Let $C_1, C_2 > 0$ be constants and $g \in C^2(\mathbb{R}^N)$ such that
$\bullet$ $\displaystyle\lim_{\varepsilon \to 0} \displaystyle\iint_{\mathbb{R}^N \times \mathbb{R}^N} \dfrac{|g(x) - g(y )|^p}{|x - y|^p}\rho_{\varepsilon}(|x - y|)dxdy \leq C_1$;
$\bullet$ $C_2\displaystyle\int_{K} |\nabla g(x)|^pdx \leq (1 + \theta)C_1$, for all $\theta > 0$ and $K \subset \mathbb{ R}^N$ compact.
It is possible to show that $$C_1\int_{\mathbb{R}^N} |\nabla g(x)|^pdx \leq \liminf_{\varepsilon \to 0} \iint_{\mathbb{R}^N \times \mathbb{ R}^N} \dfrac{|g(x) - g(y)|^p}{|x - y|^p}\rho_{\varepsilon}(|h|)dxdy?$$