A trademark search would help as a start but is not the whole answer. Smith as a common name has no protection against the other Smiths in the world. Smith Publishing is not distinctive in itself. If it is registered, there could be a serious problem for you. If it is not the whole fact picure must be analized.
Trade names are less enforceable but can be enforced against unfair competition.
Don't look for trouble. It costs less to be safe now than sorry later. Do not disclose any details of any trademark or any facts relating to your circumstances. See a lawyer. Before you post a message you must agree to the Terms of Use .