Boundedness of Hardy operators on grand variable weighted Herz spaces

: In this paper, we will introduce the idea of grand variable weighted Herz spaces ˙ K


Introduction
The notion of Herz spaces were introduced by C. Herz in [1]. Let α ∈ R, 1 ≤ p < ∞ and 1 ≤ q < ∞. The classical versions of non-homogeneous and homogeneous Herz spaces are defined by the norms respectively. Variable exponent function spaces have been widely studied and have many important applications. Some examples of works in this area are [5,6]. These variable exponent function spaces are important for studying problems in partial differential equations and applied mathematics. In particular, Herz spaces with variable exponent generalize classical Herz spaces, see [7]. The Herz-Morrey spaces MK α,λ q(·),p (R n ) generalize the idea of Herz spaces with variable exponentK α,p q(·) (R n ). These function spaces were initially defined in [8]. Lu and Zhu [9] further studied the Morrey-Herz spaces MK α(·),λ q,p(·) (R n ) and established the boundedness of integral operators on these spaces.
Izuki and Noi introduced the concept of weighted variable Herz spacesK α,r s(·) (w) in their papers [10,11]. The concept of grand Morrey spaces introduced in [12], has attracted significant attention from researchers. In [13], the idea of grand variable Herz spacesK α(·),p),θ q(·) (R n ) was introduced, and boundedness of sublinear operators were obtained. Boundedness of other integral operators on grand variable Herz spaces can be seen in [14][15][16][17][18]. In [19], the definition of grand variable Herz-Morrey spaces introduced and obtained the boundedness of Riesz potential operator in these spaces. In [20], authors obtained the boundedness of variable Marcinkiewicz integral operator on grand variable Herz-Morrey spaces. Recently, in [21], the authors proved the boundedness of fractional integral operator on grand weighted Herz spacesK α,p),θ q(·) (w) spaces. Grand weighted Herz-Morrey spaces are the generalization of grand weighted Herz spaces. In [22], Sultan et al. established the boundedness of fractional integral operator on grand weighted Herz-Morrey spaces.
Motivated by the study on grand weighted Herz spaces, our main purpose is to define grand variable weighted Herz spaces, which is the generalization of weighted Herz spaces with variable exponents. Our main purpose is to establish some boundedness results for the Hardy operators on grand variable weighted Herz spacesK α(·), ),θ q(·) (τ). Suppose that G is a measurable set in R n with Lebesgue measure |G| > 0. The characteristic function of G is denoted by χ G . It is important to note that in this paper, the symbol C represents a positive constant, which may vary in value at different occurrences.

Definition of function spaces
We first recall some necessary definitions and notations.
Definition 2.1. Let G be a measurable set in R n and r(·): G → [1, ∞) be a measurable function. We suppose that where r − := ess inf g∈G r(g), r + := ess sup g∈G r(g).
(a) Variable Lebesgue space L r(·) (G) can be defined as Norm in L r(·) (G) can be defined as The space L r(·) loc (G) can be defined as L r(·) loc (G) := f : f ∈ L r(·) (K) for all compact subsets K ⊂ G . The log-conditions may be stated as follows: where C(r) > 0. Additionally, the decay condition: There exists a number r ∞ ∈ (1, ∞), such that and also decay condition holds for some r 0 ∈ (1, ∞). We use these notations in this article: (i) The set P(G) consists of all measuable functions r(·) satisfying r − > 1 and r + < ∞.
We define the Hardy-Littlewood maximal operator M as where f ∈ L 1 loc (G). Definition 2.2. The weighted L r(·) space is defined as the set of all measurable functions f on R n such that f τ 1 r(·) ∈ L r(·) (R n ), where r(·) ∈ P(R n ) and τ is a weight. The norm of the Banach space L r(·) (τ) is denoted by where r (·) is the conjugate exponent of r(·).
We can define non-homogeneous grand weighted Herz-Morrey spaces in a similar manner.

Important lemmas
We require the following preliminary results to prove our main theorems.
Lemma 2.11. Let decay conditions at origin and infinity be fulfilled. Then respectively, where t 0 ≥ 1 and t ∞ ≥ 1 independent of k.
Proof. We will prove (2.12) and other inequality can be estimated similarly. As we can see from [23] that Now right hand side inequality of (2.12) is given as Using the decay condition we see that which determines the choice of t p − 0 = 2 knp ∞ −kn we will get our desired result.

Boundedness of Hardy operators on grand variable weighted Herz spaces
In this section, we will prove the main results of this paper.
Proof. Let f ∈K α(·), ),θ For E 1 , we use the facts that, for each ∈ Z, z ∈ R with j ≤ . Then Hölder's inequality and size condition imply .

Splitting by using Minkowski's inequality we have
For E 11 , by virtue of Lemma 2.11 we get Applying to E 11 we can get . (

3.2)
Applying the fact 2 − (1+∆) < 2 − , Fubini's theorem for series and Hölder's inequality we get, Now for E 12 using Minkowski's inequality we have The estimate for A 2 can be followed in a similar manner to E 11 with replacing q (0) by q ∞ and using n q ∞ − α ∞ > 0. For A 1 by virtue of Lemma 2.11 we obtain As α ∞ − n q ∞ < 0 we have . By using n q (0) − α(0) > 0 and Hölder's inequality we have which completes our desired results.

Conclusions
In this paper, we defined the idea of grand variable weighted Herz spaces. We proved the boundedness of Hardy operators on grand variable weighted Herz spaces by using the properties of exponents. These results hold for weighted Herz spaces with variable exponent.

Use of AI tools declaration
The authors declare they have not used Artificial Intelligence (AI) tools in the creation of this article.