The airport manager at a certain civil airfield I have knowledge of simply wrote to the complainer and suggested they take it up with the estate agent who handled their house purchase.
But it's not only low flying that gets these imbecilic NIMBYs, there was one on the Fulham Football Club website about 2 years ago. A Fulham supporter living close to Craven Cottage answered a knock on his door one evening. A lady asked him to sign a petition to get the football club closed down due to its noise. He asked how long she had lived there - it transpired they'd viewed, bought and moved in during the closed season - and how much? They'd paid £1.25M and claimed they had been unaware of the existence of the football stadium (it's only been there since 1879 and it's hardly small enough not to see it).