Are context-free languages in $a^*b^*$ closed under complement?
This characterization of bounded context-free languages is due to Ginsburg ("The Mathematical Theory of Context-Free Languages"), and appears as Corollary 5.3.1 in his book. For general $k$ there are some restrictions on the semilinear sets, but for $k \leq 2$ these restrictions are always satisfied, and so it is straightforward to deduce that the complement of such a language (within $w_1^* w_2^*$) is context-free. 

Ginsburg mentions these implications in his book.

Corollary 5.6.1 If $M_1 \subseteq w_1^*w_2^*$ and $M_2$ are [context-free] languages, $w_1$ and $w_2$ words, then $M_1\cap M_2$ is a [context-free] language.

Corollary 5.6.2 If $M_1 \subseteq w_1^*w_2^*$ and $M_2$ are [context-free] languages, $w_1$ and $w_2$ words, then $M_1 - M_2$ and $M_2-M_1$ are [context-free] languages.