Generalized fractional integral transforms with gauss function kernels as G-transforms

摘要

The paper is devoted to the study of the generalized fractional integral transforms (I_(0+)~(α,β,η)f)(x)=x~(-α-β)/(Γ(α) ∫from x=0 to x=x of (x - t)~(α-1) _2F_1(α+β, -η;α; 1 - t/x)f(t)dt and (I_-~(α,β,η)f)(x)=1/(Γ(α) ∫from x=x to x=∞ of (t - x)~(α-1) _2F_1(α+β, -η;α;1 - x/t)t~(-α-β)f(t)dt, with α,β,η ∈ C (Re(α) > 0) involving the Gauss hypergeometric function _2F_1(α+β, -η;α;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_(0+)~(α,β,η) and (I_-~(α,β,η) are proved in the space L_(v,r) of Lebesgue measurable function f on R_+ = (0, ∞) such that ∫from x=0 to x=∞ of |t~vf(t)|~r(dt)/t < ∞ (1 ≤ r < ∞), ess sup[t~v|f(t)|] < ∞ (r = ∞) t > 0, for v ∈ R = (-∞,∞), coinciding with the space L~r(0,∞) when v = 1/r. Similar results are obtained for two modifications of the transforms (I_(0+)~(α,β,η)f and (I_-~(α,β,η)f and for Liouville and Kober fractional integral transforms.