If we have
$$\|u\left(t\right)\|_X \leq \|v\left(t\right)\|_Y+\|w\left(t\right)\|_Z,$$
can we conclude that $$\operatorname{ess sup}\|u\left(t\right)\|_X \leq \operatorname{ess sup} \|v\left(t\right)\|_Y+\operatorname{ess sup}\|w\left(t\right)\|_Z,$$ where $X,Y,Z$ are Banach spaces?