Change of variable for a weak derivative

Question

Let $h \in C([0,T], \mathbb R) \cap W^{1, \infty}([0,T], \mathbb R)$ such that the following estimate is satisfied $$\frac{h'(t)}{h(t)^{3/2}} \le C f(t)$$ for $h'$ the (weak) derivative of $h$ and $f$ some function in $L^1(0,T)$. If everything were smooth, I could integrate on both side of the inequality and get $$\frac{1}{\sqrt{h(T)}} - \frac{1}{\sqrt{h(0)}} = \int_0^T\frac{h'(t)}{h(t)^{3/2}} \le C \|f\|_{L^1(0,T)}.$$ However, here we are working in Sobolev spaces so I cannot integrate the left hand side as I did. Is there a way to treat this problem in the case of Sobolev spaces ?

Answer 1

Suppose $h(t) \ge c > 0$ for all $t$. Then $t\mapsto \frac1{\sqrt t}$ is $C^1$ on the range of $h$, i.e., on $[c, \|h\|_\infty]$. We can extend it to a smooth function $\phi$ on $\mathbb R$. Then $\phi(h)$ is in $C([0,T]) \cap W^{1,\infty}(0,T)$. And the calculation you did is valid.