Let $\Omega\subset \mathbb{R}^2$ and assume that $u\in W^{2,2}_{loc}(\Omega)$. Let $B_\varepsilon(y)\subset\subset \Omega$ be a ball of radius $\varepsilon$ centered at $y\in\Omega$. I want to estimate $\int_{B_\varepsilon(y)}|\nabla u(x) - \nabla u(y)|^2 \,dx$.
Can I use something like $|\nabla u(x) - \nabla u(y)| \leq |\nabla^2 u(\xi)| |x-y|$ where $\xi$ is some point in $B_\varepsilon (y)$?
I need to show that $\|\nabla u - \nabla u(y)\|_{L^2(B_\varepsilon)} = O(\varepsilon^{1+\gamma})$ with $\gamma\in(0,1)$ but, I think, I am not be able to get that $\nabla u\in C_{loc}^{0,\gamma}(\Omega)$.