Just starting out with some references here:
Below is the first study that establishes solid damage to white matter from OSA. Note the previous studies mentioned in the narrative and references.
Citation: Macey PM; Kumar R; Woo MA; Valladares EM; Yan-Go FL; Harper RM. Brain structural changes in obstructive sleep apnea. SLEEP 2008;31(7):967-977.
The study below is recent but by the same basic team. Here we are looking at metabolites to see indicators of damage occurring and/or healing. At the same time looking at such things as the depression and anxiety apparently directly related to the damage in progress.
Citation: Yadav SK, Kumar R, Macey PM, Woo MA, Yan-Go FL, Harper RM. Insular cortex metabolite changes in obstructive sleep apnea. SLEEP 2014;37(5):951-958. Related article (commentary): 835
Another study looking at damage and results. Note the scores of references in all of these studies.
Citation: O’Donoghue FJ; Wellard RM; Rochford PD; Dawson A; Barnes M; Ruehland WR; Jackson ML; Howard ME; Pierce RJ; Jackson GD. Magnetic resonance spectroscopy and neurocognitive dysfunction in obstructive sleep apnea before and after CPAP treatment. SLEEP 2012;35(1):41-48.
Meta-analysis looks at many studies to more clearly establish what is actually going on. Executive functioning uses parts of the brain apparently harmed by OSA to get things done. So a person with moderate to severe OSA is likely to have trouble getting things done.
Citation: Olaithe M; Bucks RS. Executive dysfunction in OSA before and after treatment: a meta-analysis. SLEEP 2013;36(9):1297-1305.