The Junction BID would oversee the maintenance of the proposed plaza, as well as any events or activities in the plaza.
The Junction BID: http://www.flatbushjunctionbid.com/
The Junction BID Facebook Page: https://www.facebook.com/flatbushjunction.bid/
The Junction BID Twitter Page: https://twitter.com/thejunctionbid
The New York City Department of Transportation is responsible for 6,000 miles of city streets and 12,000 miles of city sidewalks. They also govern the placement of over 50 pedestrian plazas throughout the 5 boroughs.
NYC DOT Main Page: http://www.nyc.gov/html/dot/html/home/home.shtml
NYC DOT Pedestrian Plazas: http://www.nyc.gov/html/dot/html/pedestrians/nyc-plaza-program.shtml
For more information about the plaza process, please read this PDF.