I made up a Facebook page and I used a phrase as its name, do I own it as a trademark?

[bcnu]

You obtain certain "common law rights" by being the first to use a "distinctive" brand in association with some particular goods or services "in commerce", i.e., you're offering, giving or selling them to others and using your brand as an indication of "source or quality".

You obtain "better" rights by registering your claim to that brand for those types of goods or services.

However, you still have to prove that the name/phrase is being used as a brand, not just a decoration or some indication of identification other than trademark source or quality. For instance,"Jo's Burgers" would only be the name (identity) until and unless Jo's also starts to sell burgers under that name.