**Definition** are Banach spaces, we say is embedded in , denoted by , if and .

**Notation**: .

Here are some embedding results.

**Theorem 1** , if .

*Proof:*

since .

If , then . Therefore is continuous and vanishes at (Riemann Lebesgue Theorem).

**Theorem 2** , if .

*Proof:*

**Theorem 3 (Sobolev embedding)** If , then for .

*Proof:* The idea is to split into low frequency part and high frequency part in Fourier space, i.e.

where is the inverse Fourier transform. Note that

So if we define

Since

we have

Then we know

Since the integrand is nonnegative, we can use Tonelli theorem to change the order of integration and get

**Theorem 4 (Moray Inequality)** If , then , for .

*Proof:* Take an and first assume . By Theorem 1 we know , so we need to check for any .

we can split the later term into two parts, which is

Therefore

for . To conclude the case where , we simply just use the fact that

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