I tried to ask this in mathstack, but no one answered me. Maybe here is the best place to ask this:
Let $B = B(x_0,R) \subset \subset \Omega$ a ball in $R^n$ with $\Omega $ a domain in $R^n$ with smooth boundary and consider two functions $u,v \in W^{1,p}(\Omega) $, $(1<p \leq 2).$$(1<p \leq 2)$. From general theory we have the inequality
$$ C_1\int_B (|\nabla u| + |\nabla v|)^{p-2} |\nabla u- \nabla v|^2 \leq \int_B |\nabla u|^p - |\nabla v|^p, \ \ \ C_1 = C_1 (n,p) \ \ \ (1).$$
Suppose that
$$\int_B |\nabla u|^p - |\nabla v|^p \leq C_2(n,p)R^n \ \ \ (2) $$
The author of the paper that I am reading says that it is possible to conclude that
$$ \int_B |V(\nabla u) - V(\nabla v)|^2 \leq C_3(n,p)R^n, (*)$$
where $V(\xi) = |\xi|^{\frac{p-2}{2}}\xi, \xi \in R^n,$ from the inequality
$K^{-1} (|\xi|^2 + |\eta|^2)^{\frac{p-2}{2}} |\xi - \eta|^2 \leq |V(\xi) - V(\eta)|^2 \leq K(|\xi|^2 + |\eta|^2)^{\frac{p-2}{2}} |\xi - \eta|^2 , K=K(n,p) \ \ (3),$$K^{-1} (|\xi|^2 + |\eta|^2)^{\frac{p-2}{2}} |\xi - \eta|^2 \leq |V(\xi) - V(\eta)|^2 \leq K(|\xi|^2 + |\eta|^2)^{\frac{p-2}{2}} |\xi - \eta|^2 , \text{where } K=K(n,p) \ \ (3),$
for $\xi , \eta \in R^n-\{ 0\}$$\xi , \eta \in R^n \setminus \{ 0\}$
I am not seeing how to obtain $(*)$ from (1) , (2) and (3). Someone could help me ?
Thanks for your attention
The paper is this : http://arxiv.org/abs/1508.07447 and the problem is in page 6
