Is there a function $f\in L_{1}[0,1]$ but $ \frac{d}{dx}\Big(\int_{0}^{x}\frac{f(s)}{(x-s)^{\alpha}}ds\Big)\not\in L_{1}[0,1]? $

Question

Let $\alpha\in (0,1).$ For a given $f\in L_{1}[0,1]$ consider $$ \phi(x)=\int_{0}^{x}\frac{f(s)}{(x-s)^{\alpha}}ds, \,\,\,\,\,x\in [0,1]. $$ It is clear that if $\phi\in AC[0,1]$ then $\phi$ is differentiable a.e. and $\phi'\in L_{1}[0,1].$ However, there is a function $f\in L_{1}[0,1]$ such that $\phi\not\in AC[0,1].$ I am interested in the following question. Is there a function $f:[0,1]\rightarrow R$ such that $f\in L_{1}[0,1]$ but $\phi'\not\in L_{1}[0,1]?$

Answer 1

$$ \int_{0}^{x}\frac{f(s)}{(x-s)^{\alpha}}ds=\Gamma(1-\alpha)\frac{d^{(\alpha-1)}}{dx^{(\alpha-1)}}f(x)$$ $\Gamma$ is the Gamma function.

$\frac{d^\nu}{dx^\nu}$ denotes the fractionnal derivative of non-integer degree $\nu$ or the fractionnal antiderivative of non-integer degree $-\nu$.

Thus this question is related to fractional calculus : https://en.wikipedia.org/wiki/Fractional_calculus

There is an extensive literature about fractional calculus. At lower level, an article for the general public : https://fr.scribd.com/doc/14686539/The-Fractional-Derivation-La-derivation-fractionnaire

Also you can find a wide documentation about the above intgral referenced as Riemann-Liouville integral and/or Riemann-Liouville's transform.