Let $B=[0,T]\times Y$, here $Y$ denotes  a closed manifold. Suppose we use the product metric on this finite tube. 

**Q**: Can we have the following inequality
$$\|s\mid_{t\times Y}\|_{L^2_1(Y)}\leq Const \|s\|_{L^2_2(B)},$$
where $Const$ is independent on $t$.

$L^2_k$ denotes the $k$-th. Sobolev norm in $L^2$ sense.