Gentrification is a classic example of supply and demand from microecon. As an area becomes nicer to live in it gains more and more demand with people with high willingness and ability to pay. Supply remains fixed, because of US zoning laws, or grows more slowly than people come into the neighborhood. The price goes up because of _demand_ going up without supply going up.
Why does it gentrify in the first place? Because the initial group of "gentrifiers" took the time and money to develop the area to make it more appealing, thus increasing demand.
Economics says you can't fix it no matter how much regulation you impose because at some point all the surrounding businesses, etc. will be gentrified too and force out poor people. Even if you freeze rent, ban new businesses from coming in, etc. the existing business owners will start to cater to their new clientele simply because the demand from those customers is much higher. The only thing you _can_ do is ban people from moving to the neighborhood, at which point you've turned into the worst parts of the Soviet Union.