Generalized Fractional Integral Transforms with Gauss Function Kernels As G-Transforms

• Published in: Integral transforms and special functions 2002-01, Vol.13 (3), p.285-307
• Main Authors: Kilbas, Anatoly A., Repin, Oleg A., Saigo, Megumi
• Format: Article
• Language: English
• Subjects: Gauss Hypergeometric Function; Generalized Fractional Integral Transforms; Integral Transforms With Meijer's G-function In The Kernels; Liouville And Kober Fractional Integrals; Weighted Space Of Summable Functions
• ISSN: 1065-2469; 1476-8291
• DOI: 10.1080/10652460213516

The paper is devoted to the study of the generalized fractional integral transforms (I^{\alpha,\beta,\eta}_{0+}f)(x)={x^{-\alpha -\beta}\over \Gamma (\alpha)} \vint^{x}_{0}(x-t)^{\alpha -1}\ _{2}F_{1}\left(\alpha +\beta,-\eta;\alpha; 1-{t\over x}\right)f(t)\;{\rm d}t and (I^{\alpha,\beta,\eta}_{-}f)(x)={1\over \Gamma (\alpha)} \vint^{\infty}_{x}(t-x)^{\alpha -1}\ _{2}F_{1}\left(\alpha +\beta,-\eta;\alpha; 1-{x\over t}\right)t^{-\alpha -\beta}f(t)\;{\rm d}t , with \alpha, \beta, \eta \in {\bf C} \ ({\rm Re}(\alpha )\gt 0) involving the Gauss hypergeometric function _{2}F_{1}(\alpha +\beta,-\eta;\alpha;z) in the kernels, and of two their modifications. It is proved that the considered fractional constructions can be represented as the integral transforms involving Meijer's G -function as kernels. On the basis of these representations mapping properties such as the boundedness, the representation and the range of I^{\alpha,\beta,\eta}_{0+} and I^{\alpha,\beta,\eta}_{-} are proved in the space {\cal L}_{\nu,r} of Lebesgue measurable functions f on {\bf R}_{+}=(0,\infty) such that \vint^{\infty}_{0}\vert t^{\nu}f(t) \vert^{r}{{\rm d}t\over t}\lt \infty \;\;(1\leq r \lt\infty),\quad \mathop{\rm ess\ sup}_{t\gt 0}[t^{\nu}\vert\, f(t)\vert ]\lt\infty \;\; (r=\infty) , for \nu \in {\bf R}=(-\infty,\infty) , coinciding with the space L^{r}(0,\infty) when \nu =1/r . Similar results are obtained for two modifications of the transforms I^{\alpha,\beta,\eta}_{0+}f and I^{\alpha,\beta,\eta}_{-}f and for Liouville and Kober'fractional integral transforms.