Assume $\Psi\in W^{1,\frac{3}{2}}_{loc}(\mathbb{R^2})$, satisfies:
$$\lim\limits_{r\to 0}\frac{1}{r}\int_{B_r(x)}|\nabla\Psi(y)|^{3/2}dy=0$$
Then for each $(x_0,y_0)$, and for every $\epsilon>0$ we have:
$$\lim\limits_{\rho\to 0+}\frac{m(E)}{2\rho}=1$$
Where:
$$E=\{t\in(-\rho,\rho): \int_{x_0-\rho}^{x_0+\rho}|\frac{\partial\Psi}{\partial r} (x,y_0+t)|dx<\epsilon\}$$
How to prove this limit ? Any suggestions are welcome!