Integral representability of compact operators on L_2

Question

Let $\Omega$ be a bounded Lipschitz domain and $K: L_2(\Omega) \to L_2(\Omega)$ a compact linear operator. Does $K$ in general have an integral representation, i.e., is there an integrable function $k \in L_1(\Omega)$, s.t., $$ K(\varphi)(x) = \int_\Omega k(x, y)\varphi(y) \mathrm{d} y \text{ for all } \varphi \in L_2(\Omega)? $$ If not, what are examples for such compact operators, that do not possess an integral representation?