About fractional integrals in the space of locally integrable functions

Suitable domains to define fractional integrals of Weyl via fractional powers of operators

“…This was already indicated in [11,Remark 3.5], where we also noted that, from the merely algebraic point of view, the operator K 1 w (which we will call K) cannot be a non-negative operator in L 1 loc (R), since 1 + K is not surjective. We cannot work with the closure operator either, since K is not closable, and thus fractional powers of K do not make sense in L 1 loc (R), and obviously, we cannot establish relationships between them and fractional integrals of Weyl either.…”

“…By means of these two results we proved (see [11,Theorem 3.7]) that for α, β ∈ C + with Re β < 1, the operator…”

“…, let x in R be a point where the function K α+β w f is defined, and let a > x. At the beginning of the proof of [11,Theorem 3.7] we proved that…”

“…is convergent, then we deduce without difficulty (see [11,Lemmas 3.1,3.2 and 3.3]) that for almost every x ∈ R, the integral…”

“…However, if it is a sequentially complete locally convex space, the theory of fractional powers can provide global results applicable to the part of the operator in certain classical spaces such as the Lebesgue spaces L p . This approach to Fractional Calculus has already been considered by the authors (see [11]- [13]).…”