  This paper considers traces at the initial time for solutions of evolution
equations with local or non-local derivatives in vector-valued $L_p$ spaces
with $A_p$ weight. To achieve this, we begin by introducing a generalized real
interpolation method. Within the framework of generalized interpolation theory,
we make use of stochastic process theory and two-weight Hardy's inequality to
derive our trace and extension theorems. Our results encompass findings
applicable to time-fractional equations with broad temporal weight functions.
