What authority do the feds have to enforce local law?
To answer your question directly, plenty. In areas where federal law supersedes local law, ie. Federal law has pre-empted state law. The FBI has the authority to investigate when there are violations of civil rights (federal) or allegations of such. The department of justice then has authority prosecute/sanction when agencies are found to be abusing power.
Many people also used this argument to push back against gay marriage. As local (read state) laws varied on the subject and were enacted directly by the will (vote) of the people the federal government chose a different course for the nation. Many (you?) also believe that the federal government should enact more gun laws to restrict them on a federal level.
Now help me understand your question. Were you asking it because you did not know? Are you assuming that Trump meant that he would bring in the national guard or could it be some form of federal oversight? Also, I'd like to hear from you directly as to my question about which bothers you most. I can assume that martial law is preeminent because thats where your question is directed but correct me if I'm wrong.
I'll state my concern more clearly and toss it out to you and the thread for response. There is a very real and measurable death count to people living in the city of Chicago. For comparison sake:
US coalition deaths in Iraq 2003-2011 - Hostile fire deaths = 3,777 (A rate of about 470/year)
Gun related deaths in Chicago the last 4 years = 2,200 (A rate of about 550/year)
Comparing a 'gun controlled' city to an actual war zone and the death rate is higher here in the US is appalling. You can choose to focus on Trump or what you see as the bigger threat. Ignoring the real numbers and reality of the city may point to what a person holds more dearly.
Do the numbers make you pause and think a bit?
Does Trumps tweet say "martial law" or might he be trying to gain focus on the real problem?
Which do you (anyone...) the most? Extrapolated fear of martial law or Chicago having the death rate of a war zone?