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Weighted Estimates for Commutator of Rough -Adic Fractional Hardy Operator on Weighted -Adic Herz–Morrey Spaces
The current article investigates the boundedness criteria for the commutator of rough -adic fractional Hardy operator on weighted -adic Lebesgue and Herz-type spaces with the symbol function from weighted -adic bounded mean oscillations and weighted -adic Lipschitz spaces.
For a fixed prime , it is always possible to write a nonzero rational number in the form , where is not divisible by and is an integer. The -adic norm is defined as . The -adic norm fulfills all the properties of a real norm along with a stronger inequality:
The completion of the field of rational number with respect to leads to the field of -adic numbers . In , it can be seen that any can be represented in the formal power series form aswhere . The convergence of series (2) is followed from .
The -dimensional vector space consists of tuples , where , with the following norm:
The ball and the corresponding sphere with center at and radius in non-Archimedean geometry are given by
When , we write .
Since the space is locally compact commutative group under addition, it cements the fact from the standard analysis that there exists a translation invariant Haar measure . Also, the measure is normalized bywhere represents the Haar measure of a measurable subset of . Furthermore, one can easily show that , , for any .
The last several decades have seen a growing interest in the -adic models appearing in various branches of science. The -adic analysis has cemented its role in the field of mathematical physics (see, for example, [2–4]). Many researchers have also paid relentless attention to harmonic analysis in the -adic fields [5–11]. The present paper can be considered as an extension of investigation of Hardy-type operators started in [6, 7, 12–16].
The one-dimensional Hardy operatorwas introduced by Hardy in  for measurable functions which satisfies the inequalitywhere the constant is sharp. In , Faris proposed an extension of an operator on higher dimensional space bywhere for . In addition, Christ and Grafakos  obtained the exact value of the norm of an operator defined by (8). Over the years, Hardy operator has gained a significant amount of attention due to its boundedness properties [19–22]. For complete understanding of Hardy-type operators, we refer the interested readers to study [12, 23–29] and the references therein.
In what follows, the -dimensional -adic fractional Hardy operatorwas defined and studied for and in . When , the operator transfers to the -adic Hardy operator (see  for more details). Fu et al. in  acquired the optimal bounds of -adic Hardy operator on . On the central Morrey spaces, the -adic Hardy-type operators and their commutators are discussed in . In this link, see also [6, 7, 14, 27].
From now on, we turn our attention towards the rough kernel version of an operator which recently received a substantial attention in analysis (see for instance [11, 31–37]). The roughness of Hardy operator was first time studied by Fu et al. in . Motivated from the results of rough Hardy-type operators in Euclidean space, we define a special kind of rough fractional Hardy operator and its commutator in the -adic field.
Let , and be measurable functions and let . Then, for , we define a rough -adic fractional Hardy operator and its commutator asandwhenever
Remark 1. Obviouslyholds for every integer and prime . Since the inclusionholds and is a linear space over field , the product is correctly defined. Moreover, if a nonzero has a form and(see (2)), then there is such thatwhenever . Using (3), we obtain . Now from (16) and (17), it follows thatThus, for every nonzero , the vector belongs to the sphereFrom (12), it directly follows that for every nonzero , and using (12) and (13), we havefor every . Consequently, the operators and are correctly defined.
The aim of the present paper is to study the weighted central mean oscillations and weighted -adic Lipschitz estimates of on weighted -adic function spaces like weighted -adic Lebesgue spaces, weighted -adic Herz spaces and -adic Herz–Morrey spaces. Throughout this article, the letter represents a constant whose value may differ at all of its occurrence. Before turning to our key results, let us define and denote the relevant -adic function spaces.
2. Notations and Definitions
Suppose is a weight function on , which is non-negative and locally integrable function on . The weighted measure of is denoted and defined as . Let be the space of all complex-valued functions on such that
Definition 1. Suppose and is a weight function. The -adic space is defined as follows:where
Definition 2. (see ). Suppose , and and are weight functions. Then, the weighted -adic Herz space is defined bywhereand is the characteristic function of the sphere .
Remark 2. Obviously .
Definition 3. (see ). Let , , and be weight functions and be a non-negative real number. Then, the weighted -adic Herz–Morrey space is defined as follows:where
Remark 3. It is evident that .
Now, we define the weighted -adic Lipschitz space.
Definition 4. Suppose , and is a weight function. The -adic space is defined aswhereMuckenhoupt introduced the theory of weights on in . Let us define the weights in the -adic field.
Definition 5. A weight function , if there exists a constant free from choice of such thatFor the case , we havefor every .
Remark 4. A weight function if it undergoes the stipulation of weights.
3. Weighted Estimates of on Weighted -Adic Herz-Type Spaces
The present section discusses the boundedness of on weighted -adic Lebesgue spaces as well as on the weighted -adic Herz-type spaces. We begin the section with some useful lemmas to prove our main results.
Lemma 1. (see ). Suppose ; then, there exists constants and such thatfor measurable subset of a ball .
Remark 5. If , then it follows from Lemma 1 that there exists a constant and such that as and as .
Lemma 2. Suppose and ; then, there is a constant such that for , ,
Proof. Firstly, we considerWe assume without loss of generality that ; then, using Lemma 1, we are down to
Lemma 3. Suppose ; thenfor ,where .
Proof. Since , satisfies the conditionsfor every .
From here, we easily get
Theorem 1. Let , , ; thenholds for all , , and .
Now we state the results about the boundedness of commutator of rough -adic fractional Hardy operator on weighted -adic Herz-type spaces.
Theorem 2. Let , and let , .
If , then the inequalityholds for all , , and .
Theorem 3. Let , and let , and . If , thenholds for all , , and .
Proof. of Theorem 2. By definition, we firstly haveFor with , we getAlso, since , by the application of Hölder’s inequality together with Lemma 3, we haveTo estimate , we make use of Hölder’s inequality, Remark 5, and along with (43) and (44) to haveNow, we turn our attention towards estimating .In order to evaluate , we need the following preparation. Apply Hölder’s inequality at the outset to deduceWe imply Hölder’s inequality, inequality (47), Lemma 3, and Remark 5 to estimate .In a similar fashion, we can estimate . Using Hölder’s inequality, Lemmas 2 and 3, Remark 5, and inequality (44), we getFrom (45), (48), and (49) together with Jensen inequality, we haveConsequently,From here on in the proof we consider couple of cases, and .
Case 1. When , noticing that , we proceed as follows.
Case 2. When , applying Hölder’s inequality with , we getTherefore, the proof of theorem is completed.
4. Weighted Lipschitz Estimates for the Commutator of Rough -Adic Fractional Hardy Operator on Herz–Morrey Spaces
In this section, we obtain the weighted -adic Lipschitz estimates for the commutator of rough -adic fractional Hardy operator on -adic Lebesgue spaces and -adic Herz-type spaces. We begin the section with a useful lemma which can be proved in the similar lines as Lemma 2.
Lemma 5. Suppose and ; then, there is a constant such that for , ,
Theorem 4. Let , , , ; then,holds for all , , , and .
Now we state the results about the boundedness of commutator of rough -adic fractional Hardy operator on weighted