Sobolev embedding theorem

Remarks

Part 1 is the Gagliardo–Nirenberg–Sobolev inequality, and part 2 is Morrey's inequality. For $p < n$, the space $W^{1,p}(\R^n)$ also embeds in $L^q(\R^n)$ for every $q$ between $p$ and $p^*$, by interpolation.

The exponent $p^*$ is forced by scaling. For $u_\lambda(x) = u(\lambda x)$, one has $\norm{u_\lambda}_{L^q} = \lambda^{-n/q} \norm{u}_{L^q}$ and $\norm{\nabla u_\lambda}_{L^p} = \lambda^{1 - n/p} \norm{\nabla u}_{L^p}$. The inequality of part 1 can hold for all $\lambda > 0$ only if $-n/q = 1 - n/p$, that is $q = p^*$.

For $p = n \ge 2$, functions in $W^{1,n}(\R^n)$ need not be bounded. On the ball $\abs{x} < 1/2$ in $\R^2$, the function $\log \log (1/\abs{x})$ is in $W^{1,2}$ but unbounded near $0$.

Going deeper

For a student

Sobolev embeddings are the main tool to get regularity of weak solutions of partial differential equations. One first finds a solution in a Sobolev space by energy methods, then uses embeddings to show that it is continuous or smooth. On bounded domains with regular boundary the same embeddings hold, and the Rellich–Kondrachov theorem makes them compact for $q < p^*$.