Malls and parking lots are not public places. They are private places open to the public.
I'm uncertain what you mean by "regular laws." In all states in the U.S. that I'm aware, you can not get a ticket for running a stop sign in a parking lot but you can get arrested for drunk driving. As far as law enforcement is concerned, it's not any different from your own property. Misdemeanors can not be enforced but felonies (called "high misdemeanors in some states) can be enforced.
The only other distinction is related specifically to "private property open to the public," such as malls, stores, restaurants, etc. The governing bodies have the authority to write and enforce "public safety" laws, such as related to discrimination, food safety, handicap access, etc. (Yeah, some of those laws are a stretch.)
In the absence of any specific law to the contrary, the owner of the property has the same rights as the owner of a private home.