Supose that $w \in W^{1,1}((0,\infty)\times (0,\infty))$ (Sobolev space) need satisfy the conditions of proposition 4.9 of Prüss (evolutionari integral equations 2012, pg. 109)., $t, \ \kappa >0$, $0<\eta<1$. Can we conclude that exist $C>0$ such that
$$\int_0^{\frac{t}{\kappa}}\frac{\partial}{\partial \tau} w(t;\tau) t^\eta \tau^{-\eta}d\tau \leq C, \ \forall t>0?$$