To fully set up the problem: I have a function $$F : \mathbb R^n \times \mathbb R^+ \to \mathbb R$$ such that $F$ is positive and for any fixed $t^* > 0$, $$F(x,t^*) \in H^\infty(\mathbb R^n);$$ i.e., $F$ is infinitely regular in $x$ for fixed $t$.
I also have a constant $M > 0$ such that $$\left| \left| \frac{\partial F}{\partial t} (\cdot, t) \right| \right|_\infty \le M, \,\,\,\, t > 0.$$ In case it isn't clear, in the above line, the sup-norm is taken over spatial variables, $x$.
Is this enough to claim that $$\left| \frac{d}{dt} \left| \left| F( \cdot, t) \right| \right|_\infty \right| \le M^*, \,\,\,\, t > 0$$ for some other constant $M^*>0$? If not, are there some additional conditions I can impose on $F$ which will guarantee this?
EDIT: As copper.hat pointed out, $|| F(\cdot, t) ||_\infty$ need not necessarily be differentiable. That being said, I don't actually need differentiability of $|| F(\cdot, t) ||_\infty$. For my purposes, I only need uniform continuity of $|| F(\cdot, t) ||_\infty$. Having a bounded derivative would be, of course, a much stronger condition. Can I say anything about uniform continuity of $|| F(\cdot, t) ||_\infty$?