Consider a kernel $K:\mathbb{R}^N \to \mathbb{R}$ non negative and symmetric; that is, $K(-x) = K(x)$ for any $x \in \mathbb{R}^N \backslash \{0\}$. Moreover, consider $\gamma K \in L^1(\mathbb{R}^N)$, where $\gamma(x) = \min\{1, |x|^2\}$. Here, $u \in L^2(\mathbb{R}^N)$ with $u = 0$ in $\mathbb{R}^N \backslash \Omega$, where $\Omega$ is a bounded and smooth domain in $\mathbb{R}^N$. I need to show the following estimate
$$\int \int _{\mathbb{R} \times \mathbb{R}} K(x - y) (u(x) - u(y))u(x) dydx \leqslant C_0\|u\|_{L^2(\Omega)}^2 + C_1,$$ where $C_0$ and $C_1$ are positive constants.
It is possible to show that there exists $R > 0$ and $\varepsilon > 0$ such that $$\int_{B_R(0)^C}K(z) dz < \varepsilon.$$
But, I don't know how to use this information to show what I want.
The operator
$$L_K(x) = \int_{\mathbb{R}} K(x - y)(u(x) - u(y)) dy$$
is well known in the literature as a peridynamic non-local operator. For instance,
in the article https://arxiv.org/pdf/1612.05696.pdf the authors study this operator.