  In this paper we study the inverse problem of identifying a source or an
initial state in a time-fractional diffusion equation from the knowledge of a
single boundary measurement. We derive logarithmic stability estimates for both
inversions. These results show that the ill-posedness increases exponentially
when the fractional derivative order tends to zero, while it exponentially
decreases when the regularity of the source or the initial state becomes
larger. The stability estimate concerning the problem of recovering the initial
state can be considered as a weak observability inequality in control theory.
The analysis is mainly based on Laplace inversion techniques and a precise
quantification of the unique continuation property for the resolvent of the
time-fractional diffusion operator as a function of the frequency in the
complex plane. We also determine a global time regularity for the
time-fractional diffusion equation which is of interest itself.
