The owners of the World Trade Center buildings destroyed in the Sept. 11 attacks want billions of dollars in damages from aviation companies linked to the hijacked planes.
A federal non-jury trial started Monday in Manhattan.
The buildings' owners, World Trade Center Properties, have already received nearly $5 billion in insurance proceeds.
A lawyer for the airlines argued against awarding the company another $3.5 billion, saying the extra billions would be more than double the fair market value of the buildings that were destroyed.
But the owners say it's in line with their costs.
In court papers, both sides have accused the other of unfairly characterizing their claims.
The aviation companies include United, American and AMR Corp.