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?