# An inequality for Sobolev functions

Let $$\Omega$$ be a smooth bounded domain and $$f\in L^p(\Omega)$$, $$u\in W^{1,1}(\Omega)$$ such that $$p\geq 1$$. Then $$\int_{\Omega}|fu|\,dx\leq C\,\int_{\omega}\{|f(x)|dx(\int_{\Omega}\frac{|\nabla u(\zeta)|}{|x-\zeta|^{n-1}}\,d\zeta+||f||_{L^1(\Omega)})\}\leq\,C\{(\int_{\Omega}\int_{\Omega}\frac{|f(x)|}{|x-\zeta|^{n-\frac{1}{q}}}d\zeta\,dx)^{1-\frac{1}{n}} (\int_{\Omega}|\nabla\zeta|^n\,d\zeta\int_{\Omega}\frac{|f(x)|\,dx}{|x-\zeta|^\frac{n-1}{q}})^\frac{1}{n}+||u||_{L^1(\Omega)}||f||_{L^1(\Omega)} \}$$ for some positive constant $$C$$. In the above line, I have used the following Lemma 1, proved in Trudinger 1967 paper which states that for any $$u\in W^{1,1}(\Omega)$$, we have $$|u(x)|\leq C(n)\{\int_{\Omega}\frac{|\nabla u(\zeta)|}{|x-\zeta|^{n-1}}\,d\zeta+||u||_{L^1(\Omega)}\}.$$

• What is the exact question? You don't understand how to use the Lemma to obtain your inequality or you need a proof of the Lemma? It is not clear what you are asking. Commented Oct 14, 2018 at 11:32
• I am very sorry to bother you about the exact question. The exact question is how to get the first inequality using the Lemma stated at the end?
– Math
Commented Oct 14, 2018 at 14:16

First you use the Lemma to get \begin{align}\int_{\Omega}|fu|\,dx&\leq C\,\int_{\Omega}|f(x)|\left(\int_{\Omega}\frac{|\nabla u(\zeta)|}{|x-\zeta|^{n-1}}\,d\zeta+||u||_{L^1(\Omega)}\right)dx \\&=\int_{\Omega}\int_{\Omega}|f(x)|\frac{|\nabla u(\zeta)|}{|x-\zeta|^{n-1}}\,d\zeta dx+\int_{\Omega}|f(x)|\,dx||u||_{L^1(\Omega)}\end{align} Since $$\frac{n-1}n\left(n-\frac1q\right)+\frac1n\frac{n-1}{q}=n-1$$ you can rewrite the first term on the right-hand side as $$\int_{\Omega}\int_{\Omega}\left(\frac{|f(x)|}{|x-\zeta|^{n-1/q}}\right)^{(n-1)/n}\left(\frac{|f(x)|}{|x-\zeta|^{(n-1)/q}}\right)^{1/n}|\nabla u(\zeta)|\,d\zeta dx$$ and use Holder's inequality with exponent $$p=n$$ and $$p'=\frac{n}{n-1}$$ to bound this term from above by $$\left(\int_{\Omega}\int_{\Omega}\frac{|f(x)|}{|x-\zeta|^{n-1/q}}\,d\zeta dx\right)^{(n-1)/n}\left(\int_{\Omega}\int_{\Omega}\frac{|f(x)|}{|x-\zeta|^{(n-1)/q}}|\nabla u(\zeta)|^n\,d\zeta dx\right)^{1/n}.$$ In the last term you can use Fubini's term to separate the integrals.

• Thank you very much for the nice steps, but in the last time there is one $q$ such that $q=\frac{p}{p-1}$
– Math
Commented Oct 15, 2018 at 7:08
• OK but then there is something wrong in what you wrote. You should have $|f(x)|^p$ somewhere. Can you check if you what you wrote is correct? The only other tool you can use is math.stackexchange.com/questions/1042479/… Commented Oct 15, 2018 at 11:32
• I have attached the link from where pdf file can be accessed. May I kindly request you t have a look at the first step of the proof of Theorem 2. sci-hub.tw/10.1512/iumj.1968.17.17028
– Math
Commented Oct 15, 2018 at 11:44
• Thanks. I'll take a look tonight after work Commented Oct 15, 2018 at 11:52
• Thank You very much. I am waiting to hear from you.
– Math
Commented Oct 15, 2018 at 11:53