Uniqueness of inverse source problems for time-fractional diffusion equations with singular functions in time

We consider a fractional diffusion equations of order $\alpha \in (0,1)$ whose source term is singular in time: $({\mathrm{\partial }}_t^{\alpha }+A)u(x,t)=\mu (t)f(x)$, $(x,t)\in \mathrm{\Omega }\times (0,T)$, where $\mu$ belongs to a Sobolev space of negative order. In inverse source problems of determining $f{\mathrm{\mid }}_{\mathrm{\Omega }}$ by the data $u{\mathrm{\mid }}_{\omega \times (0,T)}$ with a given subdomain $\omega \subset \mathrm{\Omega }$ or $\mu {\mathrm{\mid }}_{(0,T)}$ by the data $u{\mathrm{\mid }}_{\{x_0\}\times (0,T)}$ with a given point $x_0\in \mathrm{\Omega }$, we prove the uniqueness by reducing to the case $\mu \in L^2(0,T)$. The key is a transformation of a solution to an initial-boundary value problem with a regular function in time.