L2-Boundedness of Integral Operators Involving3F2
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. L2-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 L2.
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 2F1 and the confluent hypergeometric functions 1F1 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 L2-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 L2.
Thearem 1
Proof. An application of Lemma 3 and Lemma 4 yields
By using Lemma 5, we conclude that
Consequently, it implies that
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