Estimates for a Rough Fractional Integral Operator and Its Commutators on p -Adic Central Morrey Spaces

: In the current paper, we obtain the boundedness of a rough p -adic fractional integral operator on p -adic central Morrey spaces. Moreover, we establish the λ -central bounded mean oscillations estimate for commutators of a rough p -adic


Introduction
In this day and age, fractional calculus is a key area because of its heaps of applications in engineering science and technology, see for instance [1,2].Moreover, fractional integral operators are major part of the mathematical analysis.These operators have been used to formulate and construct new results in the theory of inequalities.Many of the familiar inequalities and relevant results are generalized and extended via fractional integral operators [3,4].Fractional integral operator of order β is defined by where ζ n (β) = π n/2 2 β Γ(β/2) Γ((n−β)/2) .A fractional integral operator is a smooth operator and has been applied in several branches such as partial differential equations, harmonic analysis, non-linear control theory, and potential analysis, see for example [5,6] and references therein.Over the years, the boundedness properties of T β has put many researchers in the spotlight [7][8][9].
In the last few years.the field of p-adic numbers Q p is wildly used in harmonic analysis [10][11][12] and mathematical physics [13,14].Let p be a prime number.The field of the p-adic absolute value |y| p is defined by setting |0| p = 0, where γ, s, t ∈ Z, and p, s and t are coprime.
In [14], we see that any y = 0 ∈ Q p can be uniquely represented as: where p consists all n-tuples of Q p with the following norm Now, let be, respectively, the ball and sphere with radius p γ and the center at a.It is a familiar fact that Q n p is a locally compact commutative group under addition; denote by dy, the Haar measure on Q n p normalized by B 0 (0) dy = 1.Additionally, In [15], author introduced the fractional integral operator on Q n p as The explicit formula of the above operator on the p-adic field is acquired in [16,17].The fundamental properties of the fractional integral operator on local fields are given in [15].Moreover, λ central bounded mean oscillations estimate for commutators of fractional integral operator on p-adic Morrey spaces are reported in [18].Recently, the boundedness of the fractional integral operator on Morrey spaces is shown in [12,19].The current paper deals with the roughness of an operator which is a key topic in analysis in this day and age; see for instance [20,21] and the references therein.Motivated by [21], we define the rough fractional integral operator.Suppose b : and respectively, whenever and In this article, we consider the rough fractional integral operator T p β,Ω along with its commutator T p,b β,Ω and acquire the boundedness on p-adic central Morrey spaces.In the latter case, the symbol function is from the λ-central bounded mean oscillations(C ṀO s,λ )(Q n p ).The results of the paper can also be implied in locally compact Vilenkin groups and Heisenberg groups.From here on, the letter C means a constant with a different values at separate occurrence.Definition 1 ([22]).Suppose 1 < s < ∞. and λ ∈ R. The space Ḃs,λ (Q n p ) is defined as where [23]).
It is noteworthy to illustrate the importance of our main results before stating them.The following example will do a world of good in this context.
Example 1.The solution u(y, t) of the homogeneous Cauchy problem of linear evolutionary pseudodifferential equation is given by u(y, t) = (T p β u 0 )(y).For the regularity of the solution, we consider two function spaces X and Y. Since T p β is linear, then we have Here, we came across the boundedness inequality It is imperative to mention here that our operator is very helpful in finding the regularity of Cauchy problem of Schrödinger equation.

Boundedness of Rough p-Adic Fractional Integral Operator on Central Morrey Spaces
The current section deals the boundednesss of T p β,Ω on central Morrey spaces.However, in order to do this, we need a lemma which can be proved in the same way as [15].
Now, we turn towards our key result of the section.
For I I, first we have By the application of Hölder's inequality, equality (11) and λ < −β/n, we proceed as Consequently, From ( 10) and ( 13), we have the desired result.
(ii) For q = 1 we set f 2 = f − f 1 and f 1 = f χ B γ .Then by Lemma 1, we have Now, from the similar estimate as in ( 12), we have Making use of Chebyshev's inequality, we obtain Ultimately, for some γ ∈ Z and σ > 0. Hence proof is completed.

λ-Central Bounded Mean Oscillation Estimates of T p,b β,Ω on Central Morrey Spaces
The following section discusses the λ-central bounded mean oscillation estimates of T p,b β,Ω on p-adic central Morrey spaces.We need an important result before proving this.
Now are are firmly in a position to prove our key result.