in variable exponent Lebesgue spaces Lp(·) is discussed in detail in [14]. In the subsequent theorem, we establish the boundedness of the intrinsic square function in variable exponent continuous Herz spaces H p(·),q(·),κ(·) µ;(γ,δ) (Rn). Here, the parameters p(·), q(·), and κ(·) are variable, subject to the condition that Sζ is bounded on Lp(·)(Rn). It’s important to mention that when the constants q are considered, the norms within the continuous Herz space become equivalent for various values of γ and δ, particularly when 0 < γ < δ. Nonetheless, this equality doesn’t persist when q varies and is not constant. G. A. Basendwah et al. / Eur. J. Pure Appl. Math, 18 (4) (2025), 6641 8 of 16 4.1. Non-Homogeneous Herz space(when µ = 0) Theorem 4.1. Suppose that p ∈ Plog ∞ (Rn), q ∈ Plog ∞ (µ,∞) with 1 < p− < p+ < ∞, 1 < q− ≤ q+ < ∞, then every intrinsic square function is bounded on Lp(·)(Rn), is also bounded on the continualHerz spaces with variable exponent H p(·),q(·),κ(·) µ;(γ,δ) (Rn), if − n p∞ < κ∞ < n p′∞ . (4.1) Proof. Norms of H p(·),q(·),κ(·) µ;(γ,δ) (Rn) are equal for different values of µ > 0, and δ > 0 when q constant, so for simplicity we choose δ = 2 and µ = 2, ∥g∥ H p(·),q,κ(·) 2,2 = N(g)κp,q + ∥g∥Lp(·)(O(0,2+θ)) (4.2) where we can define, N(g)κp,q+ = ∥tκ∞∥g1Tt,2t∥Lp(·)∥Lq((2,∞), dt t ). (4.3) We first estimate ∥Sβg∥Lp(·)(O(0,2+θ)), where we choose θ ∈ (0, 1), and estimate for Sζg follows from the boundedness on Lp(·) of Sζg, so we have ∥Sζg∥Lp(·)(O(0,2+θ)) ≤ C∥g∥Lp(·)(O(0,2+θ)) ≤ C∥g∥ H p(·),q,κ(·) 2,2 . Now to find the estimate of N(Sζg)p∗,q,κ term, we will split the functions g(ℓ) as g(ℓ) = f0(ℓ) + ft(ℓ) + gt(ℓ) + ht(ℓ) where f0(ℓ) = g(ℓ)1O(0,1)(ℓ), ft(ℓ) = g(ℓ)1O(0, t 2 )\O(0,1)(ℓ) gt(ℓ) = g(ℓ)1O(0,8t)\O(0, t 2 )(ℓ), ht(ℓ) = g(ℓ)1Rn\O(0,8t)(ℓ), now we have pointwise inequality, |Sζg(ℓ)| ≤ |Sζf0(ℓ)|+ |Sζft(ℓ)|+ |Sζgt(ℓ)|+ |Sζht(ℓ)|. Assume that ϕ ∈ Cζ , k ∈ Z, ℓ ∈ Tk and (y, t) ∈ Γ(ℓ), then we have |g(1l) ∗ ϕt(y)| = ∣∣∣∣∣∣∣ ∫ Tl ϕt(y)g(ℓ)dℓ ∣∣∣∣∣∣∣ ≤Ct−n ∫ {ℓ∈Tl:|y−ℓ| 1 2 (|ℓ− y|+ |y − ℓ|) ≥ 1 2 |ℓ− y| ≥ 1 2 (|ℓ| − |y|) ≥1 2 (|ℓ| − 2t) ≥ 1 2 (|ℓ| − 2k−2) ≥ 1 2 (|ℓ| − 2−1|ℓ|) = |ℓ| 4 . As a result we get |Sζ(g1l)(ℓ)| = ∫ ∫ Γ(ℓ) ( sup ϕ∈Cζ |g1l ∗ ϕt(y)|2 dydt tn+1 )2  1/2 ≤ C  ∞∫ |ℓ| 4 ∫ {y:|ℓ−y| 1, 0, 0 < t < 1, (4.4) by defining, the operator Hω(t) = ∞∫ 0 K( tρ)ω(ρ) dρ ρ , as a result we get N (Sζft)p,q,κ ≤ ∥Hω∥Lq((2,∞)); dt t ) ≤ ∥ω∥Lq((2,∞)); dt t ) ≤ ∥tκ∞∥g1Tt,2t∥p(·)∥Lq((1,2)); dt t ) + ∥tκ∞∥g1Rt,2t∥p(·)∥Lq((2,∞)); dt t ) ≤ ∥g∥Lp(·)(O(0,4)\O(0,1)) + ∥tκ∞∥g1Tt,2t∥p(·)∥Lq((2,∞)); dt t ) ≤ ∥g∥ H p(·),q,κ(·) 2 . G. A. Basendwah et al. / Eur. J. Pure Appl. Math, 18 (4) (2025), 6641 11 of 16 By employing ([13], Corollary 4.5), the inequality κ∞ − n p′∞ ≤ 0, incorporating embed- ding in the last inequality O(0, 4) \ O(0, 1) ⊂ O(0, 2 + θ) ∪O(0, 4) \ O(0, 2 + θ), and with the assistance of Lemma (4). Estimation of Sζgt(ℓ). By the Lp(·) −→ Lp(·) boundedness of Sζ , we obtain ∥Sζgt)1Rt,2t∥p(·) ≤ ∥gt∥Lp(·) ≤ 2∑ j=−1 ∥g1R 2jt,2j+1t ∥p(·). N(Sζgt)p,q,κ ≤ C∥tκ∞∥g1R t 2 ,t ∥p(·)∥Lq((2,∞); dt t ) + ∥tκ∞∥g1Rt,2t∥p(·)∥Lq((2,∞); dt t ) ≤ C∥tκ∞∥g1Rt,2t∥p(·)∥Lq(·)((1,2); dt t ) + ∥tκ∞∥g1Rt,2t∥p(·)∥Lq((2,∞); dt t ) ≤ ∥g∥ H p(·),q,κ(·) 2 here we used the facts, ∥tκ∞∥g1T 2jt,2j+1t ∥p(·)∥Lq((2,∞); dt t ) ≤ ∥tκ∞∥g1Rt,2t∥p(·)∥Lq((2,∞); dt t ), j = 1, 2, and ∥g1Rt,2t∥Lp(·) ≤ ∥g1R0,4∥p(·). Estimation of Sζht(ℓ). Since ℓ ∈ Tt,2t, and Hölder’s inequality, we get |Sβht(ℓ)| ≤ C ρn ∫ |y>8t| g(ℓ)dℓ ≤ C ρn t∫ 1 dρ ρ ∫ ρ 2 <|y|<ρ g(ℓ)dℓ ≤ C tn ∞∫ 4t dρ ρ ∥g1Tρ,2ρ∥p(·)∥1Tρ,2ρ∥p′(·) ≤ C tn t∫ 1 ∥g1ρ,2ρ∥p(·)ρ n p′∞ −1 dρ. tκ∞∥|Sζht(ℓ).1Tt,2t∥Lp(·) ≤ Ct κ∞+ n p∞ ∞∫ 4t ∥g1Tρ,2ρ∥p(·)ρ n p′∞ −1 dρ ≤ ∞∫ t ( t ρ )κ∞+ n p∞ ω(ρ) dρ ρ , where ω(ρ) = ρκ∞∥g1ρ,2ρ∥p(·)1(2,∞), above inequality is the Hardy type inequality, by using the fact κ∞ + n/p∞ > 0 and Lemma (2), get ∥tκ∞∥1Rt,2tSζht(y)∥p(·)∥Lq((2,∞); dt t ≤ C∥ω∥Lq((2,∞); dt t ≤ ∥g∥ H p(·),q,κ(·) 2 . G. A. Basendwah et al. / Eur. J. Pure Appl. Math, 18 (4) (2025), 6641 12 of 16 Now, let’s explore the scenario where q varies. Once again, we set µ = 2, γ = 1, and δ = 2. The proof follows a similar pattern, thus we’ll omit the details. Our objective is to establish that ∥Sζg∥Lp∗(·)(O(0,2+θ)) +N p∗,q,κ 1,2 (Sζg) ≤ ∥g∥Lp(·)(O(0,2+θ)) +N p,q,κ λ′,δ′ (g) with λ′ < 1, δ′ > 2, N p,q,κ λ,δ (g) := ∥tκ∞∥g1Tλt,δt∥p(·)∥Lq(·)((2,∞), dt t ). The estimation for ∥Sζg∥Lp(·)(O(0,2+θ)) can be calculated similarly as we done in the case of constant q. To estimate N (Sζg) p∗,q,κ 1,2 , by splitting the function g(ℓ) as g(ℓ) = f0(ℓ) + ft(ℓ) + gt(ℓ) + ht(ℓ) where f0(ℓ) = g(ℓ)1O(0, 1 2 )(ℓ), ft(ℓ) = g(ℓ)1O(0,γ′t)\O(0, 1 2 ))(ℓ) gt(ℓ) = g(ℓ)1O(δ′t)\O(0,γ′t), ht(ℓ) = g(ℓ)1Rn\O(0,δ′t), then |Sζg(ℓ)| ≤ |Sζ(f0)(ℓ)|+ |Sζ(ft)(ℓ)|+ |Sζ(g)t)(ℓ)|+ |Sζ(ht)(ℓ)|. Estimation of Sζf0. It can be treated similarly as in the case of constant q. Estimation of Sζft. By the same estimation as in the case of constant q, and considering |ℓ− y| > (1− γ′)t we get |Sζft(ℓ)| ≤ Ct−n t∫ 1 2 ∥g1R ρ 2 ,ρ ∥p(·)ρn/p ′ ∞−1dρ, which yields tκ∞∥Sζft(ℓ).1Tt,2t∥Lp∗(·) ≤ Ct κ∞+ n p′∞ t∫ 1/2 ∥g1T ρ 2 ,ρ ∥p(·)ρ n p′∞ −1 dρ ≤ t∫ 1/2 ( t ρ )κ∞− n p′∞ ψ(ρ) dρ ρ , where ψ(ρ) = ρκ∞∥1Tρ,2ρ∥p(·). Left hand side of above equation is a Hardy type operator and by using Lemma (2) we get N p∗,q,κ 1,2 (Sζft) ≤ ∥Γψ∥Lq(·)((2,∞); dt t ) ≤ ∥ψ∥Lq(·)((2,∞); dt t ) ≤ ∥g∥ H p(·),q(·),κ(·) 2;(γ′,δ′) (Rn) , G. A. Basendwah et al. / Eur. J. Pure Appl. Math, 18 (4) (2025), 6641 13 of 16 by following same reasoning as in the case of constant q. Estimation of Sζgt. By the boundedness of the operator Sζ in the space Lp(·)(Rn) → Lp∗(·)(Rn) we obtain ∥(Sζgt)1Tt,2t∥p(·) ≤ C∥gt∥p(·) = C∥g1Rγ′t,δ′t∥p(·) which implies N p∗,q,κ 1,2 (Sζgt) ≤ C∥g1Rγ′t,δ′t∥Hp(·),q(·),κ(·) 2;(γ′,δ′) (Rn) . Estimation of Sζht. The estimate of Sζ can be obtained similarly from the constant q case, we get ∥tκ∞∥1Rt,2tSζht(ℓ)∥p(·)∥Lq(·)((2,∞); dt t ) ≤ C∥g∥ H p(·),q(·),κ(·) 2;(γ′,δ′) (Rn) . By taking into consideration we obtained our desired result. 5. Conclusion This manuscript contributes significantly to the field of mathematical analysis by in- troducing new results within the framework of continual Herz spaces. The study deepens our understanding of boundedness of the intrinsic square function on continual Herz spaces by building on previous findings. The application of these results to establish the exis- tence of the regularity solutions of some elliptic PDEs with smooth boundaries in these spaces. There are a number of exciting avenues for further study in continual weighted Herz-Morrey spaces in the future. Extending these results to two weighted continual Herz-Morrey spaces with variable exponents in solving PDEs within such spaces could be particularly valuable. 6. Ethics declarations 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. Ethics approval and consent to participate This manuscript has not and will not be submitted to more than one journal for simultaneous consideration. The submitted work is original and will not be published elsewhere. Funding Authors state no funding involved. G. A. Basendwah et al. / Eur. J. Pure Appl. Math, 18 (4) (2025), 6641 14 of 16 Availability of data No data is available for this study. References [1] L. Diening, P. Harjulehto, P. H”ast”o and M. Ružička, Lebesgue and Sobolev Spaces with Variable Exponents, Lecture Notes in Mathematics. Vol. 2017, Springer, Heidel- berg (2011). [2] V. Kokilashvili, A. Meskhi, H. Rafeiro and S. Samko, Integral Operators in Non- Standard Function Spaces. Vol. 1: Variable Exponent Lebesgue and Amalgam Spaces. Oper. Theory Adv. 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