Douglas' lemma gives a necessary and sufficient condition for two bounded operators $A$ and $B$ on a Hilbert space $H$ to get $\mathrm{Ran} A \subset \mathrm{Ran} B$ (wikipedia article, original article).

Does there exist a condition on two kernels $K_1$ and $K_2$ in order to get the inclusion of the ranges of the integral operators associated with? In order words, I am looking for a specific Douglas type lemma for integral operators which gives a condition on the kernels.

Douglas' lemma gives a necessary and sufficient condition for two bounded operators $A$ and $B$ on a Hilbert space $H$ to get $\operatorname{Ran} A \subset \operatorname{Ran} B$ (wikipedia article, original article).

Does there exist a condition on two kernels $K_1$ and $K_2$ in order to get the inclusion of the ranges of the integral operators associated with? In order words, I am looking for a specific Douglas type lemma for integral operators which gives a condition on the kernels.