Definition are Banach spaces, we say is embedded in , denoted by , if and .
Here are some embedding results.
Theorem 1 , if .
If , then . Therefore is continuous and vanishes at (Riemann Lebesgue Theorem).
Theorem 2 , if .
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
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
for . To conclude the case where , we simply just use the fact that