  We establish trace and extension theorems for evolutionary equations with the
Caputo fractional derivatives in (weighted) $L_p$ spaces. To achieve this, we
identify weighted Sobolev and Besov spaces with mixed norms that accommodate
solution spaces and their initial values well-suited for equations involving
time-fractional derivatives. Our analysis encompasses both time-fractional
sub-diffusion and super-diffusion equations. We also provide observations on
the initial behavior of solutions to time-fractional equations.
