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After some thinking, I have a terrible headache caused by the following problem. Imagine we have a function $u \colon \mathbb{R}^n \to \mathbb{R}$ such that $u \in L^2(\mathbb{R}^n)$ and$$\int_{\mathbb{R}^n \times \mathbb{R}^n} \frac{|u(x)-u(y)|^2}{|x-y|^{n+2s}}dx\, dy < +\infty$$ where $0<s<1$ is fixed. I take $R \gg 1$ and define $\eta=\eta_R$ to be $0$ on $B(0,R)$, $1$ on $\mathbb{R}^n \setminus B(0,R+1)$, and affine on $B(0,R+1)\setminus B(0,R)$, so that $\eta$ is lipschitz continuous. We want to investigate whether $$\lim_{R \to +\infty} \int_{\mathbb{R}^n \times \mathbb{R}^n} \frac{|\eta(x)u(x)-\eta(y)u(y)|^2}{|x-y|^{n+2s}}dx\, dy=0.$$

I can't even guess if the answer is positive (as in the case of truncation for $W^{1,p}$ functions) or negative.

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