Let $B=[0,T]\times Y$, here $Y$ denotes a closed manifold. Suppose we use the product metric on this finite tube.
We can 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 $t\in[0,T]$.
$L^2_k$ denotes the $k$-th. Sobolev norm in $L^2$ sense.
As Piotr Hajlasz pointed out by the standard trace theorem.
Q: Can we let the Constant does not depend on the $T$, or just depends on the volume of $B$.
Yes, that is true. This is a consequence of the classical Trace Theorem for Sobolev spaces. The proof can be found in any textbook on Sobolev spaces for example in Evans' Partial Differential Equations. The usual statement of the trace theorem deals with traces of $T:W^{1,p}(\Omega)\to L^p(\partial\Omega)$, but the proof would also give traces $T:W^{1,p}([0,T]\times Y)\to L^p(\{ t\}\times Y)$. If $u\in W^{2,p}$, then the derivative is in $W^{1,p}$ so the trace of the derivative is in $L^p$ and since the derivative of the trace is the trace of the derivative (roughly speaking) we obtain the trace $T:W^{2,p}([0,T]\times Y)\to W^{1,p}(\{ t\}\times Y)$.
