User student - MathOverflow most recent 30 from http://mathoverflow.net 2013-05-25T18:37:31Z http://mathoverflow.net/feeds/user/11604 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/49566/a-question-about-the-proofs-of-the-sobolev-embedding-theorem A question about the proofs of the Sobolev embedding theorem. student 2010-12-15T20:58:29Z 2010-12-15T23:00:38Z <p>I have seen two proofs of the Sobolev embedding theorem. One uses the Hardy-Littlewood-Sobolev inequality</p> <p><code>$\displaystyle \|f\|_{L^q({\bf R}^d)} \leq C_{p,q,d} \|f\|_{W^{1,p}({\bf R}^d)}.$</code></p> <p>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 <em>Singular Integrals and Differentiability Properties of Functions</em>.</p> <p>In the book by <em>Partial Differential Equations</em> by L.C. Evans, he proves it using <strong>only</strong> the Gagliardo-Nirenberg inequality because the case $p=1$ implies the case $p>1$.</p> <p>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?</p>