In this work we present two generalizations (see the operators I0+α,ρ,σ and Iα,ρ,σ defined below) of the classical Liouville fractional integrals. We study their boundedness as operators mapping the space ℒv,r into the spaces Lv+2+Re(α)2−2Re(ρ),r and Lv+1+−Re(ρ),r. In the end, we will apply our generalization to some particular functions.

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