Commutators of the Fractional Hardy Operator on Weighted Variable Herz-Morrey Spaces

It is important to note that taking β = 0 in (1), we get multidimensional Hardy operator defined and studied in [6, 7]. Also, (1) reduces to the one dimensional Hardy operator [8] if we choose β = 0 and n = 1. Here, we cite some important literature with regards to the study of Hardy-type operators on different function spaces which include [9–15]. The new development of variable exponent commenced with the work of Kov’aˇcik and R’akosn’ık in [16], where a class of function spaces having variable exponent was defined, and basic properties of variable exponent Lebesgue space were explored. Recently, the theory of variable exponent analysis is modeled in terms of the boundedness of the Hardy Littlewood maximal operator M [17–21]:


Introduction
Hardy operators and related commutators play an indispensable role in the theory of partial differential equations [1,2] and the characterization of function spaces [3][4][5]. Without going into much details, let us first define the fractional Hardy operators [3] Hg z ð Þ = 1 t j j n−β dt, z ∈ ℝ n / 0 f g ð1Þ and related commutators: It is important to note that taking β = 0 in (1), we get multidimensional Hardy operator defined and studied in [6,7]. Also, (1) reduces to the one dimensional Hardy operator [8] if we choose β = 0 and n = 1. Here, we cite some important literature with regards to the study of Hardy-type operators on different function spaces which include [9][10][11][12][13][14][15].
The new development of variable exponent commenced with the work of Kov'aˇcik and R'akosn'ık in [16], where a class of function spaces having variable exponent was defined, and basic properties of variable exponent Lebesgue space were explored. Recently, the theory of variable exponent analysis is modeled in terms of the boundedness of the Hardy Littlewood maximal operator M [17][18][19][20][21]: Besides, Muckenhoupt A p theory [22] is generalized in the recent span of time with regard to variable exponent spaces ( [23][24][25][26][27][28]). By taking into account the generalization of function spaces with variable exponents and the same with weights, many results like duality, boundedness of sublinear operators, the wavelet characterization, and commutators of fractional and singular integrals have been studied [29][30][31][32][33][34][35][36][37][38].
Recently, authors have studied generalized Herz space in terms of both Muckenhoupt weights and variable exponent [39][40][41]. Moreover, an idea of combining two function spaces to develop a new one is also an interesting problem in Harmonic analysis. One such problem is considered in [42] in which Herz-Morrey space was defined. Although, the weighted versions of Herz-Morrey spaces were introduced recently in [43,44].
In this piece of work, our main focus is on establishing the boundedness of commutators of fractional Hardy operators on a class of function spaces called the weighted Herz-Morrey space with variable exponents. We seek to find the boundedness of these commutators with symbol functions in BMO (bounded mean oscillation) spaces. In establishing such a boundedness, we make use of the boundedness of the fractional integral operator I β on weighted Lebesgue space which was done in [39].
In the rest of this paper, the symbol C expresses a constant whose value may differ at all of its occurrences. The Greek letter χ S denotes the characteristics function of a sphere S where S is a measurable subset of R n and |S | represents its Lebesgue measure. Before turning to our key results, let us first define the relevant variable exponent function spaces.

Preliminaries
Let us consider a measurable function pð·Þ on ℝ n having range ½1, ∞Þ. The Lebesgue space with variable exponent L pð⋅Þ ðℝ n Þ is the set of all measurable function f such that The space L pð⋅Þ ðℝ n Þ turns out to be Banach function space under the norm: We denote by P ðℝ n Þ the set of all measurable functions pð⋅Þ: ℝ n ⟶ ð1,∞Þ such that where Definition 1. Suppose pð·Þ is a real valued function on ℝ n . We say that (i) C log loc ðℝ n Þ is the set of all local log-Holder continuous functions pð·Þ satisfying (ii) C log 0 ðℝ n Þ is the set of all local log-Holder continuous function pð·Þ satisfying at the origin (iii) C log ∞ ðℝ n Þ is the set of all log-Holder continuous functions satisfying at infinity (iv) C log ðℝ n Þ = C log ∞ ∩ C log loc denotes the set of all global log-Holder continuous functions pð·Þ.
Suppose wðxÞ is a weight function on ℝ n , which is nonnegative and locally integrable on ℝ n . Let L pð⋅Þ ðwÞ be the space of all complex-valued functions f on ℝ n such thatf w 1/pð⋅Þ ∈ L pð⋅Þ ðℝ n Þ. The space L pð⋅Þ ðwÞ is a Banach function space equipped with the norm: Benjamin Muckenhoupt introduced the theory of A p ð1 < p<∞Þ weights on ℝ n in [22]. Recently, in [39,40], Izuki and Noi generalized the Muckenhoupt A p class by taking p as a variable.
In [25], the authors proved that w ∈ A pð⋅Þ if and only if M is bounded on the space L pð⋅Þ .

Some Useful Lemmas
We start this section with some useful lemmas that will be helpful in proving our main results.
Lemma 7 (see [47]). If X is Banach function space, then (i) The associated space X ′ is also Banach function space is the generalized Hölder inequality.
Lemma 8 (see [39]). Suppose X is a Banach function space. Then, we have that for all balls B, Lemma 9 (see [28,39]). Let X be a Banach function space. Suppose that the Hardy Littlewood maximal operator M is weakly bounded on X; that is, is true for σ > 0 and for all f ∈ X. Then, we have Lemma 10 (see [39,48]).
(1) Xðℝ n , WÞ is Banach function space equipped with the norm where (2) The associate space X ′ðℝ n , W −1 Þ is also a Banach function space Lemma 11 (see [39]). Let X be a Banach function space.

Main Results and their Proofs
Definition 15. Let f ∈ L 1 loc ðRnÞ and set where the supremum is taken all over the balls B ∈ ℝ n and b B = jBj −1 Ð B bðyÞdy. The function b is a bounded mean oscillation if kbk BMO < ∞ and BMOðℝ n Þ consist of all f ∈ L 1 loc ðℝ n Þ with BMOðℝ n Þ < ∞. For a comprehensive review of the BMO space, we suggest the reader to follow the books [49,50].

Lemma 16.
Let qð·Þ ∈ P ðℝ n Þ and w be an A qð·Þ weight. Then, for all b ∈ BMO and all l, Proof. First part of this lemma is a consequence of [ [41], Theorem 18]. Next, we will prove (28), for all l, In the view of (27), we have Also, it is easy to see that Combining (29), (30), and (31), we get (28).
If αð·Þ ∈ L ∞ ðℝ n Þ ∩ C log ðℝ n Þ, then Proof. The proof is similar to the proof of Proposition 17 in [44]. So, we omit the details.
Theorem 18. Let 0 < p 1 ≤ p 2 < ∞, q 2 ð·Þ ∈ P ðℝ n Þ ∩ C log ðℝ n Þ, and q 1 ð·Þ be such that 1/q 1 ð·Þ = 1/q 2 ð•Þ − β/n:. Also, let w q2 ð·Þ ∈ A 1 , b ∈ BMOðℝ n Þ, λ > 0, and αð·Þ ∈ L ∞ ðℝ n Þ ∩ C log ðℝ n Þ be log Hölder continuous at the origin, with αð0Þ ≤ αð∞Þ < λ + nδ 2 Proof. For any f ∈ M _ K αð·Þ,λ p 1 ,q 1 ð·Þ ðw q 1 ð·Þ Þ, if we denote f l = f · χ l = f · χ A l , and for each l ∈ ℤ, then it is not difficult to see that Journal of Function Spaces The generalized Hölder inequality (Lemma 7) yields the following inequality for E 1 : Applying the norm on both sides and using Lemma 16, we get Now, we turn to estimate E 2 . For this, we have Similar to the estimation for E 1 , we take the norm on both sides of above inequality and use Lemma 16 to obtain Hence, from inequalities (35), (37), and (39), one has k½b, Now using Lemma 13, we learn In the definition of the fraction integral I β , we replace f by χ Bl to obtain from which we infer that Taking the norm on both sides and using Lemmas 14 and 9, respectively, we get

Journal of Function Spaces
In view of Lemmas 8 and 9, the use of (44) into (41) results in the following inequality: Now, by virtue of the condition p 1 ≤ p 2 and Proposition 17, we have where To estimate X 1 , X 2 , and X 3 , we make use of the conditions on αð·Þ, such that for l < 0, we have and for l ≥ 0, we obtain We estimate F 1 and F 2 separately. A use of generalized inequality results in the following: Applying the weighted Lebesgue space norm on both sides and using Lemma 16, we obtain Similarly,