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You can try $\mathcal B(\varphi_1,\varphi_2)=\int \int\varphi'_1(x)\ \varphi_2'(y) (|x-y|\ln|x-y|)\ dx\ dy$ , i.e. $K$ is the second derivative $\partial_x\partial_y$ of (the distribution defined by) the locally integrable function $H(x,y)=|x-y|\ln|x-y|$ .