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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 $t\in[0,T]$.

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

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 $B$.

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.

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 $t\in[0,T]$.

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

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 $t$.

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.