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?
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$\begingroup$ I doubt that it suffices. Assume that $|f(x)-f(y)|$ behaves as $|x-y|^a$ for small $a$ and consider pairs of $x,y$ with distance about $\varepsilon$ $\endgroup$– Fedor PetrovCommented Jan 2, 2022 at 9:48
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