We have $0\le a\le b\le1$ and $T\in(0,\infty)$. We want to know when there is a positive constant $C$ such that
$$\int_0^T dt\, \int_a^b dx\, u^2(x-t)\geq C\int_0^1 dx\,u^2(x) \tag{1}$$
for all measurable functions $u\colon\mathbb R\to\mathbb R$ such that $u(x)=0$ for $x\notin(a,b)$. 

The answer is: never. Indeed, without loss of generality $a<b$. The left-hand side of (1) is 
$$\int_0^T dt\, \int_a^b dx\, u^2(x-t) \\ 
=\int_{\mathbb R} ds\, u^2(s)\int_{\mathbb R} dt\,1\{s\in(0,1),t+s\in(a,b),t\in(0,T)\} \\
= 
\int_0^1 ds\,u^2(s)w(s),$$
where 
$$w(s):=\max[0,\min(T,b-s)-\max(0,a-s)].$$
Clearly, the weight function $w$ is continuous, so that $w(b-)=w(b)=0$. 
Letting now $u:=1_{(b-h,b)}$ with $h\downarrow0$, we see that the left-hand side of (1) is $\int_{b-h}^b ds\,w(s)=o(h)$, whereas the right-hand side of (1) is $C\int_{b-h}^b ds=Ch$, so that (1) fails to hols for any real $C>0$.