## L^{2}-Boundedness of Integral Operators Involving_{3}F_{2}

### Shahid Mubeen*

*Department of Mathematics, University of Sargodha, Pakistan*

**Submission: ** May 5, 2017; **Published: ** July 11, 2017

***Corresponding author:** Shahid Mubeen, Department of Mathematics, University of Sargodha, Pakistan, Email : smjhanda@gmail.com

**How to cite this article:** Shahid M. L^{2}-Boundedness of Integral Operators Involving . Biostat Biometrics Open Acc J. 2017;2(2): 555583. DOI: 10.19080/BBOAJ.2017.02.555583

**Abstract**

In this paper, we formulate the integral operators involving hypergeometric functions as kernel. We discuss that these operators are composition of Erdlyi-Kober fractional integral operators. We also discuss the boundedness of these integral operators in L^{2}.

** Keywords: ** Fractional integral transform; Liouville and Kober frac-tional integrals; Hypergeometric functions; Integral transform with hypergeometric functions in the kernel

There have made numerous investigations pertaining to integral operators involving various hypergeometric functions _{2}F_{1} and the confluent hypergeometric functions _{1}F_{1} as kernel [1-5]. Many authors also discussed the boundedness of integral opera- tors and used their mapping properties to derive inversion processes [6].

In this paper, we use the integral representation of hypergeomet- ric functions [7]

for formulating the integral operators of the following form

Where

Here we start with a basic result that use later, see Karapetiants and Samko [8] and Okikiol

**Lemma 1**

Suppose that Ψ is a measurable and homogeneous function of degree ^{-1} for all real numbers h i.e. .

Let

then

Also as a consequence, we have the L^{2}-boundedness of generalized Erdlyi-Kober fractional integrals [10] as transcribed below.

**Lemma 2**

Let

We now prove the boundedness of the following integral operators involving homogeneous functions as kernel. These integral operators are generalization of integral operators those are studied by Love [11] and Habibullah [12].

**Lemma 3**

By using Fubini's theorem, we have the following lemma:

**Lemma 4**

**Lemma 5**

Proof. After making some substitutions in the integral representation of 3F2, we get the following integral

By replacing in place of g in Lemma 4, we have obtain

Now, we formulate integral operators involving hypergeometric functions of the type and then prove the boundedness of these integral operators in L^{2}.

**Thearem 1**

Proof. An application of Lemma 3 and Lemma 4 yields

By using Lemma 5, we conclude that

Consequently, it implies that

**References**

- Erdelyi A (1964) An integral equation involving Legendre functions. J Soc 12(1): 15-30.
- Higgins TP (1963) An inversion of integral for a Gegenbauer transformation. J Soc Indust Appl Math 11(4): 886-893.
- Anatoly AK, Saigo, Megumi, Trujillo JJ (2000) On the Meijer transform in space. Integral transforms and special functions 10(3-4): 267-282.
- Anatoly AK, Repin, Oleg A, Saigo, Megumi (2002) Generalized fractional integral transforms with Gauss function kernels as G-transforms. Integral transforms and special functions. 13(3): 285-307.
- Anatoly AK, Sebastian Nicy (2008) Generalized fractional integration of Bessel Function of the First Kind. Integral transforms and special functions 19(12): 869-883.
- Habibullah GM (1977) A note on a pair of integral operators involving Whit- taker functions. Glasgow Math J Soc 18: 99-100.
- Driver KA, Johnston SJ (2006) An integral representation of some hyper- geometric functions. Electronic Transactions on Numerical Analysis. 25: 115-120.
- Love ER (1967) Some integral equations involving hypergeometric functions. Proc Edinburgh Math Soc 15: 169-198.
- Erdelyi A (1950) On some functional representations. Univ E Politic Torino Rend Semi Math 10: 217- 234.
- Kober H (1940) On fractional integrals and derivatives. Quart J Math 11: 193-211.
- Karapetiants NK, Samko SG (1999) Multidimensional integral operators with homogeneous kernels. Fract Calc and Applied Anal, pp. 67-96.
- Okikiolu GO (1966) Bounded Linear Transformation in Space. Journal London Math Soc 41: 407-414.