Let $f:\mathbb R^n \to \mathbb R$. What (minimal) assumptions are needed, in addition to $f \in L^1 \cap L^\infty$, to have $$ \int_{\mathbb R^{2n}} \frac{|f(x)-f(y)|}{(|x-y|+\epsilon)^{n+1}} dxdy\lesssim |\log(\epsilon)| $$ to hold? Does something like $$\int_{\mathbb R^{2n}} \frac{|f(x)-f(y)|}{|x-y|^{n}}\log(|x-y|)^{\alpha} dx dy < \infty $$ suffice? Do we need more?