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
$$
\int_{\mathbb R^{2n}} \frac{|f(x)-f(y)|}{(|x-y|+\epsilon)^{n+1}} \, dx \, dy\lesssim \left|\log(\epsilon)\right|
$$
to hold?
Does an assumption 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?