I have seen two proofs of the Sobolev embedding theorem. One uses the Hardy-Littlewood-Sobolev inequality


$
\displaystyle \|f\|\_{L^q({\bf R}^d)} \leq C\_{p,q,d} \|f\|\_{W^{1,p}({\bf R}^d)}.
$


One uses (for $p>1$) the Hardy-Littlewood-Sobolev theorem on fractional integration and the Gagliardo-Nirenberg inequality for $p=1$. See for example the book by E. Stein *Singular Integrals and Differentiability Properties of Functions*.


In the book by *Partial Differential Equations* by L.C. Evans, he proves it using **only** the Gagliardo-Nirenberg inequality because the case $p=1$ implies the case $p>1$.


I find the proof in the book by Evans much more elementary and don't really know why one would use the proof using fractional integration, so I fear that I am missing something. Is the proof using fractional integration somehow stronger?



A question about the proofs of the Sobolev embedding theorem.
The classical Gagliardo-Nirenberg proof covers the limiting cases
$$\|f\|\_{L^{\frac{n}{n-1}}}\le C\|\nabla f\| \_{L^1}$$
and, for $f$ vanishing at infinity,
$$\|f\| \_{L^\infty}\le C\|f\| \_{W^{n,1}}$$
which can not be easily recovered using the HLS inequality. On the other hand, HLS allows for a unified and simple proof of all the non-critical estimates, and includes the case of fractional derivatives. So I would say that the second method is stronger, with the exception of the limiting $L^1$ and $L^\infty$ cases. This is an argument for learning both methods :)


The shortest proof uses the representation
$$ |D|^{-s}f = C\ \ |x|^{s-n} \* f$$
valid for $0 < s < n$. So you have
$$ \||D|^{-s}f\| \_{L^p} \le C \|f\| \_{L^q}$$
that is to say, the Sobolev embedding $\dot H^s\_q\subset L^p$, provided you can apply HLS, which gives you the condition $s-n/q=-n/p$. You see that with a very simple and unified argument we cover all cases including real $s$ and we obtain the homogeneous estimates without lower order terms.


