I want to ask about the observability inequality for the heat equation ( internal control), we consider the backward heat equation with Dirichlet boundary conditions: 
 
\begin{array}{c}
\varphi _{t}+\Delta \varphi =0\text{  in }\Omega \times (0,T) \\ 
\varphi =0\text{ on } \partial \text{}\Omega \times (0,T)\text{}
\\ 
\varphi (T)=\varphi _{T}\text{ }%
\end{array}
The observability inequality in some books is given by:
$$
\left\Vert \varphi (0)\right\Vert ^{2}\leq c\int_{\omega
}\int_{0}^{T}\varphi (x,t)dtdx
$$
and some times $$\left\Vert \varphi (T)\right\Vert ^{2}\leq c\int_{\omega
}\int_{0}^{T}\varphi (x,t)dtdx$$
and $$\int_{\omega }\int_{T/4}^{3T/4}\varphi (x,t)dtdx\leq c\int_{\omega
}\int_{0}^{T}\varphi (x,t)dtdx$$
To passe from the first to the second I guess that they made the substitution $s=T-t$, however, for the third it comes from the golobal Carleman estimates but I can not see the equivalence between them. Thank you.