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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?

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The first inequality you wrote down, I think, is the Gagliardo-Nirenberg-Sobolev inequality, no? –  Willie Wong Dec 15 '10 at 22:57

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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.

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But from the Gagliardo-Nirenberg, integration by parts, and the Holder inequality, you can prove all of the Sobolev inequalities involving a nonnegative integer number of derivatives. In fact, you get more; you get the full family of what are known as Gagliardo-Nirenberg inequalities. This serves most needs. I would say that you need the HLS inequality only if you need fractional derivatives for some reason. –  Deane Yang Dec 16 '10 at 0:15
    
Sure, but I don't really see why one should restrain from using HLS. If you use it properly, the proof is much simpler than the classical one, it's unified, covers fractional derivatives, can be extended without effort to cover Lorentz spaces and to prove the GN inequalities, etc. –  Piero D'Ancona Dec 16 '10 at 0:33
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I guess I'm a little skeptical. The proof of Gagliardo-Nirenberg uses only the 1-d fundamental theorem of calculus and Holder's inequality. I think just stating HLS is harder than that. –  Deane Yang Dec 16 '10 at 2:41
    
@Deane. It uses also Fubini's theoremand a generalization of it. –  Denis Serre Dec 16 '10 at 7:14
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@student. I suggest you have a look at Luc Tartar's book An introduction to Sobolev spaces and interpolation spaces. Lecture Notes of the Unione Matematica Italiana, 3. Springer, Berlin; UMI, Bologna, 2007. –  Denis Serre Dec 16 '10 at 7:15

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