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 $t$.