I've been trying to prove the Poincaré inequality via a representation formula for Sobolev functions $u\in W^{1,p}(\mathbb{R}^{n})$, $1\leq p < n$, wlog with compact support. The setting is as follows. Consider a ball $B_{r}(z)\subset \mathbb{R}^{n}$ with $r>0$ and an element $z$ of the support of $u$. It should be provable that for $x\in B_{r}(z)$ we have $$\int_{B_{r}(z)}|u(x)-u(y)| dy\leq C r^{n} \int_{B_{r}(z)}\frac{|Du(y)|}{|y-x|^{n-1}} d y$$ where $C>0$ a constant depending on the support of $u$, and usually this is done via expressing the difference $u(y)-u(x)= \int_{0}^{1}Du(ty+(1-t)x)\cdot(x-y) dt$ and then to take absolute values, finally intgrate over spheres, as it is done for example in Evans-Gariepy, 'measure theory and fine properties of functions'. Here is my question. For Sobolev functions in $\mathbb{R}^{n}$, there is a representation formula, which at least for elements $u\in C_{0}^{\infty}(\mathbb{R}^{n})$ reads as $$u(x)=\frac{1}{n\omega_{n}}\int_{\mathbb{R}^{n}}\frac{\nabla u(w)\cdot (x-w)}{|x-w|^{n}}d w$$ (pointwise in $x$, with $\omega_{n}$ the $n$-dim. Lebesgue measure of the unit ball). From this I'd like to deduce the above formula $$\int_{B_{r}(z)}|u(x)-u(y)| dy\leq C r^{n} \int_{B_{r}(z)}\frac{|Du(y)|}{|y-x|^{n-1}} d y$$ So, for $x,y\in B_{r}(z)$ I obtain $u(x)-u(y)= \int_{\mathbb{R}^{n}}\nabla u(w)\cdot\left(\frac{(x-w)}{|x-w|^{n}}-\frac{(y-w)}{|y-w|^{n}}\right)d w$. As I tried to expand the term $$\left(\frac{(x-w)}{|x-w|^{n}}-\frac{(y-w)}{|y-w|^{n}}\right)=\int_{0}^{1}Dg(t(x-w)+(1-t)(y-w))\cdot(x-y)dt$$ with the function $g(y):=\frac{y}{|y|^{n}}$ and wrote $$u(x)-u(y)= \int_{\mathbb{R}^{n}}\nabla u(w)\cdot\int_{0}^{1}Dg(t(x-w)+(1-t)(y-w))\cdot(x-y)dt dw$$ I expected some cancellations to take place, but it seems to become quite messy. Has anyone an idea or a trick to appropriately eavluate the latter integral such that the above formula follows? I have thought a lot about it, but I don't see it.
Thanks, Jim