Generalized fractional integral transform with Whittaker's kernel

In this work we present two generalizations (see the operators $I_{\text{0+}}^{\mathrm{\alpha ,\rho ,\sigma }}$ and $I_{-}^{\mathrm{\alpha ,\rho ,\sigma }}$ defined below) of the classical Liouville fractional integrals. We study their boundedness as operators mapping the space L_v,_r into the spaces ${\mathcal{L}}_{\text{v+2+}\frac{\mathrm{Re}\mathrm{(\alpha )}}{2}\text{−2}\mathrm{Re}\mathrm{(\rho ),r}}$ and ${\mathcal{L}}_{\text{v+1+−}\mathrm{Re}\mathrm{(\rho ),r}}$. In the end, we will apply our generalization to some particular functions.