Sobolev-type inequality. - MathOverflow most recent 30 from http://mathoverflow.net 2013-06-20T11:54:41Z http://mathoverflow.net/feeds/question/110078 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/110078/sobolev-type-inequality Sobolev-type inequality. Felice 2012-10-19T09:09:07Z 2012-10-20T02:36:11Z <p>Let $0&lt; \alpha&lt; n$, $1 &lt; p &lt; q &lt; \infty$ and $\frac{1}{q}=\frac{1}{p}-\frac{\alpha}{n}$. Then: $\left \| \int_{\mathbb{R}^n} \frac{f(y)dy}{|x-y|^{n-\alpha} } \right\|_{L^q(\mathbb{R}^n)}\leq$ $C\left\| f\right\| _{L^p(\mathbb{R^n})}$.</p> http://mathoverflow.net/questions/110078/sobolev-type-inequality/110115#110115 Answer by Bazin for Sobolev-type inequality. Bazin 2012-10-19T20:19:45Z 2012-10-19T20:32:02Z <p>The function $\vert x\vert^{\alpha-n}$ is radial homogeneous of degree $\alpha-n$, so its Fourier transform is radial homogeneous of degree $-(\alpha-n)-n=-\alpha$ (both locally integrable since $\alpha >0$ and $-\alpha>-n$ so both are distributions which are easily seen as temperate: Fourier transforms make sense), so your convolution operator is in fact the Fourier multiplier $\vert D_x\vert^{-\alpha}$. The question at hand is thus (with homogeneous spaces) $$\Vert u\Vert_{W^{-\alpha,q}}\lesssim \Vert u\Vert_{W^{0,p}},\quad \text{i.e. }W^{0,p}\subset W^{-\alpha,q},$$ which is a particular case of Sobolev injection since $$0>-\alpha,\quad p &lt; q,\quad \frac{1}{p}-\frac{1}{q}=\frac{\alpha}{n}.$$</p> http://mathoverflow.net/questions/110078/sobolev-type-inequality/110130#110130 Answer by Shanlin Huang for Sobolev-type inequality. Shanlin Huang 2012-10-20T02:36:11Z 2012-10-20T02:36:11Z <p>This is the standard Hardy-Littlewood-Sobolev inequality(or the theorem of fractional integration).A more direct approach is write <code>$$\int{f(x-y)|y|^{\alpha-n}dy}=\int_{|y|&lt;R}+\int_{|y|\ge R}$$</code> For the second term on the RHS,using Holder inequality,and easy to see that it's dominated by <code>$\|f\|_{L^p}R^{-\frac{q}{n}}$</code>. For the first term,one can use the majorizationgiven by the maximal function M,and to see that <code>$$|f\ast |y|^{\alpha-n}|(x)\leq C(M(f)(x)\cdot R^{\alpha}+\|f\|_{L^p}\cdot R^{-\frac{q}{n}})$$</code> Choosing a proper constant R to make the two terms above be equal,and then the desired inequality hold by intergration(note that the maximal operator is bounded on $L^p$ for <code>$1&lt;p&lt;\infty$</code>).</p>