The Elements of Fractional Calculus
Let α > 0 (and in most cases below α < 1 though this is not obligatory). Define the Riemann–Liouville left- and right-sided fractional integrals on (a, b) of order α by
and
respectively.
We say that the function \(f \in D\left( {I_{a + \left( {b - } \right)}^\alpha } \right)\) (the symbol \(D\left( \cdot \right)\) denotes the domain of the corresponding operator), if the respective integrals converge for almost all (a.a.) \(x \in \left( {a,b} \right)\) (with respect to (w.r.t.) Lebesgue measure).
The Riemann-Liouville fractional integrals on R are defined as
and
respectively.
The function \(f \in D\left( {I_ \pm ^\alpha } \right)\) if the corresponding integrals converge for a.a.\(x \in R\). According to (SKM93), we have inclusion \(L_p \left( R \right) \subset D\left( {I_ \pm ^\alpha } \right),1 \le p < \frac{1}{\alpha }.\). Moreover, the following Hardy–Littlewood theorem holds.
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(2008). Wiener Integration with Respect to Fractional Brownian Motion. In: Stochastic Calculus for Fractional Brownian Motion and Related Processes. Lecture Notes in Mathematics, vol 1929. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-540-75873-0_1
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