Inclusion of Sobolev spaces $H^2(\mathbb{R^2}) \subset W^{1,4}(\mathbb{R^2})$?

Question

How do prove the inclusion of Sobolev spaces $H^2(\mathbb{R^2}) \subset W^{1,4}(\mathbb{R^2})$? Any ideas?

Answer 1

This is an immediate consequence of the general Sobolev inequality. This implies the embedding theorem, which says that if you have $k,l,p,q,n$ satisfying $$\frac{1}{p} - \frac{k}{n} = \frac{1}{q} - \frac{l}{n}$$

then you have $W^{k,p}(\mathbb{R}^n) \subseteq W^{l,q}(\mathbb{R}^n)$.

Since $H^2(X) = W^{2,2}(X)$ by definition, you can do the arithmetic and see that this holds.