Variable λ -Central Morrey Space Estimates for the Fractional Hardy Operators and Commutators

This paper aims to show that the fractional Hardy operator and its adjoint operator are bounded on central Morrey space with variable exponent. Similar results for their commutators are obtained when the symbol functions belong to λ -central bounded mean oscillation ( λ -central BMO) space with variable exponent.


Introduction
e boundedness of operators on function spaces is one of the core issues in harmonic analysis [1][2][3]. It is mainly because many problems in the theory of partial di erential equations, in their simpli ed form, are reduced to the boundedness of operators on function spaces. It stimulates the research community to embark on such problems in this eld. In this paper, we mainly obtain the boundedness of fractional Hardy operators [4]: on variable exponent central Morrey spaces. In addition, commutators of these operators b, H β g bH β g − H β (bg), b, H * β g bH * β g − H * β (bg), with symbol functions b in variable λ-central BMO spaces are shown bounded on central Morrey spaces with variable exponent. However, before stating our main results, we need to introduce the reader to some basic de nitions and preliminary results regarding variable exponent function spaces. Notably, the function spaces with variable exponents have considerable importance in Harmonic analysis as well. Back in 1931, Orlicz [5] started the theory of variable exponent Lebesgue space. Musielak Orlicz spaces were de ned and studied in [6]. e study of Sobolev and Lebesgue spaces with variable exponents in [7][8][9][10][11] further stimulated the subject. In the meantime, λ− central Morrey space, central BMO space, and associated function spaces have attractive applications by exploring estimates for operators along with their commutators [12][13][14][15][16][17][18][19][20]. Mizuta et al. de ned the variable exponent nonhomogeneous λ-central Morrey space in [21]. e central BMO space rst appeared in [22]. Meanwhile, the authors in [23] gave the de nition of variable exponent central Morrey and λ-central BMO space along with some important results regarding the estimation of some operators. Recently, some publications [24][25][26] discussing the continuity of multilinear integral operators on these function spaces have added substantially to the existing literature on this topic. e one-dimensional Hardy operator was rstly de ned by Hardy in [27] and is considered a classical operator in operator theory. Its mathematical form can be obtained from (1) by taking n � 1 and β � 0. Later on, different authors extended the definition of the one-dimensional Hardy operator to multidimensions in [28,29]. As stated earlier, the fractional Hardy operator and its adjoint operator were introduced first in [4]. Following these publications, a flux of new results emerged discussing the boundedness of Hardytype operators and their commutators on different function spaces [30][31][32][33][34][35]. e commutator operator also enjoyed a lot of attention from different zones of the globe [4,20,[36][37][38][39][40]. However, the continuity of Hardy-type operators and their commutators on variable exponent function spaces took less attention by the research community worldwide [41][42][43][44].
e same is the case with central Morrey space with variable exponent. e present article aims to fill this gap by proving the boundedness of the fractional Hardy operator and its adjoint operator in this space. In addition, this article also includes new results discussing the boundedness of commutators generated by H β (or H * β ) and the λ-central BMO function b on the variable central Morrey space.
Let us describe the framework of this paper. In Section 2, we will remind some lemmas and propositions related to variable exponent function spaces. In Section 3 of this article, we will demonstrate the boundedness for Hardy operators and their commutators on central Morrey space with variable exponent. In Section 4, we shall investigate the similar estimates for the adjoint fractional Hardy operator and its commutators.

Function Spaces with Variable Exponents
In this section, we are going to introduce some notations and definitions related to the variable exponent function spaces. roughout this article, we denote by |B| and χ B the Lebesgue measure and characteristic function of a measurable set B ⊂ R n , respectively. Also, B j � B(0, 2 j ) � x ∈ R n : |x| ≤ 2 j with A j � x ∈ R n : 2 j− 1 < |x| ≤ 2 j and χ j � χ A j for j ∈ Z. e notation g ≈ f implies that there exist two positive constants C 1 and C 2 such that C 1 f ≤ g ≤ C 2 f. Furthermore, E ⊆ R n represents an open set and p(·): E ⟶ [1, ∞) is a measurable function, and p ′ (·) denotes the conjugate exponent of p(·) which satisfies e set P(E) consists of all p(·) and p ′ (·) such that e space L p(·) is a set of all measurable function f on the open set E, in such a way that, for positive η, which becomes a Banach function space when equipped with the Luxemburg-norm Local version of variable exponent Lebesgue space is denoted by L p(·) loc (E) and is defined by L p(·) We use B(R n ) to denote a set containing p(·) ∈ P(R n ) satisfying the condition that the Hardy-Littlewood maximal operator M: where B r � y ∈ R n : |x − y| < r is bounded on L p(·) (R n ).
Proposition 1 (see [8,45]). Let E denote an open set and p(·) ∈ P(E) fulfill the following inequalities: then p(·) ∈ B(E), where C is a positive constant independent of x and y.
where supremum is taken all over the ball B ⊂ R n and Lemma 4 (see [48]). Let p(·) ∈ P(R n ), then for all b ∈ BMO(R n ) and all l, i ∈ Z with l > i, we have Definition 2 (see [23]). Let p(·) ∈ P(R n ) and λ ∈ R. en, the variable exponent central Morrey space where Definition 3 (see [23]). Let p(·) ∈ P(R n ) and λ < 1/n. en, where While proving our main results, we control the boundedness of the fractional Hardy operator using the boundedness of the fractional integral operator I β : on variable Lebesgue space. In this regard, we need the following proposition.

Fractional Hardy Operator and Commutator
In this section, we present theorems on the boundedness of the fractional Hardy operator and commutators on central Morrey space with their proofs. Theorem 1. Let p 1 (·) ∈ P(R n ) and satisfy conditions (9) and (10) in Proposition 1. Define the variable exponent p 2 (·) by If λ 2 � λ 1 + β/n and λ 2 > − (δ 1 + δ 3 ), where δ 1 and δ 3 are the same constants as appeared in inequalities (14) and (15), then Proof. By definition of the fractional Hardy operator and Lemma 1, it is easy to see that Taking the L p 2 (·) (R n ) norm on both sides, we have rough the use of Lemma 3 and the inequality (15), it is easy to see that In view of the condition 1/p 1 ′ (x) � 1/p 2 ′ (x) − β/n and Lemma 5, the last inequality reduces to the following inequality: Since therefore, from the inequality (32), we infer that Finally, inequality (14) helps us to have Since δ 3 + δ 1 + λ 2 > 0, so we get Theorem 2. Let 0 < β < n and let p(·), q(·), r(·) ∈ P(R n ) and satisfying conditions (9) and (10) in Proposition 1 with p(·) < n/β, p ′ (·) < r(·) and (37) Let 0 < ] < 1/n and − 1/q + < μ. If μ � ] + λ + β/n, with max − (] + 1), − (δ 1 + δ 3 + β/n) < λ, where δ 1 , δ 3 are the same constants as appeared in inequalities (14) and (15), and b ∈ ‖b‖ CBMO r(·),] , then Proof. We decompose the integral appearing in the commutator operator as Let us first estimate A 1 . By taking the variable Lebesgue space norm on both sides, we get Taking into consideration the condition 1/q(·) � 1/s(·) + 1/r(·), (1/s(·) � 1/p(·) − β/n), the generalized Hölder inequality gives us the following estimation of A 1 : where σ � λ + β/n. Using the result of eorem 1, we obtain Here, it is easy to see that erefore, on account of the condition μ � ] + σ, for A 1 , we have Next, we consider A 2 for approximation: which can be decomposed further as Journal of Mathematics where We define a new variable t(·) such that 1/t(·) � 1/p ′ (·) − 1/r(·), then by the generalized Hölder inequality, we have With the Lebesgue space with variable exponent norm on both sides, the above inequality takes the following form: Hence, Finally, consider e factor (b B − b 2 j B ) in the above inequality needs to be dealt with first. So, Next, Lemma 3 helps us to write 6 Journal of Mathematics |j|. (53) In turn, A 22 satisfies the below inequality: Ultimately, our last step would be applying the norm on both sides to get Combining all the approximations of A 1 , A 2 , A 21 , A 22 , we obtained the required result

Journal of Mathematics
Next, by virtue of inequality (53), D 22 satisfies To finish the estimation, we take norm on both sides of the above inequality to obtain ≤ C‖b‖ CBMO r(·),] ‖f‖ _ B p(·),λ |B| μ χ B � � � � � � � � L q(·) . (74) In the end, combining all the estimates of D 1 , D 2 , D 21 , D 22 , we arrive at the following conclusive inequality: which is as desired.

Data Availability
Data sharing is not applicable to this article as no datasets were generated or analyzed during the current study.

Conflicts of Interest
e authors declare that they have no conflicts of interest.