About fractional integrals in the space of locally integrable functions (Q1197435)

The authors show that the largest ``natural'' function space, where the Riemann-Liouville fractional integral operator \[ J^ \alpha f(x)={1\over{\Gamma(\alpha)}} \int_ 0^ x (x-t)^{\alpha-1} f(t)dt \] and differential operator \[ D^ \alpha f(x)={1 \over{\Gamma(1-\alpha)}} {d \over dx} \int_ 0^ x (x-t)^{-\alpha} f(t)dt \] (\(0<\text{Re} \alpha<1\)) may be studied, is the space \(L_{\text{loc}}^ 1([0,\infty))\). Moreover, they show how to apply the general theory of fractional powers of linear operators to obtain the main properties of \(J^ \alpha\) and \(D^ \alpha\) directly.