Regularity conjecture for solutions of a singular fractional differential equation

Consider the boundary-value problem

$$\begin{cases} D^{\alpha}_{0+}u(t)+h(t)=0,\quad t\in(0,1),\\ u(0)=0=u(1), \end{cases}$$

where $D^{\alpha}_{0+}$ is the Riemann–Liouville fractional derivative of order $\alpha\in(1,2]$. Let $C^1_{2-\alpha}[0,1]$ denote the weighted differentiability space used in the paper, and let

$$E_\alpha=\{u\in C[0,1]:t^{\alpha-1}D^{\alpha-1}_{0+}u(t)\in C[0,1]\}.$$

Regularity conjecture. If $h\in C[0,1]$ or $h\in L^1(0,1)$, then the solution $u$ belongs to $C^1_{2-\alpha}[0,1]$. Moreover, if $h\in C(0,1]$ satisfies $(H)$, then the solution $u$ belongs to $E_\alpha[0,1]$. The claim concerns regularity of solutions when the forcing term is either regular or singular at the endpoint. The surrounding discussion states that the first assertion is known for continuous $h$ and that examples demonstrate the relevance of $E_\alpha$ for singular data; the status of the full proposition, including the $L^1$ case and the hypotheses $(H)$, is not established in the supplied text.