Fractional integral and Caputo derivative

Let $T>0$ and $\beta >0$. For $g\in L^1(0,T)$, its left fractional integral of order $\beta$ is, for almost every $t\in (0,T)$,

$I^{\beta }g(t):=\frac{1}{\mathrm{\Gamma }(\beta )}{\int }_0^t(t-s{)}^{\beta -1}g(s)ds\mathrm{.}$

The definition uses Lebesgue integration, with Indefinite Lebesgue integral on a compact interval and The class $L^1(\mu )$ of integrable functions. The kernel is integrable and ${\int }_0^t(t-s{)}^{\beta -1}ds=\frac{t^{\beta }}{\beta }$ by Integral of the fractional power kernel. Tonelli's theorem Tonelli's theorem for nonnegative measurable functions on a sigma-finite product therefore gives ${\int }_0^T{\int }_0^t(t-s{)}^{\beta -1}\mathrm{\mid }g(s)\mathrm{\mid }dsdt={\int }_0^T\mathrm{\mid }g(s)\mathrm{\mid }{\int }_s^T(t-s{)}^{\beta -1}dtds\le \frac{T^{\beta }}{\beta }\mathrm{\parallel }g{\mathrm{\parallel }}_{L^1(0,T)}<\mathrm{\infty },$ so the defining integral is finite for almost every $t$. For continuous $g$, it is finite at every $t\in [0,T]$ and $\mathrm{\mid }{I}^{\beta }g(t)\mathrm{\mid }\le \frac{t^{\beta }}{\beta \mathrm{\Gamma }(\beta )}\mathrm{\parallel }g{\mathrm{\parallel }}_{\mathrm{\infty }}=\frac{t^{\beta }}{\mathrm{\Gamma }(\beta +1)}\mathrm{\parallel }g{\mathrm{\parallel }}_{\mathrm{\infty }},$ using The real Gamma functional equation $\mathrm{\Gamma }(s+1)=s\mathrm{\Gamma }(s)$.

For $0<\alpha <1$ and an absolutely continuous function $u:[0,T]\to \mathbb{R}$ whose almost-everywhere derivative belongs to $L^1(0,T)$, its Caputo derivative of order $\alpha$ is the almost-everywhere defined function

$D_C^{\alpha }u:=I^{1-\alpha }{u}^{\mathrm{\prime }}=\frac{1}{\mathrm{\Gamma }(1-\alpha )}{\int }_0^t(t-s{)}^{-\alpha }{u}^{\mathrm{\prime }}(s)ds\mathrm{.}$

Here absolute continuity is as in Absolute continuity on a compact interval. Under Countable Choice (The Axiom of Countable Choice (${\mathrm{A}\mathrm{C}}_{\omega }$)), every such absolutely continuous $u$ has $u^{\mathrm{\prime }}\in L^1(0,T)$ by Absolutely continuous functions have integrable derivatives, so the Caputo derivative is defined on the whole absolutely continuous class under that assumption. Without Countable Choice, the same formula is defined for each absolutely continuous $u$ whose derivative is separately known to be in $L^1(0,T)$. The Gamma factors are the positive real Euler Gamma function The real Gamma function by Euler's integral.