Let $\Omega$ a compact space, $J:\Omega\times\Omega\to \mathbb R$ a symmetric function and $\theta:\Omega\to [0,1]$ a continuous function taking values in $[0,1]$.
Let ${\mathcal L}$ the operator defined on functions $f\in L^2\big(\Omega\times\{1,2\}, \mathbb R\big)$ as \begin{align} {\mathcal L}f({x, i})=&\chi_{1}({i})\bigg({\int_{\Omega}({1-\theta(y)})J(x, y)(f({y, 1})-f({x,1}))dy+\int_{\Omega}\theta({y})J({x, y})({f({y,2})-f({y, 1})}dy)}\bigg)\\ &+\chi_{2}({i})\int_{\Omega}({1-\theta({y}))} J({x, y})({f({y, 1}))-f({x,2})}dy, \end{align} where $\chi_1(x)$ denotes the characteristic function centrated in $1$, that is $\chi_1(x)=1$ if $x=1$, $\chi_1(x)=0$ if $x=2$.
How can I compute the adjoint operator of $\mathcal L$?
Thank you very much!