I listened to all two hours of the oral arguments, the transcript will be a gold mine for years to come.
Compressing it way short:
The essential argument for the Dark & Fascist State of New York was that since Heller went with a "text, history and tradition" approach, NY has shown a long history and tradition of states infringing, er, regulating the right of arms. They did not satisfactorily answer several pertinent questions. Their basic premise is that once text, history and tradition answers the restriction question, you should stop and go no further, and not even get to whether the standard should be strict or intermediate construction.
Of course, what this would do would be to import as a whole body all of the state level infringements that emerged and thrived in an era in which the 2nd amendment was an inkblot nullity, especially as pertains to the states.
The essential argument for the people was that if you conditioned the exercise of a right upon atypical need over and above the typical member of the community, you have not described a right, you have described a privilege.
The issue of permits at all was conceded as a pragmatic matter, what was really at issue was "may issue" based on atypical need, and possibly abuses of discretion through vague "good moral standing" phrases.
Edit to add: I skipped a lot of stuff, including several versions of "Oh no! Danger to police because people might be armed!" and "Drunken people shooting it out in football stadiums over their favorite sportsmen sportsing with sportsballs".
