Fix $d,t>0$ and let $a \in L^2[-d,0]$ and $Y \in L^2[-d,t]$.
Is it true via some change of variable the following relation?
$$\int_{(-t) \vee (-d)}^0 \int_{-d}^s a(\xi)Y(\xi-s)d\xi d s+\int_0^t\int_{-d}^{0} I_{[-(t-s),0]}(\xi)a(\xi)Y(s) d \xi ds=\int_0^t\int_{-d}^{0} a(\xi)Y(s+\xi) d \xi ds$$